THE INTEGRAL ON QUANTUM SUPER GROUPS OF TYPE A_{r|s}

PHÙNG HÔ` HA’ I

Abstract. We compute the integral on matrix quantum (super) groups of type A_{r|s} and derive from it the quantum analogue of (super) HCIZ integral.

Introduction

A formula for the integral on the group U(n) was obtained by Itzykson and Zuber [3]. It can be given in the following form

∫_{U(n)} tr(MUNU^{−1})^{k}[dU] = ∑_{λ∈P _{k}^{n}} \frac{d_{λ}}{r_{λ}} Φ_{λ}(M)Φ_{λ}(N)  (1)

for any hermitian matrices M and N. Φ_{λ} is the irreducible character of U(n), corresponding to λ. If ξ_{1}, ξ_{2}, ..., ξ_{n} are the eigenvalues of M then Φ_{λ}(M) = s_{λ}(ξ_{1}, ξ_{2}, ..., ξ_{n}), s_{λ} are the Schur functions. d_{λ} is the dimension of the irreducible module of the symmetric function 𝔖_{n}, r_{λ} is the dimension of the irreducible representation of U(n), corresponding to partition λ. Explicitly, d_{λ} = n! ∏_{x∈[λ]} h(x)^{−1}, r_{λ} = ∏_{x∈[λ]} c(x)h(x)^{−1}, where c(x) is the content, h(x) is the hook-length of the box x in the diagram [λ]. This formula turns out to be a special case of a formula obtained by Harish-Chandra [10]. The integral on the left-hand side of (1) is therefore referred as Harish-Chandra-Itzykson-Zuber (HCIZ) integral. A super analogue of this formula was obtained by Alfaro, Medina and Urrutia [1, 2], it reads

∫_{U(m|n)} str(MUNU^{−1})^{k}[dU] = ∑_{{μ∈P^{m},ν∈P^{n} \atop |μ|+|ν|=k−mn}} \frac{k!}{|μ|!|ν|!} \frac{d_{μ}d_{ν}}{r_{μ}r_{ν}} Φ_{λ}(M)Φ_{λ}(N),  (2)

where λ = (n^{m}) + μ ∪ ν′. The aim of this paper is to give a quantum analogue of the super HCIZ integral. In contrary to the method of Itzykson-Zuber and its super version developed by Alfaro et. al., which is analytic, our method is purely algebraic. Thus, fist we want to give an algebraic definition of the integral.

It is well-known that the integral on a compact group is uniquely defined by its left (or right) invariance with respect to the group action. Thus, the integral can be considered as a linear functional on the function algebra of the group with an invariance property. Notice that the algebra of polynomial function on a compact (Lie) group is a commutative Hopf algebra. It turns out that for any Hopf algebra, one can give the notion of integral, although, such an integral does not always exist.

Thus, by definition, a left integral on a Hopf algebra H is a linear form ∫ on H such

that (∫ ⊗id)Δ = ∫ . Analogously, one has the notion of right integral. Generally,

left and right integrals may differ. For general properties of integral on Hopf algebras, the reader is referred to [16].

Originally, the integral is defined only on compact groups. The algebraically defined integral exists however on any cosemisimple Hopf algebras and on finite dimensional Hopf algebras. In particular, Hopf algebras of functions on linear reductive groups possess integrals. This is explained by the fact that each linear reductive group possesses a compact form and the corresponding integral coincides with the analytically defined one on this compact form. Having this relationship in mind, we shall find in this work the integral on the function algebras on quantum (super) groups of type A_{r|s}, such a quantum group is understood to be a generalization of the quantum general linear super group GL_{q}(r + 1|s + 1). It is defined in terms of a Hecke symmetry of birank (r + 1, s + 1).

The author would like to thank the Max-Planck-Institut für Mathematik, Bonn, for its hospitality and financial support.

In [7] we found the integral on quantum groups ot type A_{r} = A_{r|0}. In this work we extend the result to the case of A_{r|s} for arbitrary r, s. Our main Theorem 2.2 computes the integral at every monomial in the entries of the generating matrix and its inverse. This is an advantage of our method to the analytical one. The original HCIZ formula computes the integral only at certain polynomials and it is hard to recover the value of the integral at each monomials, while it is easy to obtain the HCIZ formula from our formula.

The paper is organize as follows. In Section 1 we recall the definition of quantum (super) groups of type A_{r|s}. They are define in terms of Hecke symmetries. The reader, who is not interested in this general setting, can have in mind the standard quantum general linear super group GL_{q}(r|s) [14]. The generating matrix is denoted by Z. We insert some signs in the definition of its tensor power Z^{⊗n} in order to provide that this matrix is still multiplicative. The inverse to Z is, again, upto some signs denoted by T. The signs provide T^{t} to be multiplicative. After recalling the general definition we give two main examples of Hecke symmetries, the Drinfeld-Jimbo’s matrix of type A_{r} and it super version due to Manin.

In Section 2 we compute the integral on the entries of Z^{⊗n}T^{⊗n}. Our method bases on constructing a sequence of endomorphisms P_{n} of the tensor powers of the basic comodule. In Section 3 we introduce the notion of super characters. The reader who is not familiar with ribbon categories can skip the discussion preceding Lemma 3.1. Using the characters we establish the quantum supre HCIZ integral. In Section 4 we discuss the orthogonal relations for a class of H_{R}-comodules.

1. Quantum groups associated to Hecke symmetries

Let V be a super vector space over 𝕂, a fixed field of characteristic zero. Fix a homogeneous basis x_{1}, x_{2}, …, x_{d} of V. We shall denote the parity of the basis element x_{i} by î. An even operator R on V ⊗ V can be given by a matrix R_{ij}^{kl}: R(x_{i} ⊗ x_{j}) = x_{k} ⊗ x_{l}R_{ij}^{kl}. R is even means that the matrix elements R_{kl}^{ij} satisfy î + ĵ = k̂ + l̂. R is called Hecke symmetry if the following conditions are satisfied:

i) R satisfies the Yang-Baxter equation R_{1}R_{2}R_{1} = R_{2}R_{1}R_{2}, R_{1} := R ⊗ I, R_{2} := I ⊗ R, I is the identity matrix of degree d.

ii) R satisfies the Hecke equation (R − q)(R + 1) = 0 for some q which will be

assumed not to be a root of unity. iii) There exists a matrix P_{ij}^{kl} such that P_{jn}^{im}R_{ml}^{nk} = δ_{l}^{i}δ_{j}^{k}.

The matrix bialgebra E_{R} is define as follows. Let {z_{j}^{i}, t_{j}^{i}|1 ≤ i, j ≤ d} be a set of

variables, x̂_{j}^{i} = t̂_{j}^{i} = î + ĵ. Define

E_{R} := 𝕂 < {z_{j}^{i}|1 ≤ i, j ≤ d} > /((−1)^{ŝ(î+p̂)}R_{ps}^{kl}z_{i}^{p}z_{j}^{s} = (−1)^{l̂(q̂+k̂)}z_{q}^{k}z_{n}^{l}R_{ij}^{qn})

H_{R} := 𝕂 < \{z_{j}^{i}, t_{j}^{i}|1 ≤ i, j ≤ d\} > /( \begin{matrix}(−1)^{ŝ(î+p̂)}R_{ps}^{kl}z_{i}^{p}z_{j}^{s} = (−1)^{l̂(q̂+k̂)}z_{q}^{k}z_{n}^{l}R_{ij}^{qn}, \\ (−1)^{ĵ(ĵ+k̂)}z_{j}^{i}t_{k}^{j} = (−1)^{l̂(l̂+î)}t_{l}^{i}z_{k}^{l} = δ_{k}^{i} \end{matrix} )

(𝕂 < X > denotes the free algebra generated by elements of X). The coproduct on E_{R} and H_{R} is given by

Δ(z_{j}^{i}) = z_{k}^{i} ⊗ z_{j}^{k}, Δ(t_{j}^{i}) = t_{j}^{k} ⊗ t_{k}^{i}.

The antipode on H_{R} is given by

S(z_{j}^{i}) = (−1)^{ĵ(î+ĵ)}t_{j}^{i}, S(t_{j}^{i}) = (−1)^{î(î+ĵ)}C_{k}^{i}z_{l}^{k}C^{−1}_{j}^{l},

where C_{j}^{i} := P_{jl}^{il}. We also define D_{j}^{i} := P_{lj}^{li}. The matrices C and D play important roles in our work, they are call refection operator.

Since we are working in the category of super vector spaces, the rule of sign effects on the coproduct. More precisely, the compatibility of product and coproduct of a super bialgebra reads

Δ(a ⊗ b) = (−1)^{â_{2}b̂_{1}}a_{1}b_{1} ⊗ a_{2}b_{2}.

Therefore we have

Δ(z_{j_{1}}^{i_{1}}z_{j_{2}}^{i_{2}} ⋯ z_{j_{2}}^{i_{2}}) = sign(I, K)sign(K, J)sign(I, J)z_{k_{1}}^{i_{1}}z_{k_{2}}^{i_{2}} ⋯ z_{k_{n}}^{i_{n}} ⊗ z_{j_{1}}^{k_{1}}z_{j_{2}}^{k_{2}} ⋯ z_{j_{n}}^{k_{2}},

where I := (i_{1}, i_{2}, …, i_{n}). sign(I, J) is given recurently by

sign(i, j) = 1, sign(Ii, Ji) = (−1)^{î(|Î|+|Ĵ|)}sign(I, J),

Î:= (î_{1}, î_{2}, ..., î_{n}), |Î| denotes the sum of its terms. Hence, for convenience, we denote

Z_{J}^{I} = Z^{⊗n}_{J}^{I} := sign(I, J)z_{j_{1}}^{i_{1}}z_{j_{2}}^{i_{2}} ⋯ z_{j_{2}}^{i_{2}}.

Throughout this paper we shall always use this notation. Then we have Δ(Z_{J}^{I}) = Z_{K}^{I} ⊗ Z_{L}^{K}, Δ(T_{J}^{I}) = T_{L}^{K} ⊗ T_{K}^{I}, and S(Z_{J}^{I}) = (−1)^{|Ĵ|(|Î|+|Ĵ|)}T_{J′}^{I′}. Notice that

sign(K′, L′) = (−1)^{|K̂_{1}|(k̂_{n}+ĵ_{n})}sign(K′_{1}, L′_{1}),

where for K = (k_{1}, k_{2}, ..., k_{n}), K_{1} := (k_{1}, k_{2}, ..., k_{n−1}) and K′ := (k_{n}, k_{n−1}, ..., k_{1}).

R induces an action of the Hecke algebra H_{n} = H_{q,n} (see Appendix) on the tensor powers V^{⊗n} of V, ρ_{n}(T_{i}) = R_{i} := id_{V}^{i−1} ⊗ R ⊗ id_{V}^{n−i−1}. We shall therefore use the notation R_{w} := ρ(T_{w}). On the other hand, E_{R} coacts on V by δ(x_{i}) = x_{j} ⊗ z_{i}^{j}. Since E_{R} is a bialgebra, it acts on V^{⊗n} by means of the coproduct. These two actions are centralizers of each other in End_{𝕂}(V^{⊗n}) [6]. Hence, the algebra End^{E_{R}}(V^{⊗n}) is a factor algebra of H_{n} and (E_{R}^{n})^{∗} ≅ End_{H_{n}}(V^{⊗n}). Let us denote End^{E_{R}}(V^{⊗n}) by \overline{H_{n}}. Since H_{n} is semi-simple, provided q is not a root of unity, the algebras (E_{R}^{n})^{∗} and \overline{H_{n}} are semi-simple, too.

The double centralizer theorem implies that a simple E_{R}-comodule is the image of the operator induced by a primitive idempotent of H_{n} and, conversely, each primitive idempotent of H_{n} induces an E_{R} comodule which is either zero or simple. Irreducible representations of H_{n} are parameterized by partitions of n. Thus, up to conjugation, primitive idempotents of H_{n} are parameterized by partitions of n, too.

The primitive idempotents x_{n} := ∑_{w} T_{w}/[n]_{q}! and y_{n} := ∑_{w}(−q)^{−l(w)}T_{w}/[n]_{1/q}! induce the symmetrizer and anti-symmetrizer operators on V^{⊗n}. Let S_{n} := Imρ_{n}(x_{n}) and Λ_{n} := Imρ_{n}(y_{n}). Then one can show that S := ⨁_{n=0}^{∞} S_{n} and Λ := ⨁_{n=0}^{∞} Λ_{n} are algebras. They are called symmetric and exterior tensor algebras on the corresponding quantum space.

By definition, the Poincaré series P_{Λ}(t) of Λ is ∑_{n=0}^{∞} t^{n}dim_{𝕂}(Λ_{n}). It is proved that this series is a rational function having negative roots and positive poles. Let r be the number of its roots and s be the number of its poles. As a consequence, simple E_{R}-comodules are parameterized by hook-partitions {λ|λ ∈ Γ_{n}^{r,s}}, i.e., partitions λ, such that λ_{r+1} ≤ s. (r, s) is called the birank of R [9]. Consequently, in the algebra \overline{H_{n}} primitive idempotents are parameterized by hook-partitions from Γ_{n}^{r,s}, too.

Our main assumption on R is that

tr(C) = −[s − r]_{q},  (3)

C is the reflection operator introduced above. It can be proved that tr(C) should have the form −[x]_{q} for some integer x in the interval [−s, r]. The above equation holds for any known Hecke symmetry. It is conjectured that it holds for all Hecke symmetry.

Simple H_{R}-comodules are much more complicated. The problem of classifying all its comodule is still open.

The Hopf algebra H_{R} (resp. the bialgebra E_{R}) is called the (function algebra on) quantum group (resp. quantum semigroup) of type A_{r|s}.

The following are two main examples of Hecke symmetries. The Drinfeld-Jimbo’s R-matrix of type A_{r−1} [11], for 1 ≤ i, j, k, l ≤ r, p^{2} = q,

R_{r}_{ij}^{kl} := \{ \begin{matrix}q & if & i = j = k = l \\ q − 1 & if & k = i > j = l \\ p & if & k = j ≠ i = l \\ 0 & otherwise. \end{matrix}

Assume that all parameters are even, the Hopf algebra associated to R_{r} is called the (function algebra on) standard quantum general linear group GL_{q}(r). R_{r} is a oneparameter deformation of the permuting operator. The super version of this operator was given by Manin [14]. Assume that the variable x_{i}, i ≤ r are even and the rest s variables are odd. Define, for 1 ≤ i, j, k, l ≤ r + s, p^{2} = q,

R_{r|s}_{ij}^{kl} := \{ \begin{matrix}q & if & i = j = k = l, î = 0 \\ −1 & if & i = j = k = l, î = 1 \\ q − 1 & if & k = i < j = l \\ (−1)î ; ĵp & if & k = j ≠ i = l \\ 0 & otherwise. \end{matrix}

R_{r|s} is a deformation of the permuting operator in super symmetry. The associated Hopf algebra is called the (function algebra on) standard quantum general linear super group GL_{q}(r|s).

R_{r} has the birank (r, 0). Such an operator is called even Hecke symmetry. R_{r|s} has the birank (r, s). Generally, if R is a Hecke symmetry of rank (r, s), then H_{R} is called the (function algebra on) quantum group of type A_{r−1|s−1}. The aim of this work is to compute the integral on such a quantum super group.

The integral on quantum groups of type A_{r|0} was computed in [7]. Let us recall it here. First, remark that an arbitrary element of H_{R} can be represented as a linear combination of monomials of the form Z_{I}^{J}T_{K}^{L}. Second, remark that from definition the integral on Z_{I}^{J}T_{K}^{L} is zero if l(I) ≠ l(K), l(I) is the length of I. The integral on Z_{J}^{I}T_{L}^{K} with l(I) = l(K) = n is

∫ (Z_{I}^{J}T_{K′}^{L′}) = ∑_{w∈𝔖_{n}} q^{−l(w)}(P_{n}C^{⊗n}R_{w})_{I}^{L}R_{w}_{K}^{J},

where for K = (k_{1}, k_{2}, ..., k_{n}), K′ := (k_{n}, ..., k_{1}). P_{n} is given by P_{n} := ∏_{k=1}^{n}(L_{k} − [−r]_{q})^{−1}. L_{n} are the Murphy operators, L_{1} := 0, L_{n} = ∑_{i=1}^{n−1} q^{−i}R_{(n−i,n)}, n ≥ 2. The proof bases on the fact that P_{n} are central in \overline{H_{n}} and satisfy the equation (in \overline{H_{n+1}})

P_{n} = P_{n+1}(L_{n+1} − [−r]).

2. The Integral on H_{R}

As we have seen in the previous section, the integral on H_{R} with R even Hecke symmetry is obtained by constructing a sequence of operators P_{n} with property P_{n} = P_{n+1}(L_{n+1} − [−r]). To find a integral on H_{R} with arbitrary R we construct an analogous sequence P_{n} with this property for r replaced by r − s, where (r, s) is the birank of R. Recall that we assume tr(C) = −[s − r]_{q}. In the general case, the operator L_{n} − [s − r]_{q} is not always invertible, so that we have to find a new construction.

A full set of mutual orthogonal primitive idempotents of H_{n} was found by by Dipper and James [4]:

E_{t_{i}(λ)} = ∏_{{1≤m≤n \atop |k|≤m−1} ; k≠c_{t_{i}(λ)}(m)} \frac{L_{m} − [k]_{q}}{[c_{t_{i}(λ)}(m)]_{q} − [k]_{q}} , 1 ≤ i ≤ d_{λ}, λ ∈ P_{n}.

c_{t_{i}(λ)}(m) is the content of m in the standard tableau t_{i}(λ) (see Appendix for notations). Notice that ∑_{1≤i≤d_{λ}} E_{t_{i}(λ)} = F_{λ} – the minimal central idempotent corresponding to λ.

L_{m} satisfy the equation

∏_{k=−m−1}^{m+1}(L_{m} − [k]_{q}) = 0.

Therefore, for 1 ≤ m ≤ n,

L_{m}E_{t_{i}(λ)} = E_{t_{i}(λ)}L_{m} = c_{t_{i}(λ)}(m)E_{t_{i}(λ)}.  (4)

Using this equation and the fact that P_{n} is central we obtain the follwing decompsition

P_{n} = ∑_{λ⊢n} q_{λ}F_{λ} = ∑_{{λ⊢n \atop 1≤i≤d_{λ}}} q_{λ}E_{t_{i}(λ)},  (5)

the coefficient q_{λ} is given by

q_{λ} = ∏_{x∈[λ]} \frac{q^{r}}{[c_{λ}(x) + r]_{q}} .

For an arbitrary R we shall find P_{n} in the form analogous to (5). The sum will be taken on the set {λ ∈ Γ_{n}^{r,s}|λ_{r} ≥ s} =: Ω_{n}^{r,s}. Geometrically, this is the set of λ’s, the diagram [λ] contains the diagram [(s^{r})]. Each such λ decomposes into λ = (s^{r})+μ∪ν′, where μ ∈ P^{r}, ν ∈ P^{s}.

We define

q_{λ} := ∏_{x∈[λ]\[(s^{r})]} \frac{q^{r−s}}{[c_{λ}(x) + r − s]_{q}} ,

thus q_{λ} = q^{r|ν|−s|μ|}p_{μ}p^{−}_{ν}, where p^{o}_{ν} = ∏_{x∈ν} \frac{q^{−s}}{[−c(x)−s]}. Then, we define

P_{n} := ∑_{{λ∈Ω{r,s \atop n} \atop 1≤i≤d_{λ}}} q_{λ}E_{t_{i}(λ)} = ∑_{λ∈Ω_{n}^{r,s}} q_{λ}F_{λ}.  (6)

Recall that P_{n} are defined in the algebra \overline{H_{n}} ≅ End^{E_{R}}(V^{⊗n}), which is the factor algebra of H_{n} by the two-sided ideal generated by minimal central idempotents corresponding to partitions from P_{n} \ Γ_{n}^{r,s}. Fixing an embedding \overline{H_{n}} ↪ \overline{H_{n+1}}, \overline{H_{n}} ∋ W ⟼ W ⊗ id_{V} ∈ \overline{H_{n+1}}, we identify \overline{H_{n}} with a subalgebra of \overline{H_{n+1}}.

Lemma 2.1. The operators P_{n} are central in \overline{H_{n}} and satisfy the equation (in \overline{H_{n+1}})

P_{n+1}(L_{n+1} − [s − r]_{q}) = P_{n}.

Proof. The operator P_{n}^{s} is obviously central, for it is a sum of central elements. We check the equation above. First, notice that, if λ ∈ Ω_{n+1}^{r,s} and t_{i}(λ) is a standard

λ-tableau, then the node of [λ], containing n + 1, is removable, i.e., having removed it we still have a standard tableau. The tableau t_{i}(λ) is called essential if this node is not the node (r, s), otherwise, it is called non-essential. A tableau t_{i}(λ) is essential iff the tableau, obtained from it by removing the node containing n + 1 is again a γ-tableau with γ ∈ Ω_{n}^{r,s}.

Observe, that if t_{i}(λ) is non-essential, then c_{t_{i}(λ)} = s − r, hence

E_{t_{i}(λ)}(L_{n+1} − [s − r]_{q}) = 0,

by virtue of Equation (4). We reshuffle the terms of P_{n+1} in groups as follows

P_{n+1} = ∑_{{γ∈Ω{r,s \atop n} \atop 1≤i≤dγ}} ∑_{{λ∈Ω{r,s \atop n+1} \atop t(λ)⊃t_{i}(γ)}} q_{λ}E_{t(λ)} + ∑_{t(λ) is not eessential} q_{λ}E_{t(λ)}.

That is, we pick up into group for each t_{i}(γ), γ ∈ Ω_{n}^{r,s}, those standard tableaux t(λ), that contain t_{i}(γ) as a subtableau. The preceding observation implies that the second sum in the right-hand side of the above equation multiplied by L_{n+1} − [s − r]_{q} vanishes. Thus, it is sufficient to prove, for a fixed t_{i}(γ), γ ∈ Ω_{n}^{r,s},

∑_{{λ∈Ω{r,s \atop n+1} \atop t(λ)⊃t_{i}(γ)}} q_{λ}E_{t(λ)}(L_{n+1} − [s − r]_{q}) = q_{γ}E_{t_{i}(γ)}.

We have (L_{n+1}−[s−r]_{q})E_{t(λ)} = [c_{t(λ)}(n+1)]_{q} −[s−r]_{q} and q_{λ}(c_{λ}(x)−[s−r]_{q}) = q_{γ} whenever [γ] is obtained from [λ] by removing the node x. Since, for any two standard tableaux t(γ) and t(λ) with γ ⊂ λ as above, the number n + 1 should lie in the node x, for which [λ] \ [x] = [γ], we deduce that the equation to be proved is equivalent to

∑_{{λ∈Ω{r,s \atop n+1} \atop t(λ)⊃t_{i}(γ)}} E_{t(λ)} = E_{t_{i}(γ)}.  (7)

Since ∏_{k=−n−1}^{n+1}(L_{n+1} − [k]_{q}) = 0,

∑_{m=−n−1}^{n+1} ∏_{{k=−n−1, \atop k≠m}}^{n+1} \frac{L_{n+1} − [k]_{q}}{[m]_{q} − [k]_{q}} = 1.

Therefore

E_{t_{i}(γ)} = ∑_{m=−n−1}^{n+1} E_{t_{i}(γ)} ∏_{{k=−n−1, \atop k≠m}}^{n+1} \frac{L_{n+1} − [k]_{q}}{[m]_{q} − [k]_{q}} .

Remember that we are working in the algebra \overline{H_{n+1}}, in which E_{λ} ≠ 0 iff λ ∈ Γ_{n+1}^{r,s}. Each term on the right-hand side of the above equation is either zero or a primitive idempotent of the form E_{t(λ)} with t(λ) containing t_{i}(γ) as a subtableau. Since the left-hand side of (7) contains all primitive idempotents in \overline{H_{n+1}} that correspond to standard tableaux containing t_{i}(γ) as a subtableau, the equation (7) follows. Lemma 2.1 is therefore proved.

We are now going to give the formula for an integral on the super Hopf algebra H_{R}. First, notice that as in the case of even Hecke symmetries, the integral on Z_{J}^{I}T_{L}^{K} vanishes unless l(I) = l(K). Further the integral should vanish at the identity if

rs > 0, for H_{R} is then not cosemisimple. In fact, we shall set ∫ (Z_{I}^{J}T_{K}^{L}) = 0 if

l(I) = l(K) < rs and compute the integral only for l(I) = l(K) ≥ rs. This is in fact not a restriction, for the integral if it exists then uniquely up to a constant [15].

Theorem 2.2. The Hopf algebra H_{R} associated to a Hecke symmetry R satisfying the condition (3) possesses an integral, which is uniquely determined up to a scalar multiple. Let (r, s) be the birank of R. Then an integral can be given as follows: if

l(I) = l(K) < rs, ∫ (Z_{I}^{J}T_{K}^{L}) = 0, if l(I) = l(K) = n ≥ rs,

∫ (Z_{I}^{J}T_{K′}^{L′}) = (−1)^{|K̂|(|K̂|+|L̂|)} ∑ q^{−l(w)}(P_{n}C^{⊗n}R_{w^{−1}})_{I}^{L}R_{w}_{K}^{J},

(K′ = (k_{n}, k_{n−1}, ..., k_{1})).

Notice that since the operators C, P, R_{w} are all even, the integral vanishes unless |Î| = |L̂|, |K̂| = |Ĵ|.

Proof. (sketch) The proof of this theorem does not differ from the one for even Hecke symmetries, given in [7]. We give here only a sketch, for detail the reader is referred to [7].

It is to check that the above formula is invariant with respect to the defining relations

(−1)^{ŝ(î+p̂)}R_{ps}^{kl}z_{i}^{p}z_{j}^{s} = (−1)^{l̂(q̂+k̂)}z_{q}^{k}z_{n}^{l}R_{ij}^{qn},
(−1)ĵz_{j}^{i}t_{k}^{j} = (−1)l̂t_{l}^{i}z_{k}^{l} = δ_{k}^{i}.

Notice that the last relation, combined with the first one gives us

(−1)^{ĵ(î+ĵ)}z_{k}^{l}C_{l}^{j}t_{j}^{i} = C_{k}^{i}.

The invariance with respect to the first relation above follows from the fact that P_{n} and ∑_{w∈𝔖_{n}} q^{−l(w)}R_{w^{−1}} ⊗ R_{w} are central. The invariance with respect to the other relations is equivalent to the following equations (all signs are cancelled):

∑_{w∈𝔖_{n}} q^{−l(w)}(P_{n}R_{w^{−1}}C^{⊗n})_{I_{1}m}^{L_{1}m}R_{w}_{K}^{J} = ∑_{w∈𝔖_{n−1}} q^{−l(w)}(P_{n−1}R_{w^{−1}}C^{⊗n−1})_{I_{1}}^{L_{1}}R_{w}_{K_{1}}^{J_{1}}δ_{k_{n}}^{j_{n}},

∑_{w∈𝔖_{n}} q^{−l(w)}(P_{n}R_{w^{−1}}C^{⊗n−1})_{I_{1}i_{n}}^{L_{1}l_{n}}R_{w}_{K_{1}p}^{J_{1}n}C_{n}^{p} = ∑_{w∈𝔖_{n−1}} q^{−l(w)}(P_{n−1}R_{w^{−1}}C^{⊗n−1})_{I_{1}}^{L_{1}}R_{w}_{K_{1}}^{J_{1}}C_{i_{n}}^{l_{n}},

here, for a multi-index K = (k_{1}, k_{2}, …, k_{n}) we denote K_{1} := (k_{1}, k_{2}, …, k_{n−1}). By virtue of Lemma 2.1 this equations reduce to

∑_{w∈𝔖_{n}} q^{−l(w)}R_{w^{−1}} ⊗ tr^{n}(R_{w}) = ∑_{w∈𝔖_{(n−1,1)}} q^{−l(w)}(L_{n} − [−r]_{q})R_{w^{−1}} ⊗ R_{w},

Here the map tr^{n} : \overline{H_{n}} ⟶ \overline{H_{n−1}} is given by

tr^{n}(R_{w}) = [ \begin{matrix}R_{v_{k}···v_{n−2}w_{1}} & if k ≤ n − 1 \\ −[s − r]_{q}R_{w_{1}} & if k = n, \end{matrix}

where w = v_{k}v_{k+1} ⋯ v_{n−1}w_{1}, w_{1} ∈ 𝔖_{n−1} (cf. [7], Lemma 4.1.3).

3. Characters of H_{R} and quantum analogue of super HCIZ integral

Following [8] we define characters of comodules over H_{R}. To do this we first need a co-ribbon structure on H_{R}. By definition, a coquasitriangular super Hopf algebra is a super Hopf algebra H, equipped with a ∗-invertible even bilinear form R : H ⊗ H ⟶ 𝕂, subject to the following conditions:

r(ab, c) = (−1)^{b̂ĉ_{1}}r(a, c_{1})r(b, c_{2}) r(a, bc) = r(a_{2}, b)r(a_{1}, c)
(−1)^{â_{2}b̂_{1}}r(a_{1}, b_{1})a_{2}b_{2} = (−1)^{âb̂_{1}}b_{1}, a_{1}r(a_{2}, b_{2}).

R is called a coquasitriangular (CQT) structure on H. A CQT structure on H_{R} is given by r(z_{j}^{i}, z_{l}^{k}) = (−1)^{ĵk̂}R_{jl}^{ki}, r(t_{j}^{i}, t_{l}^{k}) = (−1)^{îl̂}R_{jl}^{ki}, r(t_{j}^{i}, z_{l}^{k}) = (−1)^{îk̂}R^{−1}_{lj}^{ik}, r(z_{j}^{i}, t_{l}^{k}) = (−1)^{îl̂}P_{jl}^{ki}.

A co-ribbon structure on a CQT Hopf algebra H is a ∗-invertible linear form t : H ⟶ 𝕂, subject to the following conditions:

t(a_{(1)})a_{(2)} = a_{(1)}t(a_{(2)}), t ∘ S = t,
t(ab) = (−1)^{(â_{1}+â_{1})(b̂_{1}+b̂_{2})}t(a_{(1)})t(b_{(1)})r(b_{(2)}a_{(2)})r(a_{(3)}, b_{(3)}).

The co-ribbon structure on H_{R} is given by t(z_{j}^{i}) = t(t_{j}^{i}) = q^{(r−s+1)/2}δ_{j}^{i}.

It H is a coribbon Hopf algebra then H-comod is a ribbon category [12], the braiding and the twist are defined as follows:

τ_{M,N}(m, n) = n_{0} ⊗ m_{0}r(m_{1}, n_{1}), θ_{M}(m) := m_{0}t(m_{1}), m ∈ M, n ∈ N.

By definition, the character of an endomorphism on a comodule M over H_{R} is the element

Φ(M) := (ev_{M} ⊗ id_{H_{R}})(id_{M^{∗}} ⊗ δ_{M})τ_{M,M^{∗}}(f ∘ θ_{M} ⊗ id_{M^{∗}})db(1_{𝕂}),

where ev_{M}, db_{M} are the canonical morphism defining a structure of dual H_{R}-comodule on M^{∗}: ev : M^{∗} ⊗ M ⟶ 𝕂, db : 𝕂 ⟶ M ⊗ M^{∗}. The character Φ(M) of M is defined to be Φ(M) := Φ(id_{M}). The character map Φ is a trace map from End(M) in to H.

Let x_{1}, x_{2}, …, x_{m} be a basis of M and a_{i}^{j}, 1 ≤ i, j ≤ m be the corresponding coefficient matrix: δ(x_{i}) = x_{j} ⊗ a_{i}^{j}. Let T_{M} _{j}^{i} := t(a_{j}^{i}) and D_{M} _{j}^{i} = r(a_{l}^{i}, S(a_{j}^{l})), f(x_{i}) := x_{j}T_{i}^{j}. Then the character of f and M is given by

Φ(f) = tr(F D_{M}T_{M}A), Φ(M) = tr(D_{M}T_{M}A).

We have the following identity

τ_{M^{⊗n},M^{∗⊗n}}(θ_{M^{⊗n}} ⊗ id_{M^{∗⊗n}}) = db_{M}^{n} ∘ db_{M}^{n−1} ∘ ⋯ ∘ db_{M} : 𝕂 ⟶ M^{⊗n} ⊗ M^{∗⊗n}, (8)

where db_{M^{⊗n}}^{i} := id_{M}^{i−1} ⊗db_{M} ⊗id_{M^{∗}}^{i−1}. Using it we can show that Φ(f ⊗g) = Φ(f)Φ(g). Moreover, let f : M^{⊗n} ⟶ M^{⊗n}. Then we have

Φ(f) = tr(D^{⊗n}T^{⊗n}F A^{⊗n}),

where F is the matrix of f with respect to the basis x_{1}, ..., x_{n}.

For each partition λ ∈ Γ_{n}^{r,s}, M_{λ} denotes the corresponding simple E_{R}-comodule. Since the map E_{R} ⟶ H_{R} is injective (cf. [7]), M_{λ} is simple H_{R}-comodule, too. M_{λ} is isomorphic to Imρ(E_{λ}) of a primitive idempotent E_{λ}. Therefore Φ(M_{λ}) = Φ(E_{λ}). Thus, setting D_{j}^{i} = r(z_{l}^{i}, S(z_{j}^{l})), we have

Lemma 3.1. The characters of M_{λ} and M_{λ}^{∗}, λ ⊢ n are given by

S_{λ} := Φ(M_{λ}) = q^{n(r−s+1)/2}tr(D^{⊗n}E_{λ}Z^{⊗n}),
S_{−λ} := Φ(M_{λ}^{∗}) = q^{n(r−s+1)/2}tr(C^{⊗n}E_{λ}\overline{T^{⊗n}}),

where \overline{T^{⊗n}}_{J}^{I} := T_{J′}^{I′}.

Notice that in the definition of Z^{⊗n}, the signs are also inserted.

Since S_{λ}, λ ∈ Γ_{n}^{r,s} belong to different simple subcoalgebras of E_{R}, they are linear independent in E_{R}, and hence in H_{R}, too.

Let K be an algebra. A K-point of a (super) bialgebra B is an algebra homomorphism A : B ⟶ K. Let A be a K-point of the bialgebra E_{R}. Then the entries of A commute by the same rule as the entries of Z. We define S_{λ}(A) := A(S_{λ}). In other words, S_{λ}(A) is S_{λ} computed at Z = A. We are now ready to formulate a formula to compute the quantum super HCIZ integral.

Theorem 3.2. Let M be a K-point of E_{R} and N be a K-point of E′_{R} = S(E_{R}) (i.e., the subbialgebra of H_{R}, generated by T). Assume that entries of M and N commute and that they commute with the entries of Z and T. Then

∫ q^{n(r−s)/2}tr(C^{⊗n}M^{⊗n}Z^{⊗n}\overline{N^{⊗n}}\overline{T^{⊗n}}) = ∑_{λ∈Ω_{n}^{r,s}} \frac{d_{λ}p_{λ}^{s}}{k_{λ}} S_{λ}(M)S_{−λ}(N).

Here S_{−λ} = S(S_{λ}), S_{−λ}(N) is computed at T = N. k_{λ} = q^{n(λ)} ∏_{x∈[λ]}[h(x)]_{q}^{−1}, n(λ) = ∑_{i} λ_{i}(i − 1), h(x) is the hook length of x in the diagram [λ]: h_{λ}(x) = λ_{i} + λ′_{j} − i − j + 1, where (i, j) is the coordinate of x in the diagram [λ].

Proof. The proof, again, does not differ from the one given in [8] for quantum HCIZ integral. The main observation is that

∑_{w∈𝔖_{n}} q^{−l(w)}R_{w^{−1}}R_{w} = ∑_{{λ⊢n \atop 1≤i,j≤d_{λ}}} \frac{1}{k_{λ}} E_{λ}^{ij},

and that Φ(E_{λ}^{ij}) = 0 for i ≠ j. Here E_{λ}^{ij} is a basis of the Hecke algebra H_{n} with the property E_{λ}^{ij}E_{μ}^{kl} = δ_{λ}^{μ}δ_{k}^{j}E_{λ}^{il}, thus, in particular, E_{λ}^{ii} are primitive idempotents. For the detailed proof the reader is referred to [8].

Example. Let us consider the case of standard quantum general linear super group GL_{q}(r|s), determined in terms of the symmetry R_{r|s} given Section 1. In this case, any diagonal matrix with commuting entries is a point of E_{R}. Thus, assume that M and N are diagonal matrix with entries commuting each other and with the entries of Z and T, M = diag(m_{1}, m_{2}, ..., m_{r+s}), N = diag(n_{1}, n_{2}, ..., n_{r+s}). Then, we have

S_{(n)} = ∑_{k=0}^{n} h_{n−k}(qm_{1}, ..., q^{r}, m_{r})e_{k}(−q^{r}m_{r+1}, ..., −q^{r−s+1}m_{r+s}),

h_{k}, e_{k} are the k^{th} complete and elementar symmetric functions in r and s variables, respectively. Hence

S_{λ}(M) = s_{λ}(qm_{1}, q^{2}m_{2}, ..., q^{r}m_{r}/ − q^{r}m_{r+1}, ..., −q^{r−s+1}m_{r+s}),

s_{λ} are the Hook-Schur functions in r + s variables (cf. [13], Ex. I.3.23). Therefore, if λ ∈ Ω_{n}^{r,s}, thus, λ = (r^{s}) + μ ∪ ν′, μ ∈ P^{r}, ν ∈ P^{s}, we have [loc.cit.]

S_{λ}(M) = (−1)^{|ν|} ∏_{i=1,j=1}^{r,s}(q^{i}m_{i} − q^{r−j+1}m_{r+j})s_{μ}({q^{i}m_{i}})s_{ν}({q^{r+1−i}m_{r+i}}),

where s_{μ} (resp. s_{ν}) are the Schur functions in r variables (resp. s variables). Analogously, we have

S_{−λ}(N) = (−1)^{|ν|} ∏_{i=1,j=1}^{r,s}(q^{r−s−i+1}n_{i} − q^{j−s}n_{r+j})s_{μ}({q^{r−s+1−i}n_{i}})s_{ν}({q^{i−s}n_{r+i}}).

The quantum super HCIZ is then given by, (n ≥ rs),

∫ tr(C^{⊗n}N^{⊗n}Z^{⊗n}M^{⊗n}T^{⊗n}) =
∑_{μ∈P^{r} ,ν∈P^{s}, ; {|μ|+|ν|=n−rs \atop λ=(s^{r})+μ∩ν′}} q^{r|ν|−s|μ|}p_{μ}p_{ν}^{−} \frac{d_{λ}}{k_{λ}} ∏_{{i=1 \atop j=1}}^{r,s}(q^{i}m_{i} − q^{r−j+1}m_{r+j})(q^{r−s−i+1}n_{i} − q^{j−s}n_{r+j})

×s_{μ}({q^{i}m_{i}})s_{ν}({q^{r+1−i}m_{r+i}})s_{μ}({q^{r−s+1−i}n_{i}})s_{ν}({q^{i−s}n_{r+i}}).

4. The Orthogonal Relations

We are now interested in the orthogonal relations. Let M_{λ}, M_{μ} be two simple comodules corresponding to partitions λ and μ of n. Let M_{λ} = ImE_{λ}, M_{μ} = ImE_{μ}.

Then, using Theorem 2.2, we have

(Φ(M_{λ}), Φ(M_{μ})) = ∫ (Φ(M_{λ})Φ(M_{μ}^{∗}))
= ∑ q^{−l(w)}tr(P_{n}C^{⊗n}E_{λ}R_{w^{−1}}E_{μ}R_{w})
= δ_{λ}^{μ}δ_{Ω^{r,s}}^{λ} \frac{q_{λ}}{k_{λ}} tr(C^{⊗n}E_{λ})
= 0.

Here δ_{Ω^{r,s}}^{λ} indicates, whether λ belongs to Ω