QUASIDETERMINANTS

I. Gelfand, S. Gelfand, V. Retakh, and R. Wilson

Abstract

The determinant is a main organizing tool in commutative linear algebra. In this review we present a theory of the quasideterminants defined for matrices over a division algebra. We believe that the notion of quasideterminants should be a main organizing tool in noncommutative algebra giving them the same role determinants play in commutative algebra.

Contents

0. Introduction

1. A general theory and main identities

2. Important example: quaternionic quasideterminants

3. Noncommutative determinants

4. Noncommutative Plücker and flag coordinates

5. Bezout and Vieta theorems

6. Noncommutative symmetric functions

7. Universal quadratic algebras associated with pseudo-roots of noncommutative polynomails and noncommutative differential polynomials

8. Noncommutative traces, determinants and cyclic vectors

9. Some applications

Introduction

The universal notion of a determinant has a long history, both visible and invisible. The determinant was a main organizing tool in commutative linear algebra and we cannot accept the point of view of a modern textbook [FIS] that “determinants ... are of much less importance than they once were”.

2000 Mathematics Subject Classification. 16W30; 15A15; 05E05.

Key words and phrases. quasideterminants, noncommutative algebra, symmetric functions.

Vladimir Retakh was partially supported by the National Science Foundation and Max-Planck-Institut für Mathematik in Bonn

Attempts to define a determinant for matrices with noncommutative entries started more than 150 years ago and include several great names. For many years the most famous examples of noncommutative objects were quaternions and blockmatrices. It is not suprising that the first noncommutative determinants or similar notions were defined for such structures.

A. Cayley [C] was the first to define, in 1845, the determinant of a matrix with noncommutative entries. He mentioned that for a quaternionic matrix A =

( \begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} ) the expressions a_{11}a_{22} − a_{12}a_{21} and a_{11}a_{22} − a_{21}a_{12} are different and

suggested to choose one of them as a determinant of matrix A. The analog of this construction for 3 × 3-matrices was also proposed in [C] and later developed in [J]. This “naive” approach works now for quantum determinants and other cases. Different forms of quaternionic determinants were considered later by E. Study [St], E.H. Moore [Mo] and F. Dyson [Dy].

There were no direct “determinantal” attacks on block-matrices (excluding evident cases) but important insights were given by G. Frobenius [Fr] and I. Schur [Schur].

A theory of determinants of matrices with general noncommutative entries was in fact originated by J.H.M. Wedderburn in 1913. In [W] he constructed a theory of noncommutative continued fractions or, in modern terms, “determinants” of noncommutative Jacobi matrices.

In 1926-1928 A. Heyting [H] and A. Richardson [Ri, Ri1] suggested analogs of a determinant for matrices over division algebras. A. Heyting is known as a founder of intuitionist logic and A. Richardson as a creator of the Littlewood-Richardson rule. Heyting tried to construct a noncommutative projective geometry. As a computational tool, he introduced the “designant” of a noncommutative matrix. The designant of a 2 × 2-matrix A = (a_{ij}) is defined as a_{11} − a_{12}a_{22}^{−1}a_{21}. The designant of an n × n-matrix is defined then by a complicated inductive procedure. The inductive procedures used by Richardson were even more complicated. It is important to mention that determinants of Heyting and Richardson in general are rational functions (and not polynomials!) in matrix entries.

The idea to have non-polynomial determinants was strongly criticized by O. Ore [O]. In [O] he defined a polynomail determinant for matrices over an imporatnt class of noncommutative rings which were later named Ore rings.

The most famous and widely used noncommutaive determinant is the Dieudonne determinant. It was defined for matrices over a division algebra R by J. Dieudonne in 1943 [D]. His idea was to consider determinants with values in R^{∗}/[R^{∗}, R^{∗}] where R^{∗} is the monoid of invertible elements in R. The properties of Dieudonne determinants are close to those of commutative ones but, evidently, Dieudonne determinants cannot be used for solving systems of linear equations.

An interesting generalization of commutative determinants belongs to F. Berezin [B, Le]. He defined determinants for matrices over so called super-commutative algebras. In particular, Berezin also understood that it is impossible to avoid rational functions in matrix entries in his definition.

Other famous examples of noncommutative determinants developed for different special cases are: quantum determinants [KS, Ma], Capelli determinants [We], determinants introduced by Cartier-Foata [CF, F] and Birman-Williams [BW], etc. As we explain later by introducing another universal notion of quasideterminants, these determinants and determinants of Dieudonne, Study, Moore, etc., are related to each other much more than one would expect.

A notion of quasideterminants for matrices over a noncommutative skew-field was introduced in [GR, GR1, GR2]. Quasideterminants are defined in the “most noncommutative case”, namely, for matrices over free skew-fields. We believe that quasideterminants should be one of main organizing tools in noncommutative algebra giving them the same role determinants play in commutative algebra. Quasideterminants have been effective in many areas including noncommutative symmetric functions [GKLLRT], noncommutative integrable systems [RS, EGR, EGR1], quantum algebras [GR, GR1, GR2, KL, Mol, Mol1], and so on [P, Sch, RSh, RRV].

The main property of quasideterminants is a “heredity principle”: let A be a square matrix over a skew-field and (A_{ij}) a block decomposition of A (into submatrices of A). Consider the A_{ij}’s as elements of a matrix X. Then the quasideterminant of the matrix X will be a matrix B, and (under natural assumptions) the quasideterminant of B is equal to a suitable quasideterminant of A. Since determinants of block-matrices are not defined, there is no analog of this principle for ordinary (commutative) determinants.

The quasideterminant is not an analog of the commutative determinant but rather of a ratio of the determinant of n × n-matrices to the determinant of an (n − 1) × (n − 1)-submatrix.

Many areas of noncommutative mathematics (Ore rings, rings of differential operators, theory of factors, “quantum mathematics”, Clifford algebras, etc) were developed separately from each other. Our approach shows an advantage of working with totally noncommutative variables (over free rings and skew-fields). It leads us to a large variety of results, and their specialization to different noncommutative areas implies known theorems with additional information.

The price one pays for this is a huge number of inversions in rational noncommutative expressions. The minimal number of successive inversions required to express an element is called the height of this element. This invariant (inversion height) reflects the “degree of noncommutativity” and it is of a great interest by itself.

Our experience shows that in dealing with noncommutative objects one should not imitate the classical commutative mathematics, but follow “the way it is” starting with basics. In this paper we concentrate on two problems: noncommutative Plücker coordinates (as a background of a noncommutative geometry) and the noncommutative Bezout and Vieta theorems (as a background of noncommutative algebra). We apply the obtained results to the theory of noncommutative symmetric functions started in [GKLLRT].

We already said that the universal notion of a determinant has a long history, both visible and invisible. The visible history of determinants comes from the fact that they are constructed from another universal objects: matrices.

The invisible history of determinants is related with the Heredity principle for matrices: matrices can be viewed as matrices with matrix entries (block matrices) and some matrix properties come from the corresponding properties of block matrices. In some cases, when the matrix entries of the block-matrix commute, the determinant of a matrix can be computed in terms of the determinants of its blocks, but in general it is not possible: the determinant of a matrix with matrix entries is not defined because the entries do not commute. In other words, the determinant does not satisfy the Heredity principle.

Quasideterminants are defined for matrices over division algebras and satisfy the Heredity Principle. Their definition can be specialized for matrices over a ring (including noncommutative rings) and can be connected with different ”famous” determinants. This reflects another general principle: in many cases noncommutative algebra can be made simpler and more natural than commutative algebra.

I. A general theory and main identities

1.1. The skew-field of rational functions of free variables. The free skewfield is a natural framework for working with noncommutative variables. The theory was developed by Amitsur, Bergman and P. M. Cohn (see, for example [Co]). We consider here skew-fields as division algebras over fields of characteristic 0, but everything can be generalized to fields of characteristics p. Here we will recall Bergman’s presentation of Amitsur’s approach. Given a set X = {x_{1}, …, x_{n}} we write F(X) for the algebra freely generated by the elements of X over rationals ℚ using the operations of addition, subtraction, multiplication and taking the inverse.

No relations are imposed so far, so that, e.g., (x−x)^{−1} exists. The algebra F(X) is called the algebra of rational formulas on X. Let R be a ring with unit. Any map α : X → R defines a partial map \overline{α} of F(X) to R by the following rules:

i) \overline{α}(m) = m, m ∈ ℤ, ii) \overline{α}(x_{i}) = α(x_{i}), i = 1, …, n, iii) if a = −b, or b + c, or bc and \overline{α}(b), \overline{α}(c) are defined, then \overline{α}(a) = −\overline{α}(b), or

\overline{α}(b + c), or \overline{α}(b) · \overline{α}(c), iv) if a = b^{−1} and \overline{α}(b) is defined and invertible in R, then \overline{α}(a) = (\overline{α}(b))^{−1}.

If ᾱ(a) is defined for a ∈ F(X) we say that α can be evaluated on a.

Now let α : X → D be a map to a skew-field D. Then ᾱ(a) is not defined if and only if a has a subformula b^{−1} such that ᾱ(b) = 0. For each α : X → D one can consider a subset E(α) of F(X), the domain of \overline{α}, consisting of the formulas on which α can be evaluated. Similarly with each a ∈ F(X) one can associate its domain dom a, a subset of D^{n} consisting of the points (α(x_{i}), i = 1, …, n) such that a ∈ E(α) is defined; a is called nondegenerate if dom a ≠ ∅.

One can show [Co] that if D is a skew-field which is an algebra over an infinite field k and a and b are nondegenerate, then dom a ∩ dom b ≠ ∅.

Given a, b ∈ F(X), we write a ∼ b if a, b are nondegenerate and a, b have the same value at each point of dom a ∩ dom b. This is an equivalence relation. The equivalence classes are called the rational functions in x_{1}, …, x_{n}.

Theorem 1.1.1. [Co]

i) Let D be a skew-field with the center k of characteristic 0. Then the equivalence classes of rational formulas of variables x_{1}, …, x_{n} form a skew-field F_{D}(X).

ii) If D is infinite dimensional over k then F_{D}(X) does not depend on the skewfield D.

Under the assumptions of i) and ii) we will identify all F_{D}(X) and use the notation F (X). For example, if X = {x} then F (X) = ℚ(x). The skew-field F (X) is called the free skew-field generated by X.

Remark. P.M. Cohn has shown ([Co] that the free skew-field is universal in the following sense. Consider the algebra k⟨x_{1}, …, x_{n}⟩ of noncommutative polynomials over the commutative field k. The algebra k⟨x_{1}, …, x_{n}⟩ is naturally embedded into the algebra F (X). The universality means that if D is any division ring and

α : k⟨x_{1}, …, x_{n}⟩ → D

is a homomorphism then there is a subring R of F (X) containing k⟨x_{1}, …, x_{n}⟩ and an extension of α to a homomorphism

β : R → D

such that if a ∈ R and β(a) ≠ 0 then a^{−1} ∈ R.

We recall the following definition (see, for example, [Re]).

Definition 1.1.2. The inversion height of a rational expression is the maximal number of nested inversions in it. The inversion height of an element in the free skew-field F (X) is the smallest inversion height of its rational expressions.

Example. The inversion height of a polynomial in x_{1}, …, x_{n} equals zero. The inversion height of the ratio of two polynomials P Q^{−1} equals 0 if P is divisible by Q and 1 otherwise. The inversion height of the rational expressions r_{1} = (1−x)^{−1} + (1 − x^{−1})^{−1} and r_{2} = x^{−1} + x^{−1}(z^{−1}y^{−1} − x^{−1})^{−1}x^{−1} equals 2. However, r_{1} = 1 and r = (x − yz)^{−1} in the free skew-field generated by x, y and z. So, the height of r_{1}, r_{2} as rational functions equals to 0 and 1 correspondigly. The rational function given by the expression (x − yw^{−1}z)^{−1} has height 2.

1.2. Definition of quasideterminants. Let A = (a_{ij}), i ∈ I, j ∈ J be a matrix over a ring R. Construct the following submatrices of A:

submatrix A^{ij}, i ∈ I, j ∈ J obtained from A by deleting its i-th row and j-th column,

row submatrix r_{i}^{j} obtained from i-th row of A by deleting the element a_{ij}, column submatrix c_{j}^{i} obtained from j-th column of A by deleting the element

a_{ij}.

Definition 1.2.1. Let I, J be finite sets with the same number of elements. We say that the quasideterminant |A|_{ij} is defined if the submatrix A^{ij} is invertible over the ring R. In this case

(1.2.1)  |A|_{ij} = a_{ij} − r_{i}^{j}(A^{ij})^{−1}c_{j}^{i}.

It is sometimes convenient to adopt a more explicit notation boxing the element a_{ij}. For A = (a_{pq}), p, q = 1, …, n

|A|_{ij} = | \begin{matrix}a_{11} & … & a_{1j} & … & a_{1n} \\ & … & & … \\ a_{i1} & … & a_{ij} & … & a_{in} \\ & … & & … \\ a_{n1} & … & a_{nj} & … & a_{nn} \end{matrix} |

We set |A|_{ij} = a_{ij} if I = {i}, J = {j}.

Remark. The definition does not require any orderings of I and J or a bijective correspondence between I and J.

If all quasideterminants |A^{ij}|_{pq} for p ∈ I ∖ {i}, q ∈ J ∖ {j}. are defined and invertible, then

(1.2.2)  |A|_{ij} = a_{ij} − ∑ a_{iq}|A^{ij}|_{pq}^{−1}a_{pj}.

Here the sum is taken over all p ∈ I ∖ {i}, q ∈ J ∖ {j}.

If A is a generic n × n-matrix (in the sense that all square submatrices of A are invertible), then there exist n^{2} quasideterminants of A. However, a non-generic matrix may have k quaisdeterminants where 0 ≤ k ≤ n^{2}. A square matrix A = (a_{ij}), i ∈ I, j ∈ J is a generic matrix over the free skew-field generated by its entries.

Examples.

1) For a 2 × 2-matrix A = (a_{ij}), i, j = 1, 2 there are up to four quasideterminants

\begin{matrix}|A|_{11} = a_{11} − a_{12} · a_{22}^{−1} · a_{21}, & |A|_{12} = a_{12} − a_{11} · a_{21}^{−1} · a_{22}, \\ |A|_{21} = a_{21} − a_{22} · a_{12}^{−1} · a_{11}, & |A|_{22} = a_{22} − a_{21} · a_{11}^{−1} · a_{12}. \end{matrix}

Each is defined if the corresponding entry is invertible in R.

2) For a matrix A = (a_{ij}), i, j = 1, 2, 3 there are 9 quasideterminants. We will write here only one

|A|_{11} = a_{11} − a_{12}(a_{22} − a_{23}a_{33}^{−1}a_{32})^{−1}a_{21} − a_{12}(a_{32} − a_{33} · a_{23}^{−1}a_{22})^{−1}a_{31}−
−a_{13}(a_{23} − a_{22}a_{32}^{−1}a_{33})^{−1}a_{21} − a_{13}(a_{33} − a_{32} · a_{22}^{−1}a_{23})^{−1}a_{31}

provided all inverses exist in R.

Remark. The examples show that if each a_{ij} is an invertible morphism V_{j} → V_{i} in an additive category then the quasideterminant |A|_{pq} can be also viewed as a morphism from the object V_{q} to the object V_{p} if the matrix of morphisms A^{pq} is invertible.

Quasideterminant is not a generalization of a determinant over a commutative ring but a generalization of a ratio of two determinants.

Example. If in formulas (1.2.1), or (1.2.2) the variables a_{ij} commute then

|A|_{pq} = (−1)^{p+q} \frac{det A}{det A^{pq}} .

We will show in section III that similar expressions for quasideterminants can be given for quantum matrices, quaternionic matrices, Capelli matrices and other cases listed in the Introduction.

In general quasideterminants are not polynomials in their entries, but rational functions. The following theorem was conjectured by I. Gelfand and V. Retakh, and proved by Reutenauer [Re] in slightly different form.

Theorem 1.2.2. Quasideterminants of the n × n-matrix with formal entries have the inversion height n − 1 over the free skew-field generated by its entries.

In the commutative case determinants are finite sums of monomials with appropriate coefficients. In noncommutative case quasideterminants are formal power series with matrix entries or their inverses as indeterminants. A simple example of this type is described below.

Let A = (a_{ij}), i, j = 1, …, n be a matrix with formal entries. Denote by I_{n} the unit n×n-matrix and by Γ_{n} the complete oriented graph with vertices {1, 2, …, n}, the arrow from i to j is labelled by a_{ij}. A path p : i → k_{1} → k_{2} → ⋯ → k_{t} → j is labelled by the word w = a_{ik_{1}}a_{k_{1}k_{2}}a_{k_{2}k_{3}} … a_{k_{t}j}.

Denote by P_{ij} the set of words labelling paths going from i to j, i.e. the set of words of the form w = a_{ik_{1}}a_{k_{1}k_{2}}a_{k_{2}k_{3}} … a_{k_{t}j}. A simple path is a path p such that k_{s} ≠ i, j for every s. Denote by P′_{ij} the set of words labelling simple paths from i to j.

Proposition 1.2.3. Let i, j be two distinct integers between 1 and n. In the ring of formal series in variables a_{ij}’s

|I_{n} − A|_{ii} = 1 − ∑_{w∈P′_{ii}} w, |I_{n} − A|_{ij}^{−1} = ∑_{w∈P_{ii}} w.

Example. For n = 2,

|I_{2} − A|_{11} = 1 − a_{11} − ∑_{p≥0} a_{12}a_{22}^{p}a_{21}.

However, in some important special cases, quasideterminants are polynomials in their entries.

Proposition 1.2.4. The following quasideterminant is a polynomial in its entries:

| \begin{matrix}a_{11} & a_{12} & a_{13} & … & a_{1n} \\ −1 & a_{22} & a_{23} & … & a_{2n} \\ 0 & −1 & a_{33} & … & a_{3n} \\ & & … \\ 0 & & … & −1 & a_{nn} \end{matrix} |_{1n} =

= a_{1n} + ∑_{1≤j_{1}<j_{2}<···<j_{k}<n} a_{1j_{1}}a_{j_{1}+1j_{2}}a_{j_{2}+1j_{3}} … a_{j_{k}+1n}.

1.3. Transformation properties of quasideterminants.

Let A = (a_{ij}) be a square matrix.

i) The quasideterminant |A|_{pq} does not depend on the permutation of rows and columns in the matrix A if the p-th row and the q-th column are not changed; ii) The multiplication of rows and columns. Let the matrix B = (b_{ij} be con-

structed from the matrix A by multiplication of its i-th row by a scalar λ from the left, i.e. b_{ij} = λa_{ij} and b_{kj} = a_{kj} for k ≠ i. Then

|B|_{kj} = \{ \begin{matrix}λ|A|_{ij} & if k = i \\ |A|_{kj} & if k ≠ i and λ is invertible. \end{matrix}

Let the matrix C = c_{ij} be constructed from the matrix A by multiplication of its j-th column by a scalar μ from the right, i.e. c_{ij} = a_{ij}μ and c_{il} = a_{il} for all i and l ≠ i. Then

|C|_{iℓ} = \{ \begin{matrix}|A|_{ij}μ & if ℓ = j \\ |A|_{iℓ} & if ℓ ≠ j and μ is invertible. \end{matrix}

iii) The addition of rows and columns. Let the matrix B be constructed by adding to some row of the matrix A its k-th row multiplied by a scalar λ from the left. Then

|A|_{ij} = |B|_{ij}, i = 1, … k − 1, k + 1, … n, j = 1, …, n.

Let the matrix C be constructed by addition to some column of the matrix A its ℓ-th column multiplied by a scalar λ from the right. Then

|A|_{ij} = |C|_{ij}, i = 1, …, n, j = 1, …, ℓ − 1, ℓ + 1, … n.

1.4 General properties of quasideterminants

Quasideterminants and inverse matrices. [GR, GR1, GR2]

Theorem 1.4.1. If A = (a_{ij}) is a generic matrix and B = (b_{ij}) equals to A^{−1} then b_{ji} = |A|_{ij}^{−1} for all i, j.

Two involutions. [GR4] For a square matrix A = (a_{ij}) denote by IA = A^{−1} inverse matrix, and by HA = (a_{ji}^{−1}) the Hadamard inverse matrix. It is evident that if IA is defined, then I^{2}A = A, and if HA is defined, then H^{2}A = A.

Let A^{−1} = (b_{ij}). According to Theorem 1.4.1 b_{ij} = |A|_{ji}^{−1}. This formula can be written in the following form.

Theorem 1.4.2. For a square matrix A with formal entries

(1.4.1)  HI(A) = (|A|_{ij})

Homological relations. [GR, GR1, GR2] The ratio of two quasideterminants of the same square matrix is the ratio of two rational functions of a smaller height. It follows from the following homological relations:

Theorem 1.4.3.

a) Row homological relations:

−|A|_{ij} · |A^{iℓ}|_{sj}^{−1} = |A|_{iℓ} · |A^{ij}|_{sℓ}^{−1} ∀s ≠ i

b) Column homological relations:

−|A^{kj}|_{it}^{−1} · |A|_{ij} = |A^{ij}|_{kt}^{−1} · |A|_{kj} ∀r ≠ j

Heredity (special case). Let A = ( \begin{matrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix} ) be a decomposition of A =

(a_{ij}), i, j = 1, …, n, into a block-matrix. Let A_{11} be a k × k-matrix and let the matrix A_{22} be invertible.

Theorem 1.4.4. |A|_{ij} = |A_{11} − A_{12}A_{22}^{−1} · A_{21}|_{ij} for i, j = 1…. , k. In other words, the quasideterminant |A|_{ij} of an n × n matrix can be computed in two steps: first, consider the quasideterminant |Ã|_{11} = A_{11} − A_{12} · A_{22}^{−1} · A_{21}of a 2 × 2-matrix Ã =

( \begin{matrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix} ); and second, consider the corresponding quasideterminant of the k×k-

matrix |Ã|_{11}.

Heredity (general case). Let A = (a_{ij}), i, j = 1, …, n, be a matrix and

A = ( \begin{matrix}A_{11} & … & A_{1s} \\ A_{s1} & … & A_{ss} \end{matrix} )

be its block decomposition. Denote by Ã = (A_{ij}), i, j = 1, …, s, the matrix with A_{ij}’s as entries. Suppose that

A_{pq} = ( \begin{matrix}a_{kℓ} & … & a_{k,ℓ+m} \\ & … \\ a_{k+m,ℓ} & … & a_{k+m,ℓ+m} \end{matrix} ) .

is a square matrix and that |Ã|_{pq} is defined.

Theorem 1.4.5.

|A|_{k′ℓ′} = ||Ã|_{pq}|_{k′ℓ′} for k′ = k, …, k + m, ℓ′ = ℓ, …, ℓ + m.

Example.

Let

A = ( \begin{matrix}a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{matrix} ) .

Set A_{11} = ( \begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} ), A_{12} = ( \begin{matrix}a_{13} & a_{14} \\ a_{23} & a_{24} \end{matrix} ), A_{21} = ( \begin{matrix}a_{31} & a_{32} \\ a_{41} & a_{42} \end{matrix} ), A_{22} = ( \begin{matrix}a_{33} & a_{34} \\ a_{43} & a_{44} \end{matrix} ).

Set Ã = ( \begin{matrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix} ). Then

|Ã|_{12} = A_{12} − A_{11}A_{21}^{−1}A_{22} = ( \begin{matrix}a_{13} − … & a_{14} − … \\ a_{23} − … & a_{24} − … \end{matrix} )

and

|A|_{13} = | \begin{matrix}a_{13} − … & a_{14} − … \\ a_{23} − … & a_{24} − … \end{matrix} |_{13} .

Quasideterminants and Kronecker tensor products.

Let A = (a_{ij}), B = (b_{αβ}) be matrices over a ring. Denote by A⊗B the Kronecker tensor product. The entries of matrix A ⊗ B are indexed by pairs (ij, αβ).

Proposition 1.4.6. If quasideterminants |A|_{ij} and |B|_{αβ} are defined then the quasideterminant |A ⊗ B|_{ij,αβ} is defined and

|A ⊗ B|_{ij,αβ} = |A|_{ij}|B|_{αβ}.

Note that in commutative case the corresponding identity looks differently: if A is a m×m-matrix and B is a n×n-matrix over a commutative ring then det(A⊗B) = (det A)^{n}(det B)^{m}.

Quasideterminants and matrix rank. Let A = (a_{ij}) be a matrix over a division algebra.

Proposition 1.4.7. If the quasideterminant |A|_{ij} be defined then the following statements are equivalent.

i) |A|_{ij} = 0, ii) the i-th row of the matrix A is a left linear combination of A; iii) the j-th column of the matrix A is a right linear combination of the other

columns of A.

Example. Let i, j = 1, 2 and |A|_{11} = 0. It means that a_{11} − a_{12}a_{22}^{−1}a_{21} = 0. It follows that a_{11} = λa_{21}, where λ = a_{12}a_{22}^{−1}. Since a_{12} = (a_{12}a_{22}^{−1})a_{22}, the first row of A is proportional to the second row.

There exist notion of linear dependence of rows and of columns for matrices over a skew-field if rows are multiplied by the elements of this skew-field from the left and columns from the right. So there exist notions of the row rank and the column rank and these ranks are equal [Ja, Co]. It follows also from Proposition 1.4.7.

By definition, an r-quasiminor of a square matrix A is a quasideterminant of an r × r-submatrix of A.

Proposition 1.4.8. The rank of the matrix A over a skew field is greater or equal to r if and only if at least one of the r-quasiminors of the matrix A is defined and is not equal to zero.

1.5 Basic identities

Row and columns decomposition. As it was pointed out in [KL], homological relations lead immediately to the following analogue of the classical expansion of a determinant by a row or a column.

Proposition 1.5.1. For any k ≠ p and any ℓ ≠ q

|A|_{pq} = a_{pq} − ∑_{j≠q} a_{pj}(|A^{pq}|_{kj})^{−1}|A^{pj}|_{kq}

|A|_{pq} = a_{pq} − ∑_{i≠p} |A^{iq}|_{pi}(|A^{pq}|_{iℓ})^{−1}a_{iq}

if all terms in these expressions are defined.

Sylvester identity. The following noncommutative analogue of Sylvester’s identity stated in [GR, GR1, GR2] is closely related with the heredity property. Let A = (A_{ij}), i, j = 1, …, n be a matrix over a noncommutative skew-field. Suppose that the submatrix A_{0} = (a_{ij}), i, j = 1, …, k, is invertible. For p, q = k + 1, …, n set

b_{pq} = | A_{0}  a_{1q} ; ⋮ ; a_{kq} ; a_{p1} … a_{pk} a_{pq} |_{pq};

B = (b_{pq}), p, q = k + 1, …, n.

We call the submatrix A_{0} a pivot for matrix B.

Theorem 1.5.2. For i, j = k + 1, …, n,

|A|_{ij} = |B|_{ij}

Example. A quasideterminant of an n × n-matrix A is equal to the corresponding quasideterminant of a 2 × 2-matrix consisting of (n − 1) × (n − 1)- quasiminors of the matrix A, or to the quasideterminant of an (n − 1) × (n − 1)-matrix consisting of 2 × 2-quasiminors of the matrix A. One can use any of these procedures for an inductive definition of quasideterminants. In fact, A. Heyting [H] essentially defined inductively quasideterminants |A|_{nn} for matrices A = (a)_{ij}, i, j = 1, …, n starting with n = 2.

Theorem 1.5.3. (Sylvester): Let A = (a_{ij}), i, j = 1, …, n, be a matrix over a commutative ring. Suppose that its submatrix A_{0} = (a_{ij}), i, j = 1, …, k, is invertible. For p, q = k + 1, …, n, set

\begin{aligned}b̃_{pq} & = det ( A_{0}  a_{1q} ; ⋮ ; a_{kq} ; a_{p1} … a_{pk} a_{pq} ) , \\ B̃ & = (b̃_{pq}), p, q = k + 1, …, n. \end{aligned}

Then

det A = \frac{det B̃}{(det A_{0})^{n−k−1}} .

Remark. In the special case when A_{0} = (a_{ij}), i, j = 2, …, n−1, Theorem 1.3.1 is an analogue of a well-known commutative identity which is called the “Lewis Carroll identity” (see, for example, [Ho]).

Inversion for quasiminors. The following theorem was stated in [GR, GR1, GR2]. For a matrix A = (a_{ij}), i ∈ I, j ∈ J, P ⊂ I, Q ⊂ J, denote by A_{P Q} the submatrix

A_{P Q} = (a_{αβ}), α ∈ P, β ∈ Q.

Let |I| = |J| and B = A^{−1} = (b_{rs}). Suppose that |P | = |Q|.

Theorem 1.5.4. Let k ∉ P, ℓ ∉ Q. Then

|A_{P ∪{k},Q∪{ℓ}}|_{kℓ} · |B_{I\P,J\Q}|_{ℓk} = 1.

Set P = I ∖ {k}, Q = J ∖ {ℓ}. Then this theorem leads to the already mentioned identity

|A|_{kℓ} · b_{ℓk} = 1.

Example. Theorem 1.5.4 implies the following identity for main quasiminors. Let A = (a_{ij}), i, j = 1, …, n be an ivertible matrix. Set A_{(k)} = (a_{ij}), i, j = 1, …, k. Set B = (b_{ij}) = A^{−1}, B^{(k)} = (b_{ij}), i, j = k, …, n. Then

|A_{(k)}|_{kk} · |B^{(k)}|_{kk} = 1.

A generalization of the homological relations. For a matrix A = (a_{ij}), i ∈ I, j ∈ J, and two subsets L ⊂ I, M ⊂ J denote by A^{L,M} the submatrix of the matrix A obtained from A by deleting the rows with the indexes ℓ ∈ L and the columns with the indexes m ∈ M.

Let A be a square matrix, L = (ℓ_{1}, …, ℓ_{k}), M = (m_{0}, …, m_{k}). Set M_{i} = M ∖ {m_{i}}, i = 0, …, k.

Theorem 1.5.5. [GR1, GR2] For p ∉ L

∑_{i=0}^{k} |A^{L,M_{i}}|_{pm_{i}} · |A|_{ℓm_{i}}^{−1} = δ_{pℓ},

∑_{i=0}^{k} |A|_{m_{i}ℓ}^{−1} · |A^{M_{i},L}|_{m_{i}p} = δ_{ℓp}.

Multiplicative properties of quasideterminants. Let X = (x_{pq}), Y = (y_{rs}) be n × n-matrices.

Theorem 1.5.6.

|XY |_{ij}^{−1} = ∑_{p=1}^{n} |Y |_{pj}^{−1}|X|_{pi}^{−1}.

1.6 Noncommutative linear algebra

In this section we use quasideterminants to formulate basic theorems in noncommutative linear algebra [GR, GR1, GR2].

Solutions of systems of linear equations.

Theorem 1.6.1. If the system

\{ \begin{matrix}a_{11}x_{1} + … & +a_{1n}x_{n} = ξ_{1} \\ … \\ a_{n1}x_{1} + … & +a_{nn}x_{n} = ξ_{n} \end{matrix}

is given and for the matrix A = (a_{ij}) the quasideterminants of A are defined and invertible, then

x_{i} = ∑_{j=1}^{n} |A|_{ji}^{−1}ξ_{j}.

for i = 1, …, n.

Cramer’s rule. Let A_{ℓ}(ξ) be an n×n-matrix constructed by replacing ℓ-th column of the matrix A by the column (ξ_{1}, …, ξ_{n}).

Theorem 1.6.2. In notations of Theorem 1.4.1, if the quasideterminants |A|_{ij} and |A_{j}(ξ)|_{ij} are defined, then

|A|_{ij}x_{j} = |A_{j}(ξ)|_{ij}.

Cayley-Hamilton theorem. Let t be a formal variable and A = (a_{ij}), i, j = 1, …, n a matrix over a skew field. We do not assume that t commutes with variables a_{ij}, i, j = 1, …, n. Let I_{n} be n × n-identity matrix, Ã_{ij} = a_{ij}I_{n}, and Ã = (Ã_{ij}) the block matrix. Let T = (δ_{ij}t) also be the block matrix, and f_{ij}(t) = |T − Ã|_{ij}. Consider the expression f̃_{ij}(t) as a matrix function of a matrix variable changing in f_{ij}(t) elements a_{ij} by matrices ã_{ij} = a_{ij} · I_{n}. For example, if n = 2, then f̃_{11}(t) = t − Ã_{11} − Ã_{12}(t − Ã_{22})^{−1}Ã_{21}.

The following theorem was stated in [GR1], [GR2]. An independent proof (similar to the original one) was given by C. Reutenauer in his Montreal lectures.

Theorem 1.6.3. For all i, j = 1, …, n

f_{ij}(A) = 0.

II. Important example: Quaternionic quasideterminants

As an example we compute here quasideterminants of quaternionic matrices.

2.1 Norms of quaternionic matrices.

Let ℍ be the algebra of quaternions. For a ∈ ℍ, a = x+yi+zj+tk, x, y, z, t ∈ ℝ, we denote by ā = x−yi−zj−tk, the conjugate of a, and by ν(a) the norm, ν(a) = aā.

Let M(n, ℍ) be the ℝ-algebra of quaternionic matrices of order n. There exists a multiplicative functional ν : M(n, ℍ) → ℝ_{≥0} such that

i) ν(A) = ν(a) if A ∈ M(1, ℍ), A = (a),

ii) ν(A) = 0 if and only if the matrix A is non-invertible,

iii) If A′ is obtained from A by adding a left-multiple of a row to another row or a right-multiple of a column to another column, then ν(A′) = ν(A).

We will show later that for a generic quaternionic matrix A = (a_{ij}), i, j = 1, …, n, a norm can be defined by induction on n by formula ν(A) = ν(|A|_{11})ν(A^{11}) and that ν(A) is a polynomial in a_{ij}’s, ā_{ij}’s.

2.2. Quasideterminants of quaternionic matrices.

This section contains results from [GRW1].

Let A = (a_{ij}) , i, j = 1, …, n, be a quaternionic matrix. Let I and J be two ordered sets of natural numbers 1 ≤ i_{1}, i_{2}, …, i_{k} ≤ n and 1 ≤ j_{1}, j_{2}, …, j_{k} ≤ n such that the i_{p}’s are distinct and the j_{p}’s are distinct. For k = 1 set m_{I,J}(A) = a_{i_{1}j_{1}}. For k ≥ 2 set

m_{I,J}(A) = a_{i_{1}j_{2}} ā_{i_{2}j_{2}}a_{i_{2}j_{3}} ā_{i_{3}j_{3}}a_{i_{3}j_{4}} … ā_{i_{k}j_{k}}a_{i_{k}j_{1}}.

If the matrix A is Hermitian, i.e. a_{ji} = ā_{ij} for all i, j then

m_{I,J}(A) = a_{i_{1}j_{2}}a_{j_{2}i_{2}}a_{i_{2}j_{3}}a_{j_{3}i_{3}}a_{i_{3}j_{4}} … a_{j_{k}i_{k}}a_{i_{k}j_{1}}.

Remark. The monomials m_{I,J}(A) are conviniently described in terms of paths in a graph. Let Γ_{n} be the complete oriented n-graph with vertices 1, …, n, the arrows from p to q being labelled by a_{pq} and ā_{qp}. The monomial m_{I,J}(A) corresponds then to a path from i_{1} to j_{1}.

To a quaternionic matrix A = (a_{pq}), p, q = 1, …, n, and to a fixed row index i and a column index j we associate a polynomial in a_{pq}’s, ā_{pq}’s which we call the (i, j)-th double permanent of A.

Definition 2.2.1. The (i, j)-th double permanent of A is the sum

π_{ij}(A) = ∑ m_{I,J}(A),

taken over all orderings I = {i_{1}, …, i_{n}}, J = {j_{1}, …, j_{n}} of {1, …, n} such that i_{1} = i and j_{1} = j .

Example. For n = 2

π_{11}(A) = a_{12}ā_{22}a_{21}.

For n = 3

π_{11}(A) = a_{12}ā_{32}a_{33}ā_{23}a_{21} + a_{12}ā_{22}a_{23}ā_{33}a_{31} + a_{13}ā_{33}a_{32}ā_{22}a_{21} + a_{13}ā_{23}a_{22}ā_{32}a_{31}.

For any submatrix B of A denote by B^{c} its complementary submatrix. The matrix B^{c} is obtained from A by omitting all rows and columns containing elements from B. If B is a k × k matrix then B^{c} is a (n − k) × (n − k) matrix.

Theorem 2.2.2. If the quasideterminant |A|_{ij} of a quaternionic matrix is defined then

(2.2.1)  ν(A^{ij})|A|_{ij} = ∑(−1)^{k(B)−1}ν(B^{c})π_{ij}(B)

where the sum is taken of all square submatrices B of A containing a_{ij} and k(B) is the order of B.

Recall that the quasideterminant |A|_{ij} is defined if the matrix A^{ij} is invertible. In this case ν(A^{ij}) is invertible and formula (2.2.1) gives an expression for |A|_{ij}.

Note that both sides in formula (2.2.1) are polynomials in the a_{ij}’s, the ā_{ij}’s with real coefficients. The right-hand side in (2.2.1) is a linear combination of monomials of lengths 1, 3, …, 2n − 1. Let μ(n) be the number of such monomials for a matrix of order n. Then μ(n) = 1 + (n − 1)^{2}μ(n − 1).

Example. For n = 2

ν(a_{22})|A|_{11} = ν(a_{22})a_{11} − a_{12}ā_{22}a_{21}.

For n = 3

ν(A^{11})|A|_{11} = ν(A^{11})a_{11} − ν(a_{33})a_{12}ā_{22}a_{21} − ν(a_{23})a_{12}ā_{32}a_{31}−
−ν(a_{32})a_{13}ā_{23}a_{21} − ν(a_{22})a_{13}ā_{33}a_{31} + a_{12}ā_{32}a_{33}ā_{23}a_{21}+
+a_{12}ā_{22}a_{23}ā_{33}a_{31} + a_{13}ā_{33}a_{32}ā_{22}a_{21} + a_{13}ā_{23}a_{22}ā_{32}a_{31}.

The example shows how to simplify the general formula for quasideterminants of matrix of order 3 (see subsection 1.2) for quaternionic matrices.

Formula (2.2.1) gives a recurrence relation for norms of quaternionic matrices. From this relation one can obtain an expression of ν(A) as a polynomial in a_{ij}’s and their conjugates (see also subsection 3.4 below).

Corollary 2.2.3.

ν(A)ν(A^{ij}) = ∑(−1)^{K(B)−1}ν(B^{c})π_{ij}(B))(∑(−1)^{K(B)−1}ν(B^{c})\overline{π_{ij}(B)}).

Example. For n = 2

ν(A) = ν(a_{11})ν(a_{22}) + ν(a_{12})ν(a_{21}) − a_{12}ā_{22}a_{21}ā_{11} − a_{11}ā_{21}a_{22}ā_{12}.

For a square quaternionic matrix A = (a_{ij}) set Q_{ij}(A) = |A|_{ij}ν(A^{ij}). Then one can rewrite formula (2.2.1) as

Q_{ij}(A) = ν(A^{ij})a_{ij} − ∑_{p≠i,q≠j} a_{iq}\overline{Q_{pq}(A^{ij})}a_{pj}.

If A is a Hermitian matrix then Q̄_{pq}(A^{ij}) = Q_{qp}(A^{ij}) and

Q_{ij}(A) = ν(A^{ij})a_{ij} − ∑_{p≠i,q≠j} a_{iq}Q_{qp}(A^{ij})a_{pj}.

When A is a complex matrix, formula (2.2.1) can be written as

det A det Ā^{ij} = (−1)^{i+j}(a_{ij} det Ā^{ij} − ∑_{p≠i,q≠j} a_{iq} det Ā^{ij} det A^{ip,jq}a_{pj}).

This implies the standard row/column expansion of the determinant.

III. Noncommutative determinants

In this section we describe some results from [GR, GR1, GR2, GRW1].

3.1. Noncommutative determinants as products of quasiminors.

Let A = (a_{ij}), i, j = 1, …, n, be a matrix over a division algebra R such that all square submatrices of A are invertible. For any orderings I = (i_{1}, …, i_{n}), J = (j_{1}, …, j_{n}) of {1, …, n} define A^{i_{1}...i_{k},j_{1}...j_{k}} to be submatrices of A obtained by deleting rows with indices i_{1}, …, i_{k} and columns with indices j_{1}, …, j_{k}. Then set

D_{I,J}(A) = |A|_{i_{1}j_{1}}|A^{i_{1}j_{1}}|_{i_{2}j_{2}}|A^{i_{1}i_{2},j_{1}j_{2}}|_{i_{3}j_{3}} … a_{i_{n}j_{n}}.

In the commutative case all D_{I,J}(A) are equal up to a sign to the determinant of A. When A is a quantum matrix all D_{I,J}(A) are equal up to a scalar to the quantum determinant of A [GR, GR1, KL]. The same is true for some other wellknown noncommutative algebras. This gives us a reason to call the D_{I,J}(A) the (I, J)-predeterminants of A. From a “categorical point of view” the expressions D_{I,I′} when I′ = (i_{2}, i_{3}, …, i_{n}, i_{1}) are particularly important. Set D_{I} = D_{I,I′}. There are n! expressions D_{I}. It is convenient to have a basic predeterminant Δ = D_{{12...n},{12...n}}.

We use the homological relations for quasideterminants to compare different D_{I,J}’s. For example, if I = (i_{1}, i_{2}, …, i_{n}), J = (j_{1}, j_{2}, …, j_{n}) and I′ = (i_{2}, i_{1}, …, i_{n}), J′ = (j_{2}, j_{1}, …, j_{n}) it is enough to compare |A|_{i_{1}j_{1}}|A^{i_{1}j_{1}}|_{i_{2}j_{2}} and |A|_{i_{2}j_{2}}|A^{i_{2}j_{2}}|_{i_{1}j_{2}}. The homological relations show that these expressions are in a sense “conjugate”:

|A|_{i_{2}j_{2}}|A^{i_{2}j_{2}}|_{i_{1}j_{1}} = |A^{i_{1}j_{1}}|_{i_{2}j_{2}}|A^{i_{2}j_{1}}|_{i_{1}j_{2}}^{−1}|A|_{i_{1}j_{1}}|A^{i_{1}j_{1}}|_{i_{2}j_{2}}^{−1}|A^{i_{2}j_{1}}|_{i_{1}j_{2}}|A^{i_{2}j_{2}}|_{i_{1}j_{1}}.

Let R be a division algebra, R^{∗} be the monoid of invertible elements in R and π : R^{∗} → R^{∗}/[R^{∗}, R^{∗}] be the canonical homomorphism. The group R^{∗}/[R^{∗}, R^{∗}] is abelian. To this group, we adjoin a zero element 0 with obvious multiplication, and denote the semi-group thus obtained by R̃. For μ ∈ R set μ̃ = π(μ) if μ ≠ 0 and μ̃ = 0 otherwise

We recall here a classical notion of the Dieudonne determinant [D, A]. There exist a unique homomorphism

det : M_{n}(R) → R̃

such that

i) det A′ = μ̃ det A for any matrix A′ obtained from A ∈ M_{n}(R) by multiplying one row of A from the left by μ,

ii) det A′′ = det A for any matrix A′′ obtained from A by adding one row to another,

iii) the identity matrix has determinant 1.

The homomorphism det is called the Dieudonne determinant.

Proposition 3.1.1. π(Δ(A)) is the Dieudonne determinant of A and π(D_{I,J}) = p(I)p(J)Δ where p(I) is the parity of ordering I.

Note that in [Dr] P. Draxl introduced a Dieudonne predeterminant, denoted δϵτ. For a generic matrix A over a division algebra there exists the Gauss decomposition A = UDL where U, D, L are upper-triangular, diagonal, and low-triangular matrices. Draxl denoted the product of diagonal elements in D from top to the bottom by δϵτ(A) and called it the Dieudonne predeterminant of A. For non-generic matrices Draxl used the Bruhat decomposition instead of Gauss decomposition.

Proposition 3.1.2. δϵτ(A) = Δ(A).

Proof. Let y_{1}, …, y_{n} be the diagonal elements in D from top to the bottom. As shown in [GR1, GR2] (see also Section 4.9 of the review), y_{k} = |A^{12...k−1,12...k−1}|_{kk}. Then δϵτ(A) = y_{1}y_{2} … y_{n} = Δ(A). □

Below we consider below special examples of noncommutative determinants.

3.2. Dieudonne determinant for quaternions.

Let A = (a_{ij}), i, j = 1, …, n be a generic quaternionic matrix. Recall that for any orderings I = (i_{1}, …, i_{n}), J = (j_{1}, …, j_{n}) of {1, …, n} we defined expressions

D_{I,J}(A) = |A|_{i_{1}j_{1}}|A^{i_{1}j_{1}}|_{i_{2},j_{2}}|A^{i_{1}i_{2},j_{1}j_{2}}|_{i_{3}j_{3}} … a_{i_{n}j_{n}}.

In the quaternionic case the Dieudonne determinant D maps

D : M_{n}(ℍ) → ℝ_{≥0}

(see [As]).

The following proposition generalizes a result from [VP].

Proposition 3.2.1. In the quaternionic case |D_{I,J}(A)|, the absolute value of D_{I,J}(A), is equal to D(A) for any I, J.

The proof of Proposition 3.2.1 follows from the homological relations for quasideterminants.

3.3. Moore determinants of Herimitian quaternionic matrices.

A quaternionic matrix A = (a_{ij}), i, j = 1, …, n is Hermitian if a_{ji} = ā_{ji} for all i, j. It follows that all diagonal elements of A are real numbers and that the submatrices A^{11}, A^{12,12}, … are Hermitian.

A determinant of Hermitian matrices was introduced by E.M. Moore in 1922 [M, MB]. Here we recall the original definition.

Let A = (a_{ij}), i, j = 1, …, n, be a matrix over a ring. Let σ be a permutation of {1, …, n}. Write σ as a product of disjoint cycles such that each cycle starts with a smallest number. Since disjoint cycles commute, we may write

σ = (k_{11} … k_{1j_{1}})(k_{21} … k_{2j_{2}}) … (k_{m1} … k_{mj_{m}})

where for each i, we have k_{i1} < k_{ij} for all j > 1, and k_{11} > k_{21} ⋯ > k_{m1}. This expression is unique. Let p(σ) be a parity of σ. Define Moore determinant M(A) by

(3.3.1)  M(A) = ∑_{σ∈S_{n}} p(σ)a_{k_{11},k_{12}} … a_{k_{1j_{1}} ,k_{11}}a_{k_{21},k_{22}} … a_{k_{mjm} ,k_{m1}}.

One can write cycles starting with the largest element. This will lead to the same result.

If A is Hermitian quaternionic matrix then M(A) is a real number. Moore determinants have nice features and are widely used (see, for example, [Al, As, Dy1]).

The following Theorem 3.3.2 shows that determinants of Hermitian quaternionic matrices can be obtained via our general approach. First we prove that for a quaternionic Hermitian matrix A determinants D_{I,I}(A) coincide up to a sign.

Recall that Δ(A) = D_{I,I}(A) for I = {1, …, n} and that Δ(A) is a pre-Dieudonne determinant in the sense of [Dr]. If A is Hermitian then Δ(A) is a product of real numbers and, therefore, Δ(A) is real.

Proposition 3.3.1. Let p(I) be the parity of the ordering I. Then Δ(A) = p(I)p(J)D_{I,J}(A).

The proof follows from homological relations for quasideterminants.

Theorem 3.3.2. Let A be a Hermitian quaternionic matrix. Then Δ(A) = M(A).

Proof. We use noncommutative Sylvester formula for quasideterminants (Theorem 1.5.2)

For i, j = 2, …, n set

B_{ij} = ( \begin{matrix}a_{11} & a_{1j} \\ a_{i1} & a_{ij} \end{matrix} ) .

B_{ij} is a Hermitian matrix. Let b_{ij} = M(B_{ij}) and c_{ij} = |B_{ij}|_{11}.

Note that B = (b_{ij}) and C = (c_{ij}) also are Hermitian matrices. It follows from (3.3.1) that M(A) = a_{nn}^{2−n}M(B). Note, that M(B) = a_{nn}^{n−1}M(C), therefore, M(A) = a_{nn}M(C).

By Theorem 1.5.2 , |A|_{11} = |C|_{11}, |A^{11}|_{22} = |C^{11}|_{22}, . . .. So,

|A^{11}|_{22}|A^{11}|_{22} … |A^{12...n−1,12...n−1}|_{n−1,n−1} = |C^{11}|_{22}|C^{11}|_{22} … |C^{12...n−1,12...n−1}|_{n−1,n−1}.

The left-hand product equals Δ(A)a_{nn}^{−1} and the right-hand product equals Δ(C), so Δ(A) = Δ(C)a_{nn} = M(A). □

3.4. Moore determinants and norms of quaternionic matrices.

For a generic quaternionic matrix A = (a_{ij}), i, j = 1, …, n, define quaternionic norm ν(A) by induction:

i) ν(A) = ν(a) if A is the 1 × 1-matrix (a),

ii) ν(A) = ν(|A|_{11})ν(A^{11}).

It follows that ν(A) ∈ ℝ_{≥0}.

Proposition 3.4.1. For generic matrices A, B i) ν(AB) = ν(A)ν(B), ii) ν(A) = Δ(A)Δ(A^{∗}) = Δ(AA^{∗}).

Since AA^{∗} is a Hermitian matrix one has the following

Corollary 3.4.2. ν(A) = M(AA^{∗}).

It follows that ν(A) is a polynomial in the a_{ij}’s and the ā_{ij}’s and so it can be defined for all matrices A by continuity. The continuation posseses the multiplicative property.

3.5. Study determinants.

An embedding of the field of complex numbers ℂ into ℍ is defined by an image of i ∈ ℂ. Chose the embedding given by x + yi ↦ x + yi + 0j + 0k, where x, y ∈ ℝ and identify ℂ with its image in ℍ. Then any quaternion a can be uniquely written as a = α + iβ where α, β ∈ ℂ.

Let M(n, F) be the algebra of matrices of order n over a field F. Define a homomorphism θ : ℍ → M(2, ℂ) by setting

θ(a) = ( \begin{matrix}α & −β̄ \\ β & ᾱ \end{matrix} ) .

The homomorpism can be extended to homomorhism of matrix algebras

θ_{n} : M(n, ℍ) → M(2n, ℂ).

Let A = (a_{ij}) ∈ M(n, ℍ), set θ_{n}(A) = (θ(a_{ij})).

In 1920, E. Study [S] defined a determinant S(A) of a quaternionic matrix A of order n by setting S(A) = det θ_{n}(A). Here det is the standard determinant of a complex matrix. The following proposition is well known (see [As]).

Proposition 3.5.1. For any quaternionic matrix A

S(A) = M(AA^{∗}).

The proof in [As] was based on properties of eigenvalues of quasideterminant matrices. Our proof based on Sylvester identity and homological relations actually shows that S(A) = ν(A) for a generic matrix A.

3.6. Quantum determinants.

Recall that A = (a_{ij}), i, j = 1, …, n is a quantum matrix over a field F if q ∈ F is given such that

a_{ik}a_{il} = q^{−1}a_{il}a_{ik} for k < l, a_{ik}a_{jk} = q^{−1}a_{jk}a_{ik} for i < j,
a_{il}a_{jk} = a_{jk}a_{il} for i < j, k < l,
a_{ik}a_{jl} − a_{jl}a_{ik} = (q^{−1} − q)a_{il}a_{jk} fori < j, k < l.

The quantum determinant det_{q} A by definition equals to

∑_{σ∈S_{n}}(−q)^{−l(σ)}a_{1σ(1)}a_{1σ(2)} … a_{1σ(n)},

where l(σ) is the number of inversions in σ.

If A is a quantum matrix then any square submatrix of A also is a quantum matrix with the same scalar q.

Theorem 3.6.1. [GR, GR1, GR2, KL]

det_{q} A = (−q)^{i−j}|A|_{ij} · det_{q} A^{ij} = (−q)^{i−j}det_{q} A^{ij} · |A|_{ij}.

Corollary 3.6.2. [GR, GR1, GR2, KL]

det_{q} A = |A|_{11}|A^{11}|_{22} … a_{nn}

and all factors in the right-hand side commute.

An essential generalizations of this result for matrices satisfying Faddeev-Reshetikhin-Takhtadjan relations was given in [ER].

3.7. Capelli determinants.

Let X = (x_{ij}), i, j = 1, …, n be a matrix of formal commutative variables and X^{T} the transposed matrix. Let D = (∂_{ij}), ∂_{ij} = ∂/∂_{ij}, be a matrix of differential operators. Then X^{T} D = (f_{ij}), where f_{ij} = ∑_{k} x_{ki}∂/∂_{kj}.

Let W be a diagonal matrix, W = diag(0, 1, 2, …, n). By definition, the Capelli determinant det_{Cap} of X^{T} D − W equals to the sum

∑_{σ∈S_{n}}(−1)^{l(σ)}f_{σ(1)1}(f_{σ(2)2} − δ_{σ(2)2} … (f_{σ(n)n} − (n − 1)δ_{σ(n)n}).

Set Z = X^{T} D − I_{n}. It was shown in [GR1, GR2] that the Capelli determinant can be expressed as a product of quasideterminants. More precisely, one has Theorem 3.7.1.

|Z|_{11}|Z^{11}|_{22} … z_{nn} = det X det D

and all the factors in the left hand side commute.

It is known [We] that the right hand side in the theorem is the Capelli determinant. The theorem can also be interpreted in a different way.

Let A = (e_{ij}), i, j = 1, … n be the matrix of the standard generators of the universal enveloping algebra U(gl_{n}). Recall that these generators satisfy the relations

[e_{ij}, e_{kl}] = δ_{jk}e_{il} − δ_{li}e_{kj}.

Let I_{n} be an identity n × n-matrix. It is well-known (see, for example, [Ho]) that coefficients of the polynomial in a central variable t

det(I_{n} + tA) := ∑_{σ∈S_{n}}(−1)^{l(σ)}(δ_{σ(1)1} + te_{σ(1)1}) … (δ_{σ(n)n} + t(e_{σ(n)n} − (n − 1)δ_{σ(n)n}))

generate the center of U(gl_{n}).

Theorem 3.7.1 can be stated in the following way [GKLLRT].

Theorem 3.7.2. det(I_{n} + tA) can be factored in the algebra of formal power series in t with coefficients in U(gl_{n})

det(I_{n} + tA) = (1 + te_{11}) | \begin{matrix}1 + t(e_{11} − 1) & te_{12} \\ te_{21} & 1 + t(e_{22} − 1) \end{matrix} | · . . .

· · · . . . | \begin{matrix}1 + t(e_{11} − n + 1) & … & te_{1n} \\ … & … & … \\ te_{n1} & … & 1 + t(e_{nn} − n + 1) \end{matrix} |

and the factors in the right-hand side commute with each other.

The above version is obtained by using the classical embedding of U(gl_{n}) into the Weyl algebra generated by (x_{ij}, ∂/∂x_{ij}), i, j = 1, …, n where e_{ij} corresponds to

f_{ij} = ∑_{k=1}^{n} x_{ki}∂/∂x_{kj}

.

3.8 Berezinians.

Let p(k) be a parity of an integer k, i.e. p(k) = 0 if k is even and p(k) = 1 if k is odd. A (commutative) super-ring over R^{0} is a ring R = R^{0} ⊕ R^{1}, such that

i) a_{i}a_{j} ∈ R^{p(i+j)} for any a_{m} ∈ R^{m}, m = 0, 1,

ii) ab = ba for any a ∈ R^{0}, b ∈ R, and cd = −dc for any c, d ∈ R^{1}.

Let A = ( \begin{matrix}X & Y \\ Z & T \end{matrix} ) be a (m + n) × (m + n)-block-matrix over a super-ring

R = R^{0} ⊕R^{1}, X be a m×m-matrix over R^{0}, T be a n×n-matrix over R^{0}, and Y, Z be matrices over R^{1}. If T is an invertible matrix, then X − Y T^{−1}Z is an invertible matrix over commutative ring R^{0}. Super-determinant, or Berezinian of A is defined by the following formula

BerA = det(X − Y T^{−1}Z) det T^{−1}.

Note that this expression belongs to R^{0}.

Theorem 3.8.1. Let R^{0} be a field. Set J_{k} = {1, 2, …, k}, k ≤ m + n and A^{k} = A^{J_{k},J_{k}}. Then BerA is a product of elements from R^{0}:

BerA = |A|_{11}|A^{(1)}|_{22} … |A^{(m−1)}|_{mm}|A^{(m)}|_{m+1,m+1}^{−1} … |A^{(m+n−1)}|_{m+n,m+n}^{−1}.

3.9 Cartier-Foata determinants.

Let A = (a_{ij}), i, j = 1, …, n be a matrix such that the entries a_{ij} and a_{kl} commute when i ≠ k. In this case Cartier and Foata [CF, F] defined a determinant of A as

det _{CF} A = ∑_{σ∈S_{n}}(−1)^{l(σ}a_{1σ(1)}a_{2σ(2)} … a_{nσ(n)}.

The order of factors in monomials a_{1σ(1)}a_{2σ(2)} … a_{nσ(n)} is insignificant.

Theorem 3.9.1. Let A = (a_{ij}), i, j = 1, …, n be a matrix such that the entries a_{ij} and a_{kl} commute when i ≠ k.

|A|_{pq} = (−1)^{p+q}det_{CF} (A^{pq})^{−1}det_{CF} A

if all expressions in the formula are defined.

IV. Noncommutative Plücker and Flag Coordiniates

In this section we described results obtained in [GR4].

4.1. Commutative Plücker coordinates. Let k ≤ n and A be a k × n-matrix over a commutative ring R. Denote by A(i_{1}, …, i_{k}) the submatrix of A consisting of columns labelled by indices i_{1}, …, i_{k}. Define p_{i_{1}...i_{k}}(A) := det A(i_{1}, …, i_{k}). Expressions p_{i_{1}...i_{k}}(A) are called Plücker coordinates of matrix A.

The Plücker coordinates p_{i_{1}...i_{k}}(A) satisfy the following properties:

i) (invariance) p_{i_{1}...i_{k}}(XA) = det Xp_{i_{1}...i_{k}}(A) for any k × k-matrix X over R, ii) (skew-symmetry) p_{i_{1}...i_{k}}(A) are skew-symmetric in indices; in particular, p_{i_{1}...i_{k}}(A) =

0 if a pair of indices coincides,

iii) (Plücker relations) Let i_{1}, …, i_{k−1} be k−1 distinct numbers which are chosen from the set 1, …, n, and j_{1}, …, j_{k+1} be k + 1 distinct numbers chosen from the same set. Then

∑_{t=1}^{k}(−1)^{t}p_{i_{1}...i_{k−1}j_{t}}(A)p_{j_{1}...j_{t−1}}j_{t+1} … j_{k+1}(A) = 0.

Example. For k = 2 and n = 4 the Plücker relations in iii) imply the famous identity

(4.1.1)  p_{12}(A)p_{34}(A) − p_{13}(A)p_{24}(A) + p_{23}(A)p_{14}(A) = 0.

A k × n-matrix over a field F is non-degenerate if at least one of its Plücker coordinates is invertible. Let M be a set of all non-degenerate k × n-matrices over the field. There is a natural action of the group GL(k, F) of invertible k × kmatrices on M defined by the multiplication from the right. The invariance of Plücker coordinates implies that they depend of GL(k, F)-orbits only.

The set of orbits of this action is canonically identified with the set of k-dimensional subspaces in n-dimensional vector space over F. The latter is called Grassmanian G_{k,n}. The Plücker coordinates are projective coordinates on G_{k,n}.

4.2. Quasi-Plücker coordinates for n × (n + 1)- and (n + 1) × n-matrices.

Let A = (a_{ij}), i = 1, …, n, j = 0, 1, …, n be a matrix over a division algebra R. Denote by A^{(k)} the submatrix of A obtained from A by removing its k-th column. Suppose that all A^{(k)} are invertible, choose any s ∈ {1, …, n} and set p_{ij}(A) = |A^{(j)}|_{si}^{−1}|A^{(i)}|_{sj}.

Proposition 4.2.1. The elements p_{ij}(A) do not depend on s. If g is an invertible n × n-matrix over R then p_{ij}(gA) = p_{ij}(A).

Proof. One can see that there exist nonzero elements x_{0}, …, x_{n} ∈ R such that

A ( \begin{matrix}x_{0} \\ … \\ x_{n} \end{matrix} ) = 0.

It follows that

A^{(j)} ( x_{0} ; …; x̂_{j} ; …; x_{n} ) = − ( \begin{matrix}a_{1j} \\ … \\ a_{nj} \end{matrix} ) x_{j}.

Cramer rules and transformations properties for quasideterminanats imply that |A^{(j)}|_{si}x_{i} = −|A^{(i)}|_{sj}x_{j}. Therefore,

(4.2.1)  p_{ij}(A) = |A^{(j)}|_{si}^{−1}|A^{(i)}|_{sj} = −x_{i}x_{j}^{−1}

does depend on s.

Let g be an invertible n × n-matrix. Note that gA ( \begin{matrix}x_{0} \\ … \\ x_{n} \end{matrix} ) = 0. It follows that

p_{ij}(gA) = −x_{i}x_{j}^{−1} = p_{ij}(A). □

In the commutative case p_{ij}(A) is a ratio of two Plücker coordinates: p_{ij}(A) = \frac{det A^{(j)}}{det A^{(i)}} . We call p_{ij}(A)’s the left quasi-Plücker coordinates of n × (n + 1)-matrix A.

Remark. The identity (4.2.1) reflects an important relation between left quasi-Plücker coordinates of the Grassmanian G_{n,n+1} of n-dimensional subspaces in R^{n+1} considered as a left R-module and the right projective coordinates on a line in R^{n+1} viewed as the right R-module. Note that in (4.2.1) the left fraction equals to the right fraction. We discuss the connection below.

Define also the right quasi-Plücker coordinates r_{ij}(B) for (n + 1) × n-matrix B = (b_{ji}). Denote by B^{(k)} the submatrix of B obtained from B by removing its k-th row. Suppose that all B^{(k)} are invertible, choose any s ∈ {1, …, n} and set r_{ij}(B) = |B^{(j)}|_{is}|B^{(i)}|_{js}^{−1}.

Proposition 4.2.2. The elements r_{ij}(B) do not depend of s. If g is an invertible n × n-matrix over R then r_{ij}(Bg) = r_{ij}(B).

Proof. The proof is dual the proof of Proposition 4.2.1. It is enough to find nonzero elements y_{0}, …, y_{n} ∈ R such that the product (y_{0}, …, y_{n})B = 0 and show that −y_{i}^{−1}y_{j} = r_{ij}(B). □

In the commutative case r_{ij}(A) = \frac{det B^{(j)}}{det A^{(i)}} .

4.3. Definition of left quasi-Plücker coordinates. Let A = (a_{pq}), p = 1, …, k, q = 1, …, n,k < n be a matrix over a division algebra R. Fix 1 ≤ i, j, i_{1}, …, i_{k−1} ≤ n such that i ∉ I = {i_{1}, …, i_{k−1}}. Let A(i, j, i_{1}, …, i_{k−1}) be a (k + 1) × k-submatrix of A with columns labelled by i, j, i_{1}, …, i_{k−1}.

For 1 ≤ s ≤ k set

p_{ij}^{I}(A) = | \begin{matrix}a_{1i}a_{1i_{1}} & … & a_{1i_{k−1}} \\ & … \\ a_{ki}a_{ki_{1}} & … & a_{ki_{k−1}} \end{matrix} |_{si}^{−1} · | \begin{matrix}a_{1j}a_{1i_{1}} & … & a_{1i_{k−1}} \\ & … \\ a_{kj}a_{ki_{1}} & … & a_{ki_{k−1}} \end{matrix} |_{sj} .

Note that p_{ij}^{I}(A) = p_{ij}(A(i, j, i_{1}, …, i_{k−1})).

Proposition 4.3.1.

i) p_{ij}^{I}(A) does not depend on s;

ii) p_{ij}^{I}(gA) = p_{ij}^{I}(A) for any invertible k × k matrix g.

Proof. Use Proposition 4.2.1 or the homological relations.

Definition 4.3.2. We call p_{ij}^{I}(A) left quasi-Plücker coordinates of the matrix A.

One may consider p_{ij}^{I} as quasi-Plücker coordinates of a k-dimensional left subspace generated in the left F -linear space F^{n} by rows of the matrix A. In the commutative case p_{ij}^{I} = \frac{p_{jI}}{p_{iI}}, where p_{α_{1}...α_{k}} is the standard Plücker coordinate.

4.4. Identities for the left Quasi-Plücker coordinates.

The following properties of p_{ij}^{I}’s trivially follow from their definition:

i) p_{ij}^{I} does not depend on reordering on elements in I; ii) p_{ij}^{I} = 0 if j ∈ I; iii) p_{ii}^{I} = 1 and p_{ij}^{I} · p_{jk}^{I} = p_{ik}^{I}.

Theorem 4.4.1. (Skew-Symmetry) Let N be a set of indices, |N| = k +1, i, j, m ∈ N. Then

p_{ij}^{N∖{i,j}} · p_{jm}^{N∖{j,m}} · p_{mi}^{N\{m,i}} = −1.

Theorem 4.4.2. (Plücker relations) Fix M = (m_{1}, …, m_{k−1}), L = (ℓ_{1}, …, ℓ_{k}). Let i ∉ M. Then

∑_{j∈L} p_{ij}^{M} · p_{ji}^{L\{j}} = 1.

Examples. Suppose that k = 2.

1) From Theorem 4.4.1 it follows that

p_{ij}^{{ℓ}} · p_{jℓ}^{{i}} · p_{ℓi}^{{j}} = −1.

In the commutative case one has p_{ij}^{\{ℓ\}} = \frac{p_{jℓ}}{p_{iℓ}} and p_{ij} = −p_{ji}, so this identity is valid.

2) From Theorem 4.4.2 it follows that for any i, j, ℓ, m

p_{ij}^{{ℓ}} · p_{ji}^{{m}} + p_{im}^{{ℓ}} · p_{mi}^{{j}} = 1.

In the commutative case this identity implies the standard identity (cf. (4.1.1))

p_{ij} · p_{ℓm} − p_{iℓ} · p_{jm} + p_{im} · p_{ℓj} = 0.

Remark. In fact the products p_{ij}^{\{ℓ\}}p_{ji}^{\{m\}} (which are equal to \frac{p_{jℓ}}{p_{iℓ}} · \frac{p_{im}}{p_{jm}} in the commutative case) are noncommutative cross-ratios.

We will consider different specializations of the relations of quasi-Plücker coordinates in a separate paper.

To prove Theorems 4.4.1 and 4..4.2 we need the following lemma. Let A = (a_{ij}), i = 1, …, k, j = 1, …, n, k < n, be a matrix over a skew-field. Denote by A_{j_{1},...,j_{ℓ}}, ℓ ≤ n, its submatrix (a_{ij}), i = 1, …, k, j = j_{1}, …, j_{ℓ}. Consider n × nmatrix

X = ( \begin{matrix}A_{1...k} & A_{k+1...n} \\ 0 & I_{n−k} \end{matrix} ) ,

where I_{m} is a m × m-identity matrix.

Lemma 4.4.3. Let j < k < i. If p_{ij}^{1...ĵ...k}(A) is defined then |X|_{ij} is defined and

(4.4.1)  |X|_{ij} = −p_{ij}^{1...ĵ...k}(A)

Proof. We have to prove

(4.4.2)  |X|_{ij} = −|A_{1...ĵ...ki}|_{si}^{−1} · |A_{1...k}|_{sj}

if the right-hand side is defined. We will prove this by induction on ℓ = n − k. We suppose that formula (2.2) is valid for l = m and prove it for ℓ = m + 1. Without loss of generality we may set j = 1, i = k + 1. By homological relations (Theorem 1.4.3)

|X|_{k+1,1} = −|X^{k+1,1}|_{s,k+1}^{−1} · |X^{k+1,k+1}|_{s1}

for an appropriate 1 ≤ s ≤ k. Here

X^{k+1,1} = ( \begin{matrix}A_{2...k+1} & A_{k+2...n} \\ 0 & E_{n−k−1} \end{matrix} ) ,

X^{k+1,k+1} = ( \begin{matrix}A_{1...k} & A_{k+2...n} \\ 0 & E_{n−k−1} \end{matrix} ) .

By the induction assumption

|X^{k+1,1}|_{s,k+1} = −|A_{23...kk+2}|_{s,k+2}^{−1} · |A_{23...k+1}|_{s,k+1},

|X^{k+1,k+1}|_{s1} = −|A_{23...kk+2}|_{s,k+2}^{−1} · |A_{1...k}|_{s1}

and |X|_{k+1,1} = −p_{k+1,1}^{23...k}. □

To prove Theorem 4.4.2 we apply the second formula in Theorem 1.5.5 to the matrix

X = ( \begin{matrix}A_{1...k} & A_{k+1...n} \\ 0 & E_{n−k} \end{matrix} )

for M = (k + 1, …, n) and any L such that |L| = n − k − 1.

By Lemma 4.4.3 |X|_{m_{i}ℓ} = −p^{1...ℓ̂...k}(A), |X^{M_{i},L}|_{m_{i}q} = −p_{m_{i}q}^{1...n\L}(A), and Theorem 4.4.2 follows from Theorem 1.5.5. □

To prove Theorem 4.4.1 consider matrix X for n = k + 1 and use homological relations.

Theorem 4.4.4. Let A = (a_{ij}), i = 1, …, k, j = 1, …, n, be a matrix with formal entries, and f(a_{ij}) a rational function over a free skew-field F generated by a_{ij}. Let f be invariant under all transformations

A → gA

where g is an invertible k × k-matrix over f. Then f is a rational function of the quasi-Plücker coordinates.

Proof. Let b_{ij} = a_{ij} for i, j = 1, … k. Consider matrix B = (b_{ij}). Then B^{−1} = (|B|_{ji}^{−1}). Set C = (c_{ij}) = B^{−1}A. Then

c_{ij} = \{ \begin{matrix}δ_{ij} & j ≤ k, \\ p_{ij}^{1...î...k}(A) & j > k. \end{matrix}

By invariance we have f(A) = f(1, C). □

4.5. Right quasi-Plücker coordinates. Consider a matrix B = (b_{ij}), i = 1, …, n; j = 1, …, k, k < n over a skew-field F. Fix 1 ≤ i, j, i_{1}, …, i_{k−1} ≤ n such that j ∉ I = (i_{1}, …, i_{k−1}). Also fix 1 ≤ t ≤ k and set

r_{ij}^{I}(B) = | \begin{matrix}b_{i1} & … & b_{ik} \\ b_{i_{1}1} & … & b_{i_{1}k} \\ & … \\ b_{i_{k−1}1} & … & b_{i_{k−1}k} \end{matrix} |_{it} · | \begin{matrix}b_{j1} & … & b_{jk} \\ b_{i_{1}1} & … & b_{i_{1}k} \\ & … \\ b_{i_{k−1}1} & … & b_{i_{k−1}k} \end{matrix} |_{jt}^{−1} .

Proposition 4.5.1.

i) r_{ij}^{I}(B) does not depend of t;

ii) r_{ij}^{I}(Bg) = r_{ij}^{I}(B) for any invertible k × k-matrix g over F

Definition 4.5.2. We call r_{ij}^{I}(B) right quasi-Plücker coordinates of the matrix B.

Proposition 4.5.1 ii) shows that r_{ij}^{I}(B) are quasi-Plücker coordinates of the right k-dimensional subspace generated by columns of B in the right vector space F^{n}.

4.6. Identities for the right quasi-Plücker coordinates.

Generating identities for the r_{ij}^{I}’s are dual to generating identities for the left quasi-Plücker coordinates p_{ij}^{I}.

i) r_{ij}^{I} does not depend on reordering on elements in I; ii) r_{ij}^{I} = 0 if i ∈ I; iii) r_{ii}^{I} = 1 and r_{ij}^{I} · r_{jk}^{I} = r_{ik}^{I}.

Theorem 4.6.1. (Skew-Symmetry) Let N be a set of indices, |N| = k +1, i, j, m ∈ N. Then

r_{ij}^{N∖{i,j}} · r_{jm}^{N∖{j,m}} · r_{mi}^{N\{m,i}} = −1.

Theorem 4.6.2. (Plücker relations) Fix M = (m_{1}, …, m_{k−1}), L = (ℓ_{1}, …, ℓ_{k}). Let i ∉ M. Then

∑_{j∈L} r_{ij}^{L\{j}}r_{ij}^{M} = 1.

4.7. The duality between quasi-Plücker coordinates. Let A = (a_{ij}), i = 1, …, k; j = 1, …, n; and B = (b_{rs}), r = 1, …, n; s = 1, …, n − k. Suppose that AB = 0. This is equivalent to the statement that the subspace generated by rows of A in the left linear space is orthogonal to the subspace generated by columns of B in the right linear space. Fix 1 ≤ i, j ≤ n, I ⊂ [1, n], i ∉ I, |I| = k − 1. Set J = ([1, n] \ I) \ {i, j}.

Theorem 4.7.1.

p_{ij}^{I}(A) + r_{ij}^{J}(B) = 0.

4.8. Quasi-Plücker coordinates for k × n-matrices for different k. Let A = (a_{αβ}), α = 1, …, k, β = 1, …, n, be any k ×n - matrix over a noncommutative skew-field and A′ a (k − 1) × n-submatrix of A. Fix 1 ≤ i, j, m, j_{1}, …, j_{k−2} ≤ n such that i, m ∉ J = {j_{1}, …, j_{k−2}} and i ≠ m.

Proposition 4.8.1.

p_{ij}^{J}(A′) = p_{ij}^{J∪{m}}(A) + p_{im}^{J}(A′) · p_{mj}^{J∪{i}}(A).

4.9. Applications of quasi-Plücker coordinates

Row and column expansion of a quasideterminant.

Some of the results obtained in [GR], [GR1], [GR2] and partially listed in section I can be rewritten in terms of quasi-Plücker coordinates. Let A = (a_{ij}), i, j = 1, …, n, be a matrix over a skew-field.

Fix 1 ≤ α, β ≤ n. Using the notations from section I let B = A^{{α},∅}, C = A^{∅,{β}} be the (n − 1) × n and n × (n − 1) submatrices of A obtained by deleting the α-th row and the β-th column. For any j ≠ β and i ≠ α set

\begin{aligned}p_{jβ} & = p_{jβ}^{1...ĵ...β̂...n}(B), \\ r_{αi} & = r_{αi}^{1...α̂...î...n}(C). \end{aligned}

Proposition 4.9.1.

i) |A|_{αβ} = a_{αβ} − ∑_{j≠β} a_{αj}p_{jβ},

ii) |A|_{αβ} = a_{αβ} − ∑_{i≠α} r_{αi}a_{iβ}

if terms in the right-hand sides are defined.

Homological relations.

Proposition 4.9.2. In the previous notations i) |A|_{ij}^{−1} · |A|_{iℓ} = −p_{jℓ} (row relations) ii) |A|_{ij} · |A|_{kj}^{−1} = −r_{ik} (column relations).

Corollary 4.9.3. In the previous notations, let (i_{1}, …, i_{s}), (j_{1}, …, j_{t}) be sequences of indices such that i ≠ i_{1}, i_{1} ≠ i_{2}, …, i_{s−1} ≠ i_{s}; j ≠ j_{1}, j_{1} ≠ j_{2}, …, j_{t−1} ≠ j_{t}.

Then

|A|_{i_{s}j_{t}} = p_{i_{s}i_{s−1}} … p_{i_{2}i_{1}}p_{i_{1}i} · |A|_{ij} · r_{jj_{1}}r_{j_{1}j_{2}} … r_{j_{t−1}j_{t}}.

Example.

For a matrix A = ( \begin{matrix}a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix} ) we have

\begin{matrix}a_{21} · a_{11}^{−1} · |A|_{11} · a_{22}^{−1} · a_{22} = |A|_{22}, \\ a_{12}a_{22}^{−1}a_{21}a_{11}^{−1}|A|_{11}a_{21}^{−1}a_{22}a_{12}^{−1}a_{11} = |A|_{11} \end{matrix}

Matrix multiplication. The following formula has been already used in the proof of Theorem 4.4.4. Let A = (a_{ij}), i = 1, …, n; j = 1, …, m, n < m; B = (a_{ij}), i = 1, …, n, j = 1, …, n; C = (a_{ik}), i = 1, …, n; k = n + 1, …, m.

Proposition 4.9.4. Let the matrix B be invertible. Then the p_{ik}^{1...î...n}(A) are defined for i = 1, … n, k = n + 1, … m, and

B^{−1}C = (p_{ik}^{1...î...n}(A)), i = 1, …, n; k = n + 1, …, m.

Quasideterminant of a product. Let A = (a_{ij}), B = (b_{ij}), i, j = 1, … n be matrices over a skew-field. Fix 1 ≤ k ≤ n. Consider the (n − 1) × n-matrix A′ = (a_{ij}), i ≠ k, and the n × (n − 1)-matrix B′′ = (b_{ij}), j ≠ k.

Proposition 4.9.5.

|B|_{kk} · |AB|_{kk}^{−1}|A|_{kk} = 1 + ∑_{α≠k} r_{kα} · p_{αk},

where

r_{kα} = r_{kα}^{1...α̂...n}(B′′) − right quasi-Plücker coordinates, and
p_{αk} = p_{αk}^{1...α̂...n}(A′) − left quasi-Plücker coordinates

if all the expressions are defined.

The proof follows from the multiplicative property of quasideterminants and Proposition 4.9.2.

Gauss decomposition. Consider matrix A = (a_{ij}), i, j = 1, …, n, over a skewfield. Let A_{k} = (a_{ij}), i, j = k, … n, B_{k} = (a_{ij}), i = 1, … n, j = k, … n, and C_{k} = (a_{ij}), i = k, … n, j = 1, … n. These are submatrices of sizes (n−k +1)×(n−k +1), n × (n − k + 1), and (n − k + 1) × n respectively.

Suppose that the quasideterminants

y_{k} = |A_{k}|_{kk}, k = 1, …, n

are defined and invertible. From [GR1], [GR2] it follows

Theorem 4.9.6.

A = ( \begin{matrix}1 & & & & x_{αβ} \\ & . & . & . \\ 0 & & & & 1 \end{matrix} ) ( \begin{matrix}y_{1} & & & & 0 \\ & . & . & . \\ 0 & & & & y_{n} \end{matrix} ) ( \begin{matrix}1 & & & & 0 \\ & . & . & . \\ z_{βα} & & & & 1 \end{matrix} ) ,

where

\begin{aligned}x_{αβ} & = r_{αβ}^{β+1...n}(B_{β}), 1 ≤ α < β ≤ n \\ z_{βα} & = p_{βα}^{β+1...n}(C_{β}). 1 ≤ α < β ≤ n \end{aligned}

An important example of Quasi-Plücker coordinates of Vandermonde matrix will be considered later.

4.10. Flag coordinates. Noncommutative flag coordinates were introduced in [GR1, GR2].

Let A = (a_{ij}), i = 1, …, k, j = 1, …, n, be a matrix over a division algebra R. Let F_{p} be the subspace of the left vector space R^{n} generated by the first p rows of A. Then F = (F_{1} ⊂ F_{2} ⊂ ⋯ ⊂ F_{k}) is a flag in R^{n}. Put

f_{j_{1}...j_{k}}(F) = | \begin{matrix}a_{1j_{1}} & … & a_{1j_{k}} \\ & … \\ a_{kj_{1}} & … & a_{kj_{k}} \end{matrix} |_{kj_{1}} .

In [GR1, GR2] functions f_{j_{1}...j_{k}}(F) were called the flag coordinates of F. Transformations properties of quasideterminants imply that f_{j_{1}...j_{k}}(F) does not depend on indices j_{2}, …, j_{k}.

Proposition 4.10.1. [GR1], [GR2] The functions f_{j_{1}...j_{m}}(F) do not change under left multiplication of A by an upper triangular matrix with units as its diagonal entries.

The functions f_{j_{1}...j_{k}}(F)’s satisfy the following relations [GR1, GR2].

Theorem 4.10.2.

f_{j_{1}j_{2}j_{3}...j_{k}}(F)f_{j_{1}j_{3}...j_{k}}(F)^{−1} = −f_{j_{2}j_{1}...j_{k}}(F)f_{j_{2}j_{3}...j_{k}}(F)^{−1},

f_{j_{1}...j_{k}}(F)f_{j_{1}...j_{k−1}}(F)^{−1}+f_{j_{2}...j_{k}j_{1}}(F)f_{j_{2}...j_{k}}(F)^{−1}+⋯+f_{j_{k}...j_{k−1}}(F)f_{j_{k}...j_{k−2}}(F)^{−1} = 0

Example. Let A = ( \begin{matrix}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{matrix} ). Then f_{12}(F)a_{11}^{−1} = −f_{21}(F)a_{12}^{−1} and

f_{12}(F)a_{11}^{−1} + f_{23}(F)a_{12}^{−1} + f_{31}(F)a_{13}^{−1} = 0.

It is easy to see that

p_{ij}^{i_{1}...i_{k−1}}(A) = (f_{ii_{1}...i_{k−1}}(F))^{−1} · f_{ji_{1}...i_{k−1}}(F).

Theorems 4.4.1 and 4.4.2 may be deduced from Theorem 4.10.2.

V. Bezout and Vieta Theorems

In this section we will discuss analogues of the Bezout and Vieta Theorems for polynomials over a noncommutative division algebra. We will prove the Vieta Theorem which was formulated in [GR3, GR4] using our noncommutative form of Sylvester’s identity. The proof of this theorem, using differential operators, appeared in [EGR]. Another noncommutative version of the Vieta Theorem, based on notions of traces and determinants, was given by A. Connes and A. Schwarz in [CS].

5.1. Vandermonde quasideterminants.

Let x_{1}, x_{2}, …, x_{k} be a set of elements of a skew-field F. For k > 1 define a Vandermonde quasideterminant

V (x_{1}, …, x_{k}) = | \begin{matrix}x_{1}^{k−1} & … & x_{k}^{k−1} \\ & … \\ x_{1} & … & x_{k} \\ 1 & … & 1 \end{matrix} |_{1k} .

We say that a sequence x_{1}, …, x_{n} ∈ F is independent if the quasideterminants V (x_{1}, …, x_{k}), k = 2, …, n, are all defined and invertible. For independent sequences x_{1}, …, x_{n} and x_{1}, …, x_{n−1}, z, where x_{1}, …, x_{n}, z ∈ F set

\begin{aligned}y_{1} & = x_{1}, z_{1} = z \\ y_{k} & = V (x_{1}, …, x_{k})x_{k}V (x_{1}, …, x_{k})^{−1}, k ≥ 2 \\ z_{k} & = V (x_{1}, …, x_{k−1}, z)zV (x_{1}, …, x_{k−1}, z)^{−1}, k ≥ 2. \end{aligned}

In the commutative case y_{k} = x_{k}, and z_{k} = z for k = 1, …, n.

5.2. Bezout and Vieta decompositions of Vandermonde quasideterminants. We will prove here:

Theorem 5.2.1. (Bezout decomposition of a Vandermonde quasideterminant) Suppose that sequences x_{1}, …, x_{n} and x_{1}, …, x_{n−1}, z are independent. Then

(5.2.1)  V (x_{1}, …, x_{n}, z) = (z_{n} − y_{n})(z_{n−1} − y_{n−1}) … (z_{1} − y_{1}).

Note that if z commutes with x_{i}, i = 1, … n then

V (x_{1}, …, x_{n}, z) = (z − y_{n})(z − y_{n−1}) … (z − y_{1}).

Theorem 5.2.2. (Vieta decomposition of a Vandermonde quasideterminant) For an independent sequence x_{1}, …, x_{n}, z

(5.2.2)  V (x_{1}, …, x_{n}, z) = z^{n} + a_{1}z^{n−1} + ⋯ + a_{n−1}z + a_{n},

where

(5.2.3)  a_{k} = (−1)^{k} ∑_{1≤i_{1}<i_{2}<···<i_{k}≤n} y_{i_{k}} · y_{i_{k−1}} … y_{1}.

In particular

a_{1} = −(y_{1} + ⋯ + y_{n}),
a_{2} = ∑_{1≤i<j≤n} y_{j}y_{i},
…
a_{n} = (−1)^{n}y_{n} … y_{1}.

5.3. Proof of Theorem 5.2.2. We will show, by induction on n, that Theorem 5.2.2 follows from Theorem 5.2.1. For n = 1 one has V (x_{1}, z) = z −x_{1} and formulas (5.2.1) and (5.2.2) are valid. Suppose that these formulas are valid for m = n − 1. By Theorem 5.2.1

V (x_{1}, …, x_{n}, z) = (z_{n} − y_{n})V (x_{1}, …, x_{n−1}, z) =
= (V (x_{1}, …, x_{n−1}, z) · z) − (y_{n} · V (x_{1}, …, x_{n−1}, z)).

By induction,

V (x_{1}, …, x_{n−1}, z) = z^{n−1} + b_{1}z^{n−2} + ⋯ + b_{n−1},

where

b_{1} = −(y_{1} + ⋯ + y_{n−1}),
…
b_{n−1} = (−1)^{n}y_{n−1} · . . . · y_{1}.

So,

V (x_{1}, …, x_{n}, z) = z^{n} + (b_{1} − y_{n})z^{n−1} + (b_{2} − y_{n}b_{1})z^{n−2} + ⋯ − y_{n}b_{n} =
= z^{n} + a_{1}z^{n−1} + ⋯ + a_{n},

where a_{1}, …, a_{n} are given by formulas (3.3). □

5.4. A division lemma. To prove Theorem 5.2.1 we need the following

Lemma 5.4.1.

V (x_{1}, …, x_{n}, z) = V (x̂_{2}, …, x̂_{n}, ẑ)(z − x_{1}),

where

\begin{aligned}x̂_{k} & = (x_{k} − x_{1})x_{k}(x_{k} − x_{1})^{−1}, k = 2, …, n \\ ẑ & = (z − x_{1})z(z − x_{1})^{−1}. \end{aligned}

Proof. By definition

V (x_{1}, …, x_{n}, z) = | \begin{matrix}x_{1}^{n} & x_{2}^{n} & … & z^{n} \\ x_{1}^{n−1} & x_{2}^{n−1} & … & z^{n−1} \\ & & … \\ x_{1} & x_{2} & … & z \\ 1 & 1 & … & 1 \end{matrix} |_{1n+1} .

Multiply the k-th row by x_{1} from the left and subtract it from the (k − 1)-th row for k = 2, …, n. The quasideterminant will not change, and so

V (x_{1}, …, x_{n}, z) = | \begin{matrix}0 & x_{2}^{n} − x_{1}x_{2}^{n−1} & … & z^{n} − x_{z}^{n−1} \\ 0 & x_{2}^{n−1} − x_{1}x_{2}^{n−2} & & z^{n−1} − x_{1}z^{n−2} \\ ⋮ & ⋮ & & ⋮ \\ 0 & x_{2} − x_{1} & & z − x_{1} \\ 1 & 1 & & 1 \end{matrix} |_{1n+1} =

= | \begin{matrix}0 & (x_{2} − x_{1})x_{2}^{n−1} & … & (z − x_{1})z^{n−1} \\ ⋮ & ⋮ & & ⋮ \\ 0 & x_{2} − x_{1} & & z − x_{1} \\ 1 & 1 & & 1 \end{matrix} |_{1n+1} .

Applying Sylvester’s theorem to the last quasideterminant with the element of index (n1) as a pivot one has

V (x_{1}, …, x_{n}, z) = | \begin{matrix}(x_{2} − x_{1})x_{2}^{n−1} & … & (z − x_{1})z^{n−1} \\ … & & … \\ x_{2} − x_{1} & & z − x_{1} \end{matrix} |_{1n} .

Multiply the k-th column on the right by (x_{k+1} − x_{1})^{−1} for k = 1, …, n − 1 and the last column by (z−x_{1})^{−1}. According to elementary properties of quasideterminants,

V (x_{1}, …, x_{n}, z) =

= | \begin{matrix}(x_{2} − x_{1})x_{2}^{n−1}(x_{2} − x_{1})^{−1} & … & (z − x_{1})z^{n−1}(z − x_{1})^{−1} \\ ⋮ & & ⋮ \\ 1 & & 1 \end{matrix} |_{1n} (z − x_{1}) =

= | \begin{matrix}x̂_{2}^{n−1} & … & ẑ^{n−1} \\ ⋮ & & ⋮ \\ 1 & … & 1 \end{matrix} |_{1n} · (z − x_{1}) = V (x̂_{2}, …, x̂_{n}, ẑ) · (z − x_{1}).

5.5. Proof of Theorem 5.2.1. We proceed by induction on n. By Lemma 5.4.1, Theorem 5.2.1 is valid for n = 2. Also by Lemma 5.4.1

(5.5.1)  V (x_{1}, …, x_{n}, z) = V (x̂_{2}, …, x̂_{n}, ẑ)(z − x_{1}).

Suppose that our theorem is valid for m = n − 1. Then

V (x̂_{2}, …, x̂_{n}, ẑ) = (z′_{n} − y′_{n}) … (z′_{2} − y′_{2}),

where

\begin{aligned}z′_{2} & = ẑ, \\ y′_{2} & = x̂_{2}, \\ z′_{k} & = V (x̂_{2}, …, x̂_{k−1}, ẑ)ẑV^{−1}(x̂_{2}, …, x̂_{k−1}, ẑ), \\ y′_{k} & = V (x̂_{2}, …, x̂_{k})x̂_{k}V^{−1}(x̂_{2}, …, x̂_{k}) for k = 3, …, n. \end{aligned}

It is enough to show that

z′_{k} = z_{k}, and y′_{k} = y_{k} for k = 2, …, n.

For k = 2, this is obvious. By Lemma 5.4.1

V (x̂_{2}, …, x̂_{k−1}, ẑ) = V (x_{1}, …, x_{k−1}, z)(z − x_{1})^{−1},

and by definition ẑ = (z − x_{1})z(z − x_{1})^{−1}. So,

z′_{k} = {V (x_{1}, …, x_{k−1}, z)(z − x_{1})^{−1}}(z − x_{1})z(z − x_{1})^{−1}·
·{(z − x_{1})V^{−1}(x_{1}, …, x_{k−1}, z)} = z_{k} for k = 3, …, n.

Similarly, y′_{k} = y_{k} for k = 3, …, n and from (3.4) we have

V (x_{1}, …, x_{n}, z) = (z_{n} − y_{n}) … (z_{2} − y_{2})(z_{1} − y_{1}).
□

5.6. Another expression for the coefficients in Vieta decomposition. Another expression for the coefficients a_{1}, … a_{n} in Vieta decomposition of V (x_{1}, …, x_{n}, z) may be obtained from Proposition 1.5.1.

Theorem 5.6.1. [GKLLRT]:

V (x_{1}, …, x_{n}, z) = z^{n} + a_{1}z^{n−1} + ⋯ + a_{n},

where for k = 1, …, n

(5.6.1)  a_{k} = − | \begin{matrix}x_{1}^{n} & … & x_{n}^{n} \\ & … \\ x_{1}^{n−k+1} & … & x_{n}^{n−k+1} \\ x_{1}^{n−k−1} & … & x_{n}^{n−k−1} \\ & … \\ 1 & … & 1 \end{matrix} |_{1n} · | \begin{matrix}x_{1}^{n−1} & … & x_{n}^{n−1} \\ & … \\ x_{1}^{n−k} & … & x_{n}^{n−k} \\ & … \\ 1 & … & 1 \end{matrix} |_{kn}^{−1} .

From Theorem 5.6.1 we will get the Bezout and Vieta formulas expressing the coefficients of the equation

(5.6.2)  z^{n} + a_{1}z^{n−1} + ⋯ + a_{n} = 0

5.7. The Bezout and Vieta Theorems.

Lemma 5.7.1. Suppose that x_{1}, …, x_{n} is an independent set of roots of the equation (5.6.2). Then the coefficients a_{1}, …, a_{n} may be written in the form (5.6.1).

Proof. Consider a system of right linear equations

x_{i}^{n} + a_{1}x_{i}^{n−1} + ⋯ + a_{n−1}x_{i} + a_{n} = 0, i = 1, …, n

with unknowns a_{1}, …, a_{n} and use Cramer’s rule. □

Theorem 5.7.2. (Noncommutative Bezout Theorem). Let x_{1}, …, x_{n} be an independent set of roots of equation (5.6.2). In notations of Theorem 5.2.1

z^{n} + a_{1}z^{n−1} + ⋯ + a_{n} = (z_{n} − y_{n}) … (z_{1} − y_{1}).

Proof. Use Lemma 5.7.1, Theorem 5.6.1 and Theorem 5.2.1. □

Theorem 5.7.2. (Noncommutative Vieta Theorem) [GR3] Let x_{1}, …, x_{n} be an independent set of roots of equation (3.6). Then coefficients a_{1}, …, a_{n} of the equation are given by formulas (5.2.3).

Proof Use Lemma 3.2.1, Theorem 3.1.4 and Theorem 3.1.2 □

VI. Noncommutative symmetric functions

A general theory of noncommutative symmetric functions was developed in [GK-LLRT]. In fact, [GKLLRT] was devoted to the study of different systems of multiplicative and linear generators in a free algebra Sym generated by any system of noncommuting variables Λ_{i}, i = 1, 2, . . .. In [GKLLRT] these variables were called elementary symmetric functions but the theory was developed independently of the origin of Λ_{i}’s.

In this section we apply the general theory to noncommutative symmetric functions generated by specific Λ_{i}’s. As in the commutative case, they depend of a set of roots of a polynomial equation.

6.1. Formal noncommutative symmetric functions.

This theory was started in [GKLLRT] and developed in several papers (see, for example, [KLT LST]). An extensive review was given in [Thi]. Here we just remind some basic constructions.

The algebra Sym is a free graded associative algebra over a field F generated by an infinite sequence of indeterminants (Λ_{k}), deg Λ_{k} = k, k ≥ 1. The homogeneous component of deg n is denoted by Sym_{n}.The direct sum ⊕_{n≥1}Sym_{n} is denoted by Sym_{+}. Initially Λ_{k}’s were regarded as as the elementary symmetric functions of some virtual set of arguments. A natural set of arguments was found later [GR3, GR4].

Recall that the algebra of commutative symmetric functions Sym in variables t_{1}, t_{2}, t_{3}, … is defined as the inductive limit of algebras of symmetric polynomails Sym_{N} in commuting variables t_{1}, …, t_{N}.

Let I_{N} be an ideal Sym generated by all Λ_{k}, k > N. The canonical map π_{N} of Sym onto the algebra Sym_{N} of symmetric polynomials in commuting variables t_{1}, …, t_{N} is given by the formulas

(6.1.1a)  π_{N}(Λ_{k}) = ∑_{i_{1}<···<i_{k}} t_{i_{1}} … t_{i_{k}} for k ≤ N,

(6.1.1b)  π_{N}(Λ_{k}) = 0 for k > N.

Obviously, π_{N}(Λk) is the k-th elementary symmetric function in N variables and π_{N}(I_{N}) = 0. The inductive limit of π_{N} gives the canonical homomorphism π : Sym → Sym.

Let t be a formal variable commutating with all Λ_{k}’s. Set Λ_{0} = 1. Define the generating series

λ(t) := ∑_{k≥0} Λ_{k}t^{k}.

Definition 6.1.1. The complete homogeneous symmetric functions are defined by

σ(t) := ∑_{k≥0} S_{k}t^{k} = λ(−t)^{−1}.

The power sums symmetric functions of the first kind Ψ_{k} are defined by

∑_{k≥1} Ψ_{k}t^{k−1} := σ(t)^{−1} \frac{d}{dt}σ(t).

The power sums symmetric functions of the second kind Φ_{k} are defined by

∑_{k≥1} Φ_{k}t^{k−1} := \frac{d}{dt} log σ(t).

By using formula (6.1.1) one can prove that π_{N}(S_{k}) is the k-th complete symmetric function and π_{N}(Ψ_{k}) = π_{N}(Φ_{k}) is the k-th power symmetric function in N commuting variables.

Definition 6.1.1 leads to the following quasideterminantal formulas.

Proposition 6.1.2. For every k ≥ 1, one has

S_{k} = (−1)^{k−1} | \begin{matrix}Λ_{1} & Λ_{2} & … & Λ_{k−1} & Λ_{k} \\ 1 & Λ_{1} & … & Λ_{k−2} & Λ_{k−1} \\ 0 & 1 & … & Λ_{k−3} & Λ_{k−2} \\ & & … \\ 0 & 0 & … & 1 & Λ_{1} \end{matrix} | ,

Λ_{k} = (−1)^{k−1} | \begin{matrix}S_{1} & 1 & 0 & … & 0 \\ S_{2} & S_{1} & 1 & … & 0 \\ S_{3} & S_{2} & S_{1} & 1 & … & 0 \\ & & … \\ S_{k} & S_{k−1} & S_{k−2} & … & S_{1} \end{matrix} | ,

kS_{k} = | \begin{matrix}Ψ_{1} & Ψ_{2} & … & Ψ_{k−1} & Ψ_{k} \\ −1 & Ψ_{1} & … & Ψ_{k−2} & Ψ_{k−1} \\ 0 & −2 & … & Ψ_{k−3} & Ψ_{k−2} \\ & & … \\ 0 & 0 & … & −n + 1 & Ψ_{1} \end{matrix} | ,

kΛ_{k} = | \begin{matrix}Ψ_{1} & 1 & 0 & … & 0 \\ Ψ_{2} & Ψ_{1} & 2 & … & 0 \\ Ψ_{3} & Ψ_{2} & Ψ_{1} … & 0 \\ & & … \\ Ψ_{k} & Ψ_{k−1} & Ψ_{k−2} & … & Ψ_{1} \end{matrix} | ,

Ψ_{k} = | \begin{matrix}Λ_{1} & 2Λ_{2} & … & (k − 1)Λ_{k−1} & kΛ_{k} \\ 1 & Λ_{1} & … & Λ_{k−2} & Λ_{k−1} \\ 0 & 1 & … & Λ_{k−3} & Λ_{k−2} \\ & & … \\ 0 & 0 & … & 1 & Λ_{1} \end{matrix} | ,

Ψ_{k} = | \begin{matrix}S_{1} & 1 & 0 & … & 0 \\ 2S_{2} & S_{1} & 1 & … & 0 \\ 3S_{3} & S_{2} & S_{1} & 1 & … & 0 \\ & & … \\ kS_{k} & S_{k−1} & S_{k−2} & … & S_{1} \end{matrix} | .

Each of the four sequences (Λ_{k}), (S_{k}), (Ψ_{k}), and (Φ_{k}) constitues a set of generators in Sym. Therefore, aach of the four sets of products F_{i_{1}} … F_{i_{N}}, i_{1}, …, i_{N} ≥ 1, where F_{k} equals to Λ_{k}, S_{k}, Ψ_{k}, or Φ_{k}, k = 1, 2, … constitues a linear basis in Sym_{+}. Another important example of a linear basis in Sym_{+} is given by ribbon Schur functions.

6.2. Ribbon Schur Function.

Let I = (i_{1}, …, i_{k}), i_{1}, …, i_{k} ≥ 1 be an ordered set.

Definition 6.2.1. [GKLLRT] The ribbon Schur function R_{I} is defined by

R_{I} = (−1)^{k−1} | \begin{matrix}S_{i_{1}} & S_{i_{1}+i_{2}} & S_{i_{1}+i_{2}+i_{3}} & … & S_{i_{1}+···+i_{k}} \\ 1 & S_{i_{2}} & S_{i_{2}+i_{3}} & … & S_{i_{2}+···+i_{k}} \\ 0 & 1 & S_{i_{3}} & … & S_{i_{3}+···+i_{k}} \\ & & & … \\ 0 & 0 & 0 & … & S_{i_{k}} \end{matrix} | .

In [GKLLRT] a similar description of R_{I} as quasideterminants of matrices with entries Λ_{k}’s was given.

Proposition 6.1.2 implies that R_{I} = S_{k} if I = {k} and R_{I} = Λ_{k} if i_{1} = ⋯ = i_{k} = 1. The homomorphisms π_{N} map R_{I}’s into ribbon Schur functions was introduced by MacMahon [M].

Theorem 6.2.2. [GKLLRT] Ribbon Schur functions R_{I}’s constitue a linear basis in Sym.

The homological relations for quasideterminants imply the following multiplication rule, whose commutative version is due to MacMahon [M]. Let I = (i_{1}, …, i_{r}), J = (j_{1}, …, j_{s}), i_{p} ≥ 1, j_{q} ≥ 1 for all p, q. Set I+J = (i_{1}, …, i_{r−1}, i_{r}+j_{1}, j_{2}, …, j_{s}) and I · J = (i_{1}, …, i_{r}, j_{1}, …, j_{s}).

Theorem 6.2.3. [GKLLRT]

R_{I}R_{J} = R_{I+J} + R_{I·J}.

In commutative case ribbon Schur functions are not linearly independent. It reflects the fact that the kernel of π : Sym → Sym is generated by linear combinations of R_{I}’s. A description of Kerπ is given by the following theorem.

Let I = (i_{1}, …, i_{r} and J = (j_{1}, …, j_{s}). We say that I ≤ J if i_{1} = j_{1}+j_{2}+⋯+j_{t_{1}}, i_{2} = j_{t_{1}+1} + ⋯ + j_{t_{2}}, …, i_{s} = j_{t_{s−1}+1} + ⋯ + j_{s}. Recall that I is an ordered set. Denote by u(I) the corresponding unordered set.

Theorem 6.2.4. The kernel of π is linearly generated by differences

Δ_{J,J′} = ∑_{I≤J} R_{I} − ∑_{I′≤J} R_{I′},

where u(J) = u(J′).

Example. Let J = (12), J′ = (21). Then Δ_{J,J′} = (R_{12} + R_{3}) − (R_{21} + R_{3}) = R_{12} − R_{21} and π(R_{12} − R_{21}) = 0.

A naturality of ribbon Schur functions R_{I} can be explained via the following construction.

6.3. Algebras with two multiplications.

The relations between functions Λ_{k} and functions S_{k} may be illuminated by defining two associative products on the ideal Sym_{+}. Recall, that the monomials (Λ_{i_{1}} … Λ_{i_{r}}) as well as monomials (S_{i_{1}} … S_{i_{r}}) constitue linear basises for the ideal.

Definition 6.3.1. Define a linear map ∗_{1} : Sym_{+} ⊗ Sym_{+} → Sym_{+} by

(Λ_{i_{1}} … Λ_{i_{r}}) ∗_{1} (Λ_{j_{1}} … Λ_{j_{s}}) = Λ_{i_{1}} … Λ_{i_{r−1}}Λ_{i_{r}+j_{1}}Λ_{j_{2}} … Λ_{j_{s}}

and a linear map ∗_{2} : Sym_{+} ⊗ Sym_{+} → Sym_{+} by

(S_{i_{1}} … S_{i_{r}}) ∗_{1} (S_{j_{1}} … S_{j_{s}}) = S_{i_{1}} … S_{i_{r−1}}S_{i_{r}+j_{1}}S_{j_{2}} … S_{j_{s}}.

Write ab = a ∗ b for a, b ∈ Sym_{+}. Then it is clear that a ∗ (b ∗_{i} c) = (a ∗ b) ∗_{i} c, a ∗_{i} (b ∗ c) = (a ∗_{i} b) ∗ c and a ∗_{i} (b ∗_{i} c) = (a ∗_{i} b) ∗_{i} c for all a, b, c ∈ Sym_{+}, i = 1, 2. When it is convinient we will write ∗ = ∗_{0}.

In fact, ∗, ∗_{1} and ∗_{0} are closely related.

Lemma 6.3.2. ∗ = ∗_{1} + ∗_{2}.

Proof.

λ(−t)^{−1} = (1 − ∑_{i>0} Λ_{i}t^{i})^{−1} = 1 + ∑_{j>0} ∑_{i_{1}+...i_{l}=j}(−1)^{l+j}Λ_{i_{1}} … Λ_{i_{l}}t^{j}.

Since Λ_{i} = Λ_{1} ∗_{1} Λ_{1} ∗_{1} ⋯ ∗_{1} Λ_{1}, where there are i − 1 occurances of ∗_{1}, we have that the coefficient of t^{j} in λ(−t)^{−1} is

∑_{u_{1},...u_{j−1}∈{0,1}}(−1)^{k+1}Λ_{1} ∗_{u_{1}} Λ_{1} ∗_{u_{2}} ⋯ ∗_{u_{j−1}} Λ_{1},

where k is the number of u_{t} equal to 0.

Since 1 + ∑ S_{i}t^{i} = λ(−1)^{−1} we have that

S_{i} = ∑_{u_{1},...u_{j−1}∈{0,1}}(−1)^{k+1}Λ_{1} ∗_{u_{1}} Λ_{1} ∗_{u_{2}} ⋯ ∗_{u_{j−1}} Λ_{1}.

It follows that S_{i} ∗_{1} S_{j} = S_{i}S_{j} − S_{i+j} and hence that ∗_{1} = ∗ − ∗_{2}, as required.

This gives an isomorphism of vector spaces

θ : Sym_{+} = ∑_{l≥1}(Sym_{+})_{l} → F ⟨U⟩ = ∑_{l≥0} F ⟨U⟩_{l}

where U is the two dimensional vector space with basis {u_{0}, u_{1}}.

This isomorphism shifts degree, restricting to an isomorphism between (Sym_{+})_{l+1} and F ⟨U⟩_{l}.

The monomial u_{l_{1}} … u_{i_{l}} in F ⟨U⟩_{l} is mapped to Λ_{1}∗_{i_{1}}Λ_{1}∗_{i_{2}}⋯∗_{i_{l}}Λ_{1} ∈ (Sym_{+})_{l+1}. If u_{2} = u_{0} − u_{1}, and ϕ is the automorphism of U defined by ϕ(u_{0}) = u_{0}, ϕ(u_{1}) =

u_{2}, then ϕ extends to an endomorphism of F ⟨U⟩. Then we see that that

θ^{−1}ϕθ(Λ_{i_{1}} … Λ_{i_{r}}) = S_{i_{1}} … S_{i_{r}},

recovering Proposition 4.13 of [GKLLRT].

Similarly, the identity

a^{n−1} + (−1)^{n}b^{n−1} + ∑_{k=1}^{n−1}(−1)^{n−k}a^{k−1}(a + b)b^{n−k−1} = 0,

valid in any associative algebra, becomes upon setting a = u_{2}, b = u_{1} and applying θ^{n−1}, 0 = ∑_{k=0}^{n}(−1)^{n−k}Λ_{k}S_{n−k}.

These identities may be written in terms of quasideterminants (Proposition 6.1.2).

Write each finite sequence ω of symbols 1 and 2 as (2 … 2)(1)(2 … 2)(1) … bracketing maximal number of 2’s and each 1. Labell the sequence as I_{ω} = (i_{1}, …, i_{k}) where i_{m} is the number of elements in the m-th bracket. For example, sequences 2212 and 12212 will be labelled as 2, 1, 1 and and 1, 2, 1, 1 correspondingly. The sequence 222 is labelled as I = (3) and the sequence 111 is labelled as I = (111).

The bijective correspondences between monomials in F ⟨U⟩ and finite sequences in 1, 2’s and between the finite sequences and ordered sets I = (i_{1}, …, i_{k}). Thus, to any monomial u_{j_{1}} … u_{j_{l}} ∈ F ⟨U⟩ corresponds a ribbon Schur function R_{I} establishing a bijective correspondence between two natural linear basises.

6.4. Quasi-Schur functions.

Quasi-Schur functions were defined in [GKLLRT]. They are not elements in Sym but in the free-skew field generated by S_{1}, S_{2} . . .. Let I = (i_{1}, …, i_{k}).

Definition 6.4.1. Define Š_{I} by setting

(6.4.1)  Š_{I} = (−1)^{k−1} | \begin{matrix}S_{i_{1}} & S_{i_{2}+1} & … & S_{i_{k}+k−1} \\ S_{i_{1}−1} & S_{i_{2}} & … & S_{i_{k}+k−2} \\ & & … \\ S_{i_{1}−k+1} & S_{i_{2}−k+2} & … & S_{i_{k}} \end{matrix} | .

The expression Š_{I} is called the generalized quasi-Schur function. A generalized quasi-Schur function Š_{I} is called the quasi-Schur function if I = (i_{1}, …, i_{k}) is a partition, i.e. weakly increasing sequence of nonnegative sequences.

In particular, Š_{k} = S_{k} and Š_{1^{k}} = Λ_{k} where 1^{k} := (1 … 1) with k occurances of 1.

Definition 6.4.1 is a noncommutative analog of Jacobi-Trudi formula written in French notations. In the commutative case Š_{I} for a partition I is the ratio of two Schur functions S_{I}/S_{J} where J = (i_{1} − 1, …, i_{k−1} − 1). It shows that in general Š is not polynomial in S_{k}’s.

Remark. The homological relations and trasformation properties of quasideterminants imply that any generalized quasi-Schur function Š_{I} can be expressed as a rational function in the quasi-Schur functions. For example

Š_{42} = − | \begin{matrix}S_{4} & S_{3} \\ S_{3} & S_{2} \end{matrix} | = − | \begin{matrix}S_{3} & S_{4} \\ S_{2} & S_{3} \end{matrix} | = | \begin{matrix}S_{3} & S_{4} \\ S_{2} & S_{3} \end{matrix} | S_{3}^{−1}S_{2} = Š_{33}S_{3}^{−1}S_{2}.

6.5. Symmetric functions in noncommutative variables. Fix n independent indeterminants x_{1}, x_{2}, …, x_{n} and construct new variables y_{1}, …, y_{n} which are rational functions in x_{1}, …, x_{n}. Recall that

V (x_{1}, …, x_{k}) = | \begin{matrix}x_{1}^{k−1} & … & x_{n}^{k−1} \\ x_{1}^{k−2} & … & x_{n}^{k−2} \\ & … \\ 1 & … & 1 \end{matrix} | .

Set y_{1} = x_{1},

y_{2} = V (x_{1}, x_{2})x_{2}V (x_{1}, x_{2})^{−1} = (x_{2} − x_{1})x_{2}(x_{2} − x_{1})^{−1},
…
y_{n} = V (x_{1}, …, x_{n})x_{n}V (x_{1}, …, x_{n})^{−1}

In the commutative case x_{i} = y_{i}, i = 1, … n. In the noncommutative case they are obviously different.

Remark. Consider the free skew-field R generated by x_{1}, …, x_{n}. Define on R differential operators ∂_{i} by formulas ∂_{i}x_{j} = δ_{ij} and satisfying Leibniz rule ∂_{i}(fg) = ∂_{i}(f)g + f∂_{i}(g), for i = 1, … n.

It easy to see that ∂_{i}y_{j} ≠ δ_{ij}. Consider, however, ∂ = ∂_{1} + ⋯ + ∂_{n}. Then

∂(V (x_{1}, …, x_{k})) = 0, k = 2, … n

and

∂(y_{i}) = ∂(x_{i}) = 1, i = 1, …, n.

QUASIDETERMINANTS 45

Elementary symmetric functions. Defintion 6.5.1. The functions

Λ_{1}(x_{1}, …, x_{n}) = y_{1} + y_{2} ⋯ + y_{n},
Λ_{2}(x_{1}, …, x_{n}) = ∑_{i<j} y_{j}y_{i},
…
Λ_{n}(x_{1}, …, x_{n}) = y_{n} … y_{1}

are called elementary symmetric functions in x_{1}, …, x_{n}.

In the commutative case these functions are the standard elementary symmetric functions of x_{1}, …, x_{n}. By the Noncommutative Vieta Theorem (Theorem 5.7.1) Λ_{i}(x_{1}, …, x_{n}) = (−1)^{i}a_{i}, i = 1, …, n where x_{1}, …, x_{n} are roots of the equation

x^{n} + a_{1}x^{n−1} + ⋯ + a_{n−1}x + a_{n} = 0.

This implies Proposition 6.5.2. The functions Λ_{i}(x_{1}, …, x_{n}) are symmetric in x_{1}, …, x_{n}.

Denote by \overline{Sym}_{n} the subalgebra of of algebra of rational functions in x_{1}, …, x_{n} generated by Λ_{k}(x_{1}, …, x_{n}), k = 1, …, n. Construct a surjective homomorphism

(6.5.1)  ϕ : Sym → \overline{Sym}_{n}

by setting ϕ(Λ_{k}) = Λ_{k}(x_{1}, …, x_{n}). Theorem 6.5.3. The kernel of ϕ is generated by Λ_{k} for k > n. Remark. The order of y_{1}, …, y_{n} is essential in the definition of Λ_{i}(x_{1}, …, x_{n}), i = 1, …, n. For example, Λ_{2}(x_{1}, x_{2}) = y_{2}y_{1} is symmetric in x_{1}, x_{2} but the product y_{1}y_{2} is not symmetric. To see this set Δ = x_{2} − x_{1}. The product y_{1}y_{2} is symmetric in x_{1}, x_{2} if and only if x_{1}Δ^{2} = Δ^{2}x_{1}. Complete symmetric functions. Definition 6.5.4. The functions

S_{k}(x_{1}, …, x_{n}) = ∑_{i_{1}≤i_{2}≤···≤i_{k}} y_{i_{1}} … y_{i_{k}}, k = 1, 2, 3, …

are called complete symmetric functions in x_{1}, …, x_{n}.

In the commutative case these functions are the standard complete symmetric functions in x_{1}, …, x_{n}.

Let t be a formal variable commuting with x_{i}, i = 1, … n. Define the generating functions

λ(t) = 1 + Λ(x_{1}, …, x_{n})t + ⋯ + Λ_{n}(x_{1}, …, x_{n})t^{n}

and

σ(y) = 1 + ∑_{i} S_{i}(x_{1}, …, x_{n})t^{i} = λ(−t)^{−1}

.

Proposition 6.5.5.

σ(t)λ(−t) = 1.

In the commutative case the S_{k}(x_{1}, …, x_{n})’s are the standard complete symmetric functions.

Proposition 6.5.6. Functions S_{k}(x_{1}, …, x_{n}) are symmetric in x_{1}, …, x_{n}.

Proof. Apply Proposition 6.5.5, Theorem 6.5.3 and Proposition 6.1.2.

Remark. The order of y′s in the definition of S_{k}’s is essential: S_{2}(x_{1}, x_{2}) = y_{1}^{2} + y_{1}y_{2} +y_{2}^{2} is symmetric in x_{1}, …, x_{n} but y_{1}^{2} +y_{2}y_{1} +y_{2}^{2} is not symmetric. Otherwise, the product y_{1}y_{2} will be also symmetric. The remark after Theorem 6.5.3 shows that this is not true.

Ribbon Schur Functions.

We define ”ribbon Schur functions with arguments” R_{I}(x_{1}, …, x_{n}) similar to Definition 6.2.1. One have to replace R_{I} by R_{I}(x_{1}, …, x_{n}) and S_{k} by S_{k}(x_{1}, …, x_{n}). Evidently, ribbon Schur functions R_{I}(x_{1}, …, x_{n}) are symmetric in x_{1}, …, x_{n} and form a linear basis in \overline{Sym}_{n}.

Proposition 6.5.7. R_{I}(x_{1}, …, x_{n}) = ϕ(R_{I}) where ϕ is defined by formula (6.5.1).

To express R_{I}(x_{1}, …, x_{n}) as a sum of monomials in y_{1}, …, y_{n} we need a vocabulary. Let w = a_{i_{1}} … a_{i_{k}} be a word in ordered letters a_{1} < ⋯ < a_{n}. An integer m is called a descent of w if 1 ≤ m ≤ k − 1 and i_{m} > i_{m+1}.

Let J = (j_{i}, …, j_{k}) be a set of positive integers.

Theorem 6.5.8.

(6.5.2)  R_{J}(x_{1}, …, x_{n}) = ∑ y_{i_{1}} … y_{i_{m}},

where the sum is running over all words w = y_{i_{1}} … y_{i_{m}} whose descent set is exactly equal to {j_{1}, j_{1} + j_{2}, …, j_{1} + j_{2} + ⋯ + j_{k−1}}.

The proof of the theorem was essentially given in [GKLLRT] in Section VII.

6.6. Main Theorem for noncommutative symmetric functions.

In the commutative case the main theorem of the theory of symmetric functions says that every symmetric polynomial of n variables is a polynomial of (elementary) symmetric functions of these variables. Its analogue for a noncommutative case is given by the following theorem. Denote Λ_{k}(x_{1}, …, x_{n} as Λ_{k}(X), k = 1, …, n.

Theorem 6.6.1. [W]. Let a polynomial P (y_{1}, …, y_{n}) over ℚ be symmetric as a function in x_{1}, …, x_{n}. Then P (y_{1}, …, y_{n}) = Q(Λ_{1}, …, Λ_{n}) where Q is a noncommutative polynomial over ℚ.

Remark. Recall that P (y_{1}, …, y_{n}) is a polynomial in y’s and not in x’s. Roughly speaking, noncommutative symmetric polynomials are not polynomials in the natural variables.

Corollary 6.6.2. Any polynomial with rational coefficient function P (y_{1}, …, y_{n}) is symmetric in x_{1}, …, x_{n} if and only iff P (y_{1}, …, y_{n}) as a function of x_{1}, …, x_{n} is a linear combination of R_{J}(x_{1}, …, x_{n}).

6.7. Quasi-Plücker coordinates of Vandermonde matrices and symmetric functions.

We study here right quasi-Plücker coordinates r_{ij}^{i_{1},...,i_{n−1}}(V_{n}) where V_{n} = (x_{j}^{i}), i ∈ ℤ, j = 1, …, n is Vandermonde matrix, x_{j}, j = 1, …, n are noncommuting variables. Matrices V_{n} have n columns and infinetly many rows. The following proposition shows an importance of such coordinates.

Proposition 6.7.1.

r_{n+k−1,n−1}^{0,1,...,n−2}(V_{n}) = S_{k}(x_{1}, …, x_{n}), k = 0, 1, 2, …,
r_{n,n−k}^{0,1,...,n−k−1,n−k+1,...,n−1}(V_{n}) = (−1)^{k−1}Λ_{k}(x_{1}, …, x_{n}), k = 0, 1, …, n.

Examples.

S_{k}(x_{1}, x_{2}) = | \begin{matrix}x_{1}^{k+1} & x_{2}^{k+1} \\ 1 & 1 \end{matrix} | | \begin{matrix}x_{1} & x_{2} \\ 1 & 1 \end{matrix} |^{−1} = (x_{2}^{k+1} − x_{1}^{k+1})(x_{2} − x_{1})^{−1} =

= y_{2}^{k} + y_{2}^{k−1}y_{1} + ⋯ + y_{2}y_{1}^{k−1} + y_{1}^{k},

where y_{1} = x_{1}, y_{2} = (x_{2} − x_{1})x_{2}(x_{2} − x_{1})^{−1}.

Λ_{1}(x_{1}, x_{2}) = | \begin{matrix}x_{1}^{2} & x_{2}^{2} \\ 1 & 1 \end{matrix} | | \begin{matrix}x_{1} & x_{2} \\ 1 & 1 \end{matrix} |^{−1} = (x_{2}^{2} − x_{1}^{2})(x_{2} − x_{1})^{−1} = y_{1} + y_{2},

Λ_{2}(x_{1}, x_{2}) = − | \begin{matrix}x_{1}^{2} & x_{2}^{2} \\ x_{1} & x_{2} \end{matrix} | | \begin{matrix}x_{1} & x_{2} \\ 1 & 1 \end{matrix} |^{−1} = (x_{2}^{2} − x_{1}x_{2})(1 − x_{1}^{−1}x_{2})^{−1} = y_{2}y_{1}.

Remark. Formulas for S_{k} in Proposition 1 are valid for all k ∈ ℤ.

An important ”periodic” property of quasi-Plücker coordinates of Vandermonde matrices is given by the following proposition.

Proposition 6.7.2.

r_{ij}^{i_{1},...,i_{n−1}}(V_{n}) = r_{i+k,j+k}^{i_{1}+k,...,i_{n−1}+k}(V_{n})

for any k ∈ ℤ.

Proposition 6.7.1 can be generalized as follows. Recall that in section 6.4 we defined generalized quasi-Schur functions. Let I = {i_{1}, i_{2}, …, i_{m}}.

Proposition 6.7.3. Let I = {i_{1}, i_{2}, …, i_{m}} and i_{1} ≥ i_{2} ≥ ⋯ ≥ i_{m}. Set J = {0, 1, …, n − m − 1, n − m + i_{1}, …, n − 2 + i_{m−1}}. Then

r_{n−1+i_{1},n−m}^{J}(V_{n}) = Ŝ_{I}.

Examples. In particular,

r_{n+k−1,n−1}^{01...n−2} = S_{k}

and

r_{n−1,n−k}^{01...n−k−1,n−k+1,...,n−2} = Λ_{k}.

VII. Universal quadratic algebras associated with pseudo-roots of noncommutative polynomials and noncommutative differential polynomials

7.1. Pseudo-roots of noncommutative polynomials.

During the last years the authors introduced and studied universal algebras associated with pseudo-roots of noncommutative polynomials. The results appeared in our papers [GRW, GGR, GGRWS].

Let C be an algebra with a unit and P (t) ∈ C[t] be a polynomial (where t is a formal variable commuting with elements of C). We say that an element c ∈ C is a pseudo-root of P (t) if there exist polynomials L_{c}(t), R_{c}(t) ∈ C[t] such that P (t) = L_{c}(t)(t − c)R_{c}(t). If P (t) = a_{0}t^{n} + a_{1}t^{n−1} + ⋯ + a_{n−1}t + a_{n} and c is a pseudo-root of P (t) with R_{c}(t) = 1 then

a_{0}c^{n} + a_{1}c^{n−1} + ⋯ + a_{n−1}c + a_{n} = 0,

i.e. c is a root of the polynomial P (x) = a_{0}x^{n} + a_{1}x^{n−1} + ⋯ + a_{n−1}x + a_{n} (where x is a non-commuting indeterminant). Our theory shows that an investigation of noncommutative polynomials is impossible without studying their pseudo-roots.

Let x_{1}, …, x_{n} be roots of a generic monic polynomial P (x) = x^{n} +a_{1}x^{n−1} +⋯+ a_{n} over an algebra C. There are two important classical problems: a) to express the coefficients a_{1}, …, a_{n} via the roots, and b) to determine all factorizations of P (x), or P (t).

When C is a division algebra, the first problem was solved in [GR3,GR4] using the theory of quasideterminants and presented here in Section V. Let V (x_{i_{1}}, …, x_{i_{k}}) be the Vandermonde quasideterminant defined by the sequence x_{i_{1}}, …, x_{i_{k}} ([GR3, GR5].) For any ordering {i_{1}, …, i_{n}} of {1, …, n}, elements x_{∅,i_{1}} = x_{i_{1}},

x_{{i_{1},i_{2},...,i_{k−1}},i_{k}} = V (x_{i_{1}}, …, x_{i_{k}})x_{i_{k}}V (x_{i_{1}}, …, x_{i_{k}})^{−1}, k = 2, …, n,

in C were constructed such that for every m = 1, …, n,

(7.1.1)  (−1)^{m}a_{m} = ∑_{j_{1}>j_{2}>...j_{m}} y_{j_{1}}y_{j_{2}} … y_{j_{m}},

where y_{1} = x_{i_{1}}, y_{k} = x_{{i_{1},...,i_{k−1}},i_{k}}, k = 2, …, n.

It is surprising that the left-hand side in formula (7.1.1) does not depend on the ordering of {1, …, n} but the right-hand side a priori depends on the ordering. The independence of the right-hand side in (7.1.1) on the ordering of {1, …, n} was a key point in the theory of noncommutative symmetric functions developed in [GR3, GR4] and presented here in Section VI.

The element x_{{i_{1},...,i_{k−1}},i_{k}} has an interesting structure. It is symmetric in x_{i_{1}}, …, x_{i_{k−1}}. It is a rational function in x_{i_{1}}, …, x_{i_{k}} containing k − 1 inversions in the generic case. In other words, x_{{i_{1},...,i_{k−1}},i_{k}} is a rational expression of height k − 1.

Set A_{k} = {i_{1}, …, i_{k−1}} for k = 2, …, n. Recall that formulas (7.1.1) are equivalent to the decomposition

(7.1.2)  P (t) = (t − x_{A_{n},i_{n}})(t − x_{A_{n−1},i_{n−1}}) … (t − x_{i_{1}}).

This formula shows that elements x_{A,i} are pseudo-roots of P (t), and in the generic case the polynomial P (t) has at least n! different pseudo-roots. We study all such pseudo-roots together by constructing a universal algebra of pseudo-roots Q_{n}.

7.2. Universal algebra of pseudo-roots.

It is easy to see that elements x_{A,i}, i ∉ A satisfy simple relations:

(7.2.1a)  x_{A∪{i},j} + x_{A,i} = x_{A∪{j},i} + x_{A,j},

(7.2.1b)  x_{A∪{i},j} · x_{A,i} = x_{A∪{j},i} · x_{A,j},

for all A ⊆ {1, …, n}, i, j ∉ A.

In order to avoid inversions and to make our construction independent of the algebra C, we define the universal algebra of pseudo-roots Q_{n} over a field F to have generators z_{A,i}, A ⊆ {1, …, n}, i ∉ A, and relations corresponding to (7.2.1) (with x replaced by z).

This is a natural algebra containing a free associative algebra generated by the z_{i} = z_{∅,i}, i = 1, …, n.

We studied the internal structure of algebra Q_{n}. It has a natural derivation and a natural anti-involution. For each pair A, B ⊆ {1, …, n}, with A ∩ B = ∅ we define an element z_{A,B} ∈ Q_{n} such that z_{∅,∅} = 0 and z_{A∪{i},B} − z_{A,B∪{i}} = z_{A,B} for i ∉ A, B.

When B contains more than one element, the element z_{A,B} is “invisible” in the commutative case, i.e., under the natural commutative specialization of Q_{n}, its image is zero.

In [GRW] it was proved that

z_{A,i} − z_{i} = ∑_{∅≠C⊆A} z_{∅,C∪{i}}.

Under the natural commutative specialization

(7.2.2)  π_{n} : Q_{n} → F [t_{1}, …, t_{n}], π(z_{A,i}) = t_{i}, i = 1, …, n,

the image of z_{A,i} −z_{i} equals zero. This shows that the elements from the right hand side in (7.2.2) measure the “noncommutativity” of z_{A,i}. Moreover, the “degree of noncommutativity” carried by z_{∅,B} depends on the size of B.

Remark. One can try to construct a “noncommutative logic” using these elements; roughly speaking, the element z_{A,B} is marked “yes” for indices from A, “no” for indices from B and “do not know” for indices from {1, …, n} \ (A ∪ B).

7.3. Basises in the universal algebra of pseudo-roots.

One can see that elements z_{A,∅}, and similarly, the elements z_{∅,A}, for all A ⊆ {1, …, n}, A ≠ ∅, constitute a basis for the subspace of Q_{n} spanned by all generators. These elements satisfy simple quadratic relations.

The algebras Q_{n} seem to be very interesting. They may be viewed as a Galois type extension of the algebra of symmetric functions in n variables. Our study of Q_{n} is based on our construction of a basis for Q_{n} [GRW]. It is a hard combinatorial problem.

For A ⊆ {1, ..., n} let min(A) denote the smallest element of A. Then set A^{(0)} = A, A^{(1)} = A\{min(A)}, A^{(i+1)} = (A^{(i)})^{(1)}. Set r(A) = z_{∅,A}.

Theorem 7.3.1. The set of all monomials

r_{A_{1}^{(0)}}r_{A_{1}^{(1)}} … r_{A_{1}^{(j_{1})}}r_{A_{2}^{(0)}}r_{A_{2}^{(1)}} … r_{A_{2}^{(j_{2})}} … r_{A_{l}^{(0)}}r_{A_{l}^{(1)}} … r_{A_{l}^{(j_{l})}}

where A_{1}, …., A_{l} ⊆ {1, …, n} and, for 1 ≤ i ≤ l − 1, either A_{i+1} ⊈ A_{i} or |A_{i+1}| ≠ |A_{i}| − j_{i} − 1, is a basis in Q_{n}.

Remark. There are natural problems related to this construction. It would be interesting to study in details combinatorial properties of the basis and to give constructions of similar bases in Q_{n} using different techniques including noncommutative Gröbner bases and Bergman’s Diamond lemma.

7.4. Algebra of noncommutative symmetric polynomials as a subalgebra in Q_{n}.

For each ordering I = (i_{1}, …, i_{n}) of {1, …, n}, there is a natural free subalgebra, denoted Q_{n,I}, of Q_{n} generated by {z_{{i_{1},...,i_{k}},∅} | 1 ≤ k ≤ n}. Using the basis theorem in [GRW], we described arbitrary intersections of the Q_{n,I}. In particular, we proved the following theorem. Let S_{n} be a subalgebra in Q_{n} generated by all coefficients a_{m} (see formula (7.1.1.)). One can identify S_{n} with algebra \overline{Sym}_{n} of noncommutative symmetric functions in x_{1}, …, x_{n}.

Theorem 7.4.1. The intersection of all subalgebras Q_{n,I} coincides with algebra S_{n}.

This is a pure noncommutative phenomenon: under the commutative specialization π_{n} all algebras Q_{n,I} map to the algebra of all polynomials and algebra S_{n} maps to the algebra of symmetric funtions.

7.5. The dual algebra Q_{n}^{!}.

Since Q_{n} is a quadratic algebra, it is natural to ask if Q_{n} is a Koszul algebra. This was proved by S. Serconek and R. Wilson [SW].

The algebra Q_{n} has a natural graded structure Q_{n} = ∑_{i≥0} Q_{n,i} where Q_{n,i} is the span of all products of i generators. As usual, we denote the Hilbert series of Q_{n} by H(Q_{n}, t) = ∑_{i≥0} dim(Q_{n,i})t^{i}.

In [GGRWS] we computed the Hilbert series of Q_{n} and of its dual quadratic algebra Q_{n}^{!}. In particular, we proved:

Theorem 7.5.1.

H(Q_{n}, t) = \frac{1 − t}{1 − t(2 − t)^{n}} ,

H(Q_{n}^{!}, t) = \frac{1 + t(2 + t)^{n}}{1 + t} .

It follows that the dual algebra Q_{n}^{!} is finite dimensional. It also has a rich and interesting structure and I am going to investigate it. Clearly

H(Q_{n}, t) · H(Q_{n}^{!}, −t) = 1.

This also follows from the Koszulity of Q_{n}.

7.6. Quotient algebras of Q_{n}.

There are at least two reasons to study quotient algebras of Q_{n}. The noncommutative nature of Q_{n} can be studied by looking at quotients of Q_{n} by ideals generated by some z_{∅,A}. These quotients are “more commutative” than Q_{n}. For example, the quotient of Q_{n} by the ideal generated by all z_{∅,A} with |A| ≥ 2 is isomorphic to the algebra of commutative polynomials in n variables.

To consider more refined cases, we need to turn to a “noncommutative combinatorial topology.” In our approach the algebra Q_{n} corresponds to an n-simplex Δ_{n}, and we consider quotients of Q_{n} by ideals generated by some z_{∅,A} corresponding to subcomplexes of Δ_{n}. In [GGR] we described generators and relations for those quotients. Special attention was given to the quotients of Q_{n} corresponding to 1-dimensional subcomplexes of Δ_{n} (they are close to algebras of commutative polynomials). However, we need to study other quotient algebras of this type and to be able to “glue” together such quotient algebras. This will lead to a construction of a “noncommutative combinatorial topology.”

It would also be interesting to study quotients of algebras Q_{n} associated with special types of polynomials (such as x^{n} = 0 or polynomials with multiple roots).

Example. Let F be a field. We consider here F -algebra Q_{2} and its quotients. This algebra is generated by z_{1}, z_{2}, z_{1,2}, z_{2,1}. It corresponds to a polynomial P (t) = t^{2} − pt + q with p = z_{1} + z_{1,2} = z_{2} + z_{2,1}, q = z_{1,2}z_{1} = z_{2,1}z_{2}.

There is an “invisible element” z_{∅,12} = z_{1,2} −z_{2} = z_{2,1} −z_{1}. This element satisfies the relation z_{∅,12}(z_{1} − z_{2}) = z_{1}z_{2} − z_{2}z_{1}.

There are three quotient algebras corresponding to special cases of P (t):

i) The algebra corresponding to the polynomial t^{2} equals to Q_{2}/(p, q). This algebra is isomorphic to F < z_{1}, z_{2} > /(z_{1}^{2}, z_{2}^{2}).

ii) The algebra corresponding to a polynomial with multiple roots equals Q_{2}/(z_{1}− z_{2}). It is isomorphic to a free algebra with generators z_{1} and z_{1,2}.

iii) The algebra Q_{2}/(z_{∅,12}) is isomorphic to an algebra of commutative polynomials with two variables.

We are going to investigate such algebras for noncommutative polynomials of any degree.

7.7. Quadratic algebras associated with differential noncommutative polynomials.

In [GRW] similarly to algebras Q_{n}, we constructed universal algebras of pseudoroots of noncommutative differential polynomials.

Let F be a field and (C, D) a differential division algebra with a unit and a derivation D. Sometimes we use the notation Da = a′. Consider an operator P (D) : C → C, P = a_{0}D^{n} + a_{1}D^{n−1} + ... + a_{n}, a_{i} ∈ C for i = 0, 1, …, n. The action of P on C is given by the formula P (ϕ) = a_{0}D^{n}ϕ + a_{1}D^{n−1}ϕ + ... + a_{n}ϕ.

We say that an element c ∈ A is a pseudo-root of P (D) if there exist polynomials L_{c}(D) and R_{c}(D) with coefficients from A such that P (D) = L_{c}(D)(D − c)R_{c}(D). We say that c is a root of P (D) if R_{c}(D) = 1. We do not assume here that differential polynomials have any solutions.

Suppose that a_{0} = 1 and the differential polynomial P (D) has n different roots f_{1}, …, f_{n}. Following [EGR], for any ordering (i_{1}, …, i_{n}) of {1, …, n}, in a generic case we constructed in [GRW] pseudo-roots f_{i_{1},i_{2}}, ..., f_{i_{1},...i_{n−1},i_{n}} such that

P (D) = (D − f_{i_{1},...,i_{n−1},i_{n}}) … (D − f_{i_{1},i_{2}})(D − f_{i_{1}}).

The elements f_{i_{1},...,i_{k−1},i_{k}} for k = 2, …, n do not depend on the ordering in (i_{1}, …, i_{k−1}).

Set f_{∅,i} = f_{i}. It was proved in [GRW] that for any A ⊂ {1, …, n} such that |A| < n − 1, and for any i, j ∉ A we have

(7.7.1a)  f_{A∪i,j} + f_{A,i} = f_{A∪j,i} + f_{A,j},

(7.7.1b)  f_{A∪i,j}f_{A,i} − f′_{A,i} = f_{A∪j,i}f_{A,j} − f′_{A,j}.

Based on these formulas one can define universal algebras DQ_{n} of pseudo-roots of noncommutative differential polynomials. They are defined by elements f_{A,i} for i ∉ A satisfying relations (7.7.1).

We are planning to develop a theory of algebras DQ_{n} and to use it in noncommutative integrable systems. The theory must include a theory of symmetric functions related to algebras DQ_{n}, constructions of bases, studies of quotient algebras of

VIII. Noncommutative traces, determinants and eigenvalues

In this section we discuss noncommutative traces, determinants and eigenvalues. Classical (commutative) determinants play a key role in representation theory. Frobenius developed his theory of group characters by studying factorizations of group determinants (see [L]). Therefore, one cannot start a noncommutative representation theory without looking at possible definition of noncommutative determinants and traces. The definition of a noncommutative determinant given in this Section is different from the definition given in Section III.

8.1. Determinants and cyclic vectors.

Let A be an m × m-matrix over an algebra R with a unit. A vector v ∈ R^{m} is an A-cyclic vector if v, Av, …, A^{m−1}v is a basis in R^{m} considered as a right R-module. In this case there exist Λ_{i}(v, A) ∈ R, i = 1, …, m, such that

(−1)^{m}vΛ_{m}(v, A) + (−1)^{m−1}(Av)Λ_{m−1}(v, A) + ⋯ − (A^{m−1}v)Λ_{1}(v, A) + A^{m}v = 0.

We call Λ_{m}(v, A) the determinant of (v, A) and Λ_{1}(v, A) the trace of (v, A). We may express Λ_{i}(v, A) ∈ R, i = 1, …, m, as quasi-Plücker coordinates of the m × (m + 1) matrix with columns v, Av, …, A^{n}v (following [GR4]).

When R is commutative Λ_{m}(v, A) is the determinant of A and Λ_{1}(v, A) is the trace of A.

This definition of noncommutative determinants and traces was essentially used in [GKLLRT] for a unit vector v. In this case Λ_{i}(v, A) are quasi-Plücker coordinates of the corresponding Krylov matrix.

Example. Let A = (a_{ij}) be an m × m-matrix and v = (1, 0, …, 0)^{T}. Denote by a_{ij}^{(k)} the corresponding entries of of A^{k}. Then

Λ_{m}(v, A) = | \begin{matrix}a_{11}^{(m)} & a_{12}^{(m)} & … & a_{1m}^{(m)} \\ a_{11}^{(m−1)} & a_{12}^{(m−1)} & … & a_{1m}^{(m−1)} \\ & … & … \\ a_{11} & a_{12} & … & a_{1m} \end{matrix} | .

For m = 2 the “noncommutative trace” Λ_{1} equals a_{11} + a_{12} + a_{12}a_{22}^{−1}a_{21} and the “noncommutative determinant” Λ_{2} equals a_{12}a_{22}^{−1}a_{21}a_{11} − a_{12}a_{21}.

It was shown in [GKLLRT] that when A is a quantum matrix then Λ_{m} equals det_{q} A and when A is a Capelli matrix, Λ_{m} equals Capelli determinant.

A construction of a noncommutative determinant and a noncommutative trace via cyclic vectors in a special case was later used in [Ki].

When R is noncommutative, the expressions Λ_{i}(v, A) ∈ R, i = 1, …, m, depend on vector v. However, they provide some information about A. For example, the following statement is true.

Proposition 8.1.1. If the determinant Λ_{m}(v, A) equals zero, then matrix A is not invertible.

We will study properties of Λ_{i}(v, A) for i = 1, …, m in a separate paper.

8.2. Noncommutative determinants and noncommutative eigenvalues. One can also express Λ_{i}(v, A) ∈ R via left eigenvalues of A.

Definition 8.2.1. A row-vector u = (u_{1}, …, u_{m}) is a left eigenvector of A if there exists λ ∈ R such that uA = λu.

We call λ a left eigenvalue of A corresponding to vector u. Note, that the eigenvalue λ is defined up to a conjugation: if α ∈ R, then (αu)A = αλα^{−1}(αu).

Proposition 8.2.2. Suppose that u = (u_{1}, …, u_{m}) is a left eigenvector of A with eigenvalue λ, v = (v_{1}, …, v_{m})^{T} is a cyclic vector of A, and the scalar product u · v = u_{1}v_{1} + … u_{m}v_{m} = 1.

The eigenvalue λ satisfies the equation

(8.2.1) (−1)^{m}Λ_{m}(v, A) + (−1)^{m−1}λΛ_{m−1}(v, A) + ⋯ − λ^{m−1}Λ_{1}(v, A) + λ^{m} = 0.

Equation (8.2.1) and the corresponding Vieta theorem (see Section V) show that if the matrix A has m distinct left eigenvectors u^{i} such that u^{i}·v = 1 for i = 1, …, m and m distinct corresponding eigenvalues λ_{1}, …, λ_{m}, then λ_{1}, …, λ_{m} completely determine Λ_{i}(v, A) ∈ R, i = 1, …, m, as “noncommutative elementary symmetric functions” by formulas similar to formulas from Definition (6.5.1).

We are planning to investigate relations between noncommutative traces, determinants and eigenvalues in greater detail and to construct a corresponding “universal algebra” similar to Section VII.

8.3. Multiplicativity of determinants.

In the commutative case the multiplicativity of determinants and the additivity of traces are related to computations of determinants and traces with diagonal block-matrices. In the noncommutative case we suggest to consider the following construction.

Let A be an m × m-matrix and D an n × n-matrix over an algebra R with unit. Let v ∈ R^{m} be a cyclic vector for A and w ∈ R^{n} be a cyclic vector for D. Suppose

that vector z = (v, w)^{T} ∈ R^{m+n} is a cyclic vector for matrix C = ( \begin{matrix}A & 0 \\ 0 & D \end{matrix} ).

Then one can express functions Λ_{k}(z, C), k = 1, …, m+n via functions Λ_{i}(v, A), i = 1, …, m and Λ_{j}(w, D), j = 1, …, n. In the commutative case these formulas imply the multiplicativity of determinants and the additivity of traces.

We are going to study relations between symmetric functions associated with matrices A and D and with the block-matrix C.

IX. Some applications

In this Section we present some results from [GR1, GR2, GR4].

9.1. Continued fractions and almost triangular matrices.

Consider an infinite matrix A over a skew-field:

A = ( \begin{matrix}a_{11} & a_{12} & a_{13} & … & a_{1n} … \\ −1 & a_{22} & a_{23} & … & a_{2n} … \\ 0 & −1 & a_{33} & … & a_{3n} … \\ 0 & 0 & −1 & … & … \end{matrix} )

It was pointed out in [GR1], [GR2] that the quasideterminant |A|_{11} can be written as a generalized continued fraction

|A|_{11} = a_{11} + ∑_{j_{1}≠1} a_{1j_{1}} \frac{1}{a_{2j_{1}} + ∑_{j_{2}≠1,j_{1}} a_{2j_{2}} \frac{1}{a_{3j_{2}} +...}} .

Let

A_{n} = ( \begin{matrix}a_{11} & a_{12} & … & a_{1n} \\ −1 & a_{22} & … & a_{2n} \\ 0 & −1 & … & a_{3n} \\ & & … \\ … & 0 & −1 & a_{nn} \end{matrix} ) .

The following proposition was formulated in [GR1], [GR2].

Proposition 9.1.1. |A_{n}|_{11} = P_{n}Q_{n}^{−1}, where

(9.1.1)  P_{n} = ∑_{1≤j_{1}<···<j_{k}<n} a_{1j_{1}}a_{j_{1}+1j_{2}}a_{j_{2}+1j_{3}} … a_{j_{k}+1n}

and

(9.1.2)  Q_{n} = ∑_{2≤j_{1}<···<j_{k}<n} a_{2j_{1}}a_{j_{1}+1j_{2}}a_{j_{2}+1j_{3}} … a_{j_{k}+1n}.

Proof. From the homological relations one has

|A_{n}|_{11}|A_{n}^{1n}|_{21}^{−1} = −|A_{n}|_{1n}|A_{n}^{11}|_{2n}^{−1}.

We will apply formula (1.2.2) to compute |A_{n}|_{1n}, |A_{n}^{11}|_{2n}, and |A_{n}^{1n}|_{21}. It is easy to see that |A_{n}^{1n}|_{21} = −1. To compute the two other quasideterminants, we have to invert triangular matrices. Setting P_{n} = |A_{n}|_{1n} and Q_{n} = |A_{n}^{11}|_{2n} we arrive at formulas (9.1.1), (9.1.2). □

Remark. In the commutative case Proposition 9.1.1 is well-known. In this case P_{n} = |A_{n}|_{1n} = (−1)^{n} det A_{n} and Q_{n} = (−1)^{n−1} det A_{n}^{11}.

Formulas (9.1.1), (9.1.2) imply the following result of [GR1], [GR2]:

Corollary 9.1.2. The polynomials P_{k} for k ≥ 0 and Q_{k} for k ≥ 1 are related via the formulas:

(9.1.3)  P_{k} = ∑_{s=0}^{k−1} P_{s}a_{s+1,k},  P_{0} = 1,

and

(9.1.4)  Q_{k} = ∑_{s=1}^{k−1} Q_{s}a_{s+1,k},  Q_{1} = 1.

Corollary 9.1.3. Consider the matrix A_{n}. Suppose that for any i ≠ j and any p, q we have

a_{ij}a_{pq} = a_{pq}a_{ij}

and

a_{jj}a_{ii} − a_{ii}a_{jj} = a_{ij}, 1 ≤ i < j ≤ n.

Then

(9.1.5)  P_{n} = |A_{n}|_{1n} = a_{nn}a_{n−1n−1} … a_{11}.

The proof follows from formula (9.1.3).

Corollary 9.1.4. [GR1], [GR2] For a Jacobian matrix

A = ( \begin{matrix}a_{1} & 1 & 0 & … \\ −1 & a_{2} & 1 \\ 0 & −1 & a_{3} & … \end{matrix} )

we have:

|A|_{11} = a_{1} + \frac{1}{a_{2} + \frac{1}{a_{3}+...}} ,

and

P_{k} = P_{k−1}a_{k} + P_{k−2}, k ≥ 2; P_{0} = 1, P_{1} = a_{1},
Q_{k} = Q_{k+1}a_{k} + Q_{k−2}, k ≥ 3; Q_{1} = 1, Q_{2} = a_{2}.

In this case P_{k} is a polynomial in a_{1}, …, a_{k} and Q_{k} is a polynomial in a_{2}, …, a_{k}.

9.2 Continued fractions and formal series. In the notation of the previous subsection the infinite continued fraction |A|_{11} may be written as a ratio of formal series in the letters a_{ij} and a_{ii}^{−1}. Namely, set

P_{∞} = ∑_{\begin{matrix}1≤j_{1}<j_{2}···<j_{k}<r−1 \\ r=1,2,3,... \end{matrix}} a_{1j_{1}}a_{j_{1}+1j_{2}} … a_{j_{k}+1r}a_{rr}^{−1} · . . . · a_{11}^{−1} =

= 1 + a_{12}a_{22}^{−1}a_{11}^{−1} + a_{13}a_{33}^{−1}a_{22}^{−1}a_{11}^{−1} + a_{11}a_{23}a_{33}^{−1}a_{22}^{−1}a_{11}^{−1} + …,

and

Q_{∞} = a_{11}^{−1} + ∑_{\begin{matrix}2≤j_{1}<j_{2}···<j_{k}<r−1 \\ r=2,3... \end{matrix}} a_{2j_{1}}a_{j_{1}+1j_{2}} … a_{j_{k}+1r}a_{rr}^{−1} · . . . · a_{11}^{−1} =

= a_{11}^{−1} + a_{23}a_{33}^{−1}a_{22}^{−1}a_{11}^{−1} + a_{24}a_{44}^{−1}a_{33}^{−1}a_{22}^{−1}a_{11}^{−1} + ….

As no monomial appears more than once in these sums, these are rigorously defined formal series.

The following theorem was proved in [PPR]. Another proof was given in [GR4].

Theorem 9.2.1.

|A|_{11} = P_{∞} · Q_{∞}^{−1}.

Proof. Set b_{ij} = a_{ij}a_{jj}^{−1} and consider matrix B = (b_{ij}), i, j = 1, 2, 3, . . .. According to a property of quasideterminants |A|_{11} = |B|_{11}a_{11}.

Applying the noncommutative Sylvester Theorem to B with matrix (b_{ij}), i, j ≥ 3, as the pivot, one has

|B|_{11} = 1 + |B^{21}|_{12}|B^{11}|_{22}^{−1}a_{11}^{−1}.

It follows

(9.2.1)  |A|_{11} = (a_{11}|B^{11}|_{22}a_{11}^{−1} + |B^{21}|_{12}a_{11}^{−1})(|B^{11}|_{22}a_{11}^{−1})^{−1}.

From [GKLLRT, Proposition 2.4] it follows that the first factor in (9.2.1) equals P_{∞}, and the second one equals Q_{∞}^{−1}. □

9.3. Noncommutative Rogers-Ramanujan continued fraction. The following application of Theorem 9.2.1 to Rogers-Ramanujan continued fraction was given in [PPR]. Consider a continued fraction with two formal variables x and y:

A(x, y) = \frac{1}{1 + x\frac{1}{1+x \frac{1}{1+...} y} y} .

It is easy to see that

A(x, y) = | \begin{matrix}1 & x & & · \\ −y & 1 & x & & · & & 0 \\ & −y & 1 & x & & · \\ & & & 1 & & & · \\ & 0 & & ... & ... & & · \end{matrix} |_{11}^{−1} =

= | \begin{matrix}1 & x & 0 \\ −1 & y^{−1} & xy^{−1} \\ 0 & −1 & y^{−1} & xy^{−1} \\ & & −1 & y^{−1} & ... \end{matrix} |_{11}

Theorem 9.2.1 implies:

Corollary 9.3.1. A(x, y) = P · Q^{−1}, where Q = yP y^{−1} and

P = 1 + ∑_{\begin{matrix}k≥1 \\ n_{1},...,n_{k}≥1 \end{matrix}} y^{−n_{1}}xy^{−n_{2}}x … y^{−n_{k}}xy^{k+n_{1}+n_{2}+···+n_{k}}.

Following [PPR] suppose that xy = qyx, where q commutes with x and y. Set z = yx.

Then Corollary 9.3.1 implies Rogers-Ramanujan continued fraction identity

A(x, y) = \frac{1}{1 + _{\frac{qz}{1+ \frac{q^{2}z}{1+...}}}} =

= \frac{1 + ∑_{k≥1} \frac{q^{k(k+1})}{(1−q)...(1−q^{k})}z^{k}}{1 + ∑_{k≥1} \frac{q^{k^{2}}}{(1−q)...(1−q^{k})}z^{k}} .

9.4. Determinants of almost triangular matrices.

Consider now a general almost triangualr matrix

B = ( \begin{matrix}a_{11} & a_{12} & … & a_{1n} \\ b_{21} & a_{22} & … & a_{2n} \\ 0 & b_{32} & … & a_{3n} \\ & & … \\ … & 0 & b_{nn−1} & a_{nn} \end{matrix} ) .

In this subsection we suppose that all b_{i+1i}’s are invertible.

Denote by B_{k}, k = 1, 2, …, n − 1, submatrices of B obtained by deleting the first k rows and columns from matrix B.

There is one more definition of a determinant-like expression for an almost triangular matrix given by the formula

D(1, 2, …, n) = |B|_{11}b_{21}^{−1}|B_{1}|_{22}b_{32}^{−1}|B_{2}|_{33}b_{43}^{−1} … b_{nn−1}^{−1}a_{nn}.

The homological relations imply the following proposition [GR1], [GR2].

Proposition 9.4.1.

(−1)^{n+1}D(1, 2, … n) = |B|_{1n} =

= ∑_{1≤j_{1}<···<j_{k}<n}(−1)^{k+1}a_{1j_{1}}b_{j_{1}+1j_{1}}^{−1}a_{j_{1}+1j_{2}}b_{j_{2}+1j_{2}}^{−1}a_{j_{2}+1j_{3}} … b_{j_{k}+1j_{k}}^{−1}a_{j_{k}+1n}.

Remark. When b_{i+1i} = 1, i = 1, …, n − 1 and all a_{ij}’s commute with each other, then the expression D(1, 2, …, n) is the determinant of B.

It is interesting to determine other quasideterminants of an almost triangular matrix.

Proposition 9.4.2. [GR1], [GR2] Set D(∅) = 1. For i ≤ j we have

|B|_{ij} = D(1, …, i − 1)^{−1}D(1, …, n)D(j + 1, …, n)^{−1}.

9.5. Quasideterminants and characterisric functions of graphs.

Let A = (a_{ij}), i, j = 1, …, n, and let a_{ij}’s be formal noncommuting variables. Fix p, q ∈ {1, …, n} and a set J ⊂ {1, …, p̂, …, n} × {1, …, q̂, …, n} such that |J| = n−1 and both the projections of J onto {1, …, p̂, …, n} and {1, …, q̂, …, n} are surjective. Introduce new variables b_{kl}, k, l = 1, …, n, such that b_{kl} = a_{kl} for (l, k) ∉ J, b_{kl} = a_{lk}^{−1} for (l, k) ∈ J. Let F_{J} be a ring of formal series in variables b_{kl}.

Proposition 9.5.1. The quasideterminant |A|_{ij} is defined in the ring F_{J} and is given by the formula

(9.5.1)  |A|_{ij} = b_{ij} − ∑(−1)^{s}b_{ii_{1}}b_{i_{1}i_{2}} … b_{i_{s}j}.

The sum is taken over all sequences i_{1}, …, i_{s} such that i_{k} ≠ i, j, k = 1, …, s.

Proposition 9.5.2. The inverse to |A|_{ij} is also defined in the ring F_{J} and is given by the following formula

(9.5.2)  |A|_{ij} = b_{i}j − ∑(−1)^{s}b_{ii_{1}}b_{i_{1}i_{2}} … b_{i_{s}j}.

The sum is taken over all sequences i_{1}, …, i_{s}.

All relations between quasideterminants including Sylvester identity can be deduced from formulas (9.5.1) and (9.5.2).

Formulas (9.5.1) and (9.5.2) can be interpreted via graph theory. Let Γ_{n} be a complete oriented graph with vertices 1, …, n and edges e_{kl}, where k, l = 1, …, n. Introduce a bijective correspondence beween edges of the graph and elements b_{kl}: e_{kl} ↦ b_{kl}. Then there exist a bijective correspondence between monomials b_{ii_{1}}b_{i_{1}i_{2}} … b_{i_{s}j} and paths from vertex i to vertex j.

References

[A] E. Artin, Geometric Algebra, Reidel, 1987.

[Al] S. Alesker, Non-commutative Linear Algebra and Plurisubharmonic Functions of Quaternionic Variables, arXiv: math.CV/010429 (2002).

[As] H. Aslaksen, Quaternionic Determinants, Math. Intelligencer 15 (1996), no. 3, 57-65.

[B] F.A. Berezin, Analysis with Anticommuting variables, Kluwer, 1987.

[BW] J. Birman and R. Williams, Knotted periodic orbits I: Lorenz knots, Topology 22 (1983), 47-62.

[C] A. Cayley, On Certain Results Relating to Quaternions, Phil. Mag. 26 (1845), 141-145.

[CF] P. Cartier and D. Foata, Problemes Combinatoires de Commutation et Rearrangements, Lecture Notes in Math. 85 (1969).

[CS] A. Connes and A. Schwarz, Matrix Vieta Theorem Revisited, Lett. Math. Phys. 39 (1997), 349-353.

[Co] P.M. Cohn, Skew-Field Constructions, London Math. Soc. Lecture Note Series 27 (1977).

[D] J. Diedonne, Les Determinantes sur un Corps Non-Commutatif, Bull. Soc. Math. France 71 (1943), 27-45.

[Dr] P.K. Draxl, Skew Fields, London Math. Soc. Lecture Note Series 81 (1983).

[Dy] F.J. Dyson, Quaternion Determinants, Helv. Phys. Acta 45 (1972), 289-302.

[Dy1] F.J. Dyson, Correlations between Eigenvalues of Random Matrices, Comm. Math. Physics 19 (1970), 235-250.

[EGR] P. Etingof, I. Gelfand, and V. Retakh, Factorization of Differential Operators, Quasideterminants, and Nonabelian Toda Field Equations, Math. Res. Letters 4 (1997), no. 2-3.

[EGR1] P. Etingof, I. Gelfand, and V. Retakh, Nonabelian Integrable Systems, Quasideterminants, and Marchenko Lemma, Math. Res. Letters 5 (1998), no. 1.

[ER] P. Etingof and V. Retakh, Quantum Determinants and Quasideterminants, Asian J. of Mathematics 3 (1999), no. 2, 345-352.

[F] D. Foata, A Noncommutative Version of the Matrix Inversion Formula, Advances in Math. Soc. 31 (1979), 330-349.

[Fr] G. Frobenius, Über Lineare Substituenen und Bilinear Formen, J. Reine Angew. Math. 84 (1877), 1-63.

[FIS] S.H. Friedberg, A.J. Insel, and L.E. Spence, Linear Algebra, Third Edition.

[GGR] I. Gelfand, S. Gelfand, and V. Retakh, Noncommutative Algebras Associated to Complexes and Graphs,, Selecta Math (2001), 525-531.

[GGRSW] I. Gelfand, S. Gelfand, V. Retakh, S. Serconek, and R. Wilson, Hilbert Series of Quadratic Algebras Associated with Decompositions of Noncommutative Polynomials, math.QA/0109007.

[GKLLRT] I. Gelfand, D. Krob, A. Lascoux, B. Leclerc, V. Retakh, and J.-Y. Thibon, Noncommutative Symmetric Functions, Advances in Math 112 (1995), no. 2, 218-348.

[GR] I. Gelfand and V. Retakh, Determinants of Matrices over Noncommutative Rings, Funct. Anal. Appl. 25 (1991), no. 2, 91-102.

[GR1] I. Gelfand and V. Retakh, A Theory of Noncommutative Determinants and Characteristic Functions of Graphs, Funct. Anal. Appl. 26 (1992), no. 4, 1-20.

[GR2] I. Gelfand and V. Retakh, A Theory of Noncommutative Determinants and Characteristic Functions of Graphs. I, Publ. LACIM, UQAM, Montreal 14 (1993), 1-26.

[GR3] I. Gelfand and V. Retakh, Noncommutative Vieta Theorem and Symmetric Functions, Gelfand Mathematical Seminars 1993-95, Birkhauser (1996).

[GR4] I. Gelfand and V. Retakh, Quasideterminants, I, Selecta Math 3 (1997), no. 4.

[GRW] I. Gelfand, V. Retakh, and R. Wilson, Quadratic-linear Algebras Associated with Decompositions of Noncommutative Polynomials and Differential Polynomials, Selecta Math 7 (2001), 493-523.

[GRW1] I. Gelfand, V. Retakh, and R. Wilson, Quaternionic Quasideterminants and Determinants, math.QA/0206211 (2002).

[GV] V. Goncharenko and A. Veselov, Darboux Transformations for Multidemensional Schrödinger Operators, J. Phys. A: Math. Gen. 31 (1998), 5315-5326.

[H] A. Heyting, Die Theorie der Linearen Gleichungen in einer Zahlenspezies nit Nichtkommutativer Multiplication, Math. Ann. 98 (1927), 465-490.

[Ho] R. Howe, Perspectives in Ivariant Theory: Schur Duality, Multiplicity-Free Actions and Beyond, Isr. Math. Conf. Proc. 8 (1995).

[Ho1] R. Howe, Remarks on Classical Invariant Theory, Trans. Amer. Math. Soc 313 (1989), 539-570.

[J] C. Joly, Supplement to OEvres of Hamilton (1900).

[Ja] N. Jacobson, Lectures in Abstract Algebra. Vol. II, 1953.

[Ki] A. Kirillov, Introduction to Family Algebras, Moscow Math. J. 1 (2001), no. 1, 49-63.

[KL] D. Krob and B. Leclerc, Minor Identities for Quasi-Determinants and Quantum Determinants, Comm. Math. Phys. 169 (1995), no. 1, 1-23.

[KLT] D. Krob, B. Leclerc, and J.-Y. Thibon, Noncommutative Symmetric Functions II: Transformations of Alphabets, Int. J. Algebra Comput. 7, 181-264.

[KS] P.P. Kulish, E.K. Sklyanin, Quantum Spectral Transform Method. Recent developments, Lecture Notes in Physics 151 (1982), 61-119.

[L] T.Y. Lam, Representations of Finite Groups: A Hundred Years, Part I, Notices of the AMS 45 (1998), 361-372.

[Le] D.A. Leites, A Certain Analog of the Determinant, Russian Math. Surv. 35 (1975), no. 3, [LST] B. Leclerc, T. Scharf, and J.-Y. Thibon, Noncommutative Cyclic Characters of Symmetric

Groups, J. of Combinatorial Theory, Series A 75 (1996), 55-69.

[M] P.A. MacMacon, Combinatorial Analysis, Cambridge Univ. Press, 1915.

[Ma] Y. Manin, Multiparameter Quantum deformations of the Linear Supergroup, Comm. Math. Phys. 113 (1989), 163-175.

[Mol] A. Molev, Noncommutative Symmetric Functions and Laplace Operators for Classical Lie Algebras, Lett. Math. Phys. 35 (1995), no. 2, 135-143.

[Mol1] A. Molev, Gelfand-Tsetlin Bases for Representations of Yangians, Lett. Math. Phys. 30 (1994), no. 2, 53-60.

[MB] E.H. Moore and R.W. Barnard, General Analysis, 1935.

[Mo] E.H. Moore, On the Determinant of an Hermitian Matrix of Quaternionic Elements, Bull. Amer. Math. Soc. 28 (1922), 161-162.

[O] O. Ore, Liner Equations in Noncommutative Rings, Ann. Math. 32 (1931), 463-477.

[P] A. Polischuk, Triple Massey Products on Curves, Fay’s Trisecant Identity and Tangents to the Canonical Embedding, preprint math.AG/0109007 (2001).

[PPR] I. Pak, A. Postnikov and V. Retakh, Noncommutative Lagrange Inversion, preprint (1995).

[RS] A. Razumov, M. Saveliev, Maximally Nonabelian Toda Systems, Nuclear Physics B 494 (1997), no. 3.

[RRV] V. Retakh, C. Reutenauer, and A. Vaintrob, Noncommutative Rational Functions and Farber’s Invariants of Boundary Links, in: Differential Topology, Infinite-dimensional Lie Algebras and Applications. Amer Math Soc Transl. 194 (1999), 237-246.

[RSh] V. Retakh and V. Shander, Schwarz Derivative for Noncommutative Differential Algebras, Advances in Soviet Math. 17 (1993).

[Re] C. Reutenauer, Inversion Height in Free Fields, Selecta Mathematica 2 (1996), no. 1.

[Ri] A.R. Richardson, Hypercomplex Determinants, Messenger of Math. 55 (1926), 145-152.

[Ri1] A.R. Richardson, Simultaneous Linear Equations over a Division Algebra, Proc. London Math. Soc. 28 (1928), 395-420.

[SW] S. Serconek and R.L. Wilson, in preparation.

[Sch] M. Schork, The bc-System of Higher Rank Revisited, J. Phys. A: Math. Gen. 35 (2001), 2627-2637.

[Schur] I. Schur, Über die Charakteristischen Wurzeln einer Linearen Substitution mit einer Anwendung auf die Theorie der Integralgleichungen, Math. Ann. 66 (1909), 488-510.

[St] E. Study, Zur Theorie der Lineare Gleichungen, Acta Math. 42 (1920), 1-61.

[Thi] J.-Y. Thibon, Lectures on Noncommutative Symmetric Functions, MSJ Mem. 11 (2001), 39-94.

[VP] P. Van Praag, Sur le Norme Reduite du Determinant de Dieudonne les Matrices Quaterniennes, J. Algebra 136 (1991), 265-274.

[W] J.H.M. Wedderburn, On Continued Fractions in Noncommutative Quantities, Ann. Math. 15 (1913), 101-105.

[We] H. Weyl, Classical Groups, Their Invariants and Representations, Princeton Univ. Press, 1946.

[Wi] R.L. Wilson, Invariant Polynomials in the Free Skew Field, Selecta Math. 7 (2001), 565-586.

I. G.: Department of Mathematics, Rutgers University, Piscataway, NJ 08854 S. G.: American Mathematical Society, P.O.Box 6248 Providence, RI 02940 V. R.: Department of Mathematics, Rutgers University, Piscataway, NJ 08854 R. W.: Department of Mathematics, Rutgers University, Piscataway, NJ 08854

E-mail address:

I. G.:  igelfand@math.rutgers.edu
S. G.:  sxg@ams.org
V. R.:  vretakh@math.rutgers.edu
R. W.:  rwilson@math.rutgers.edu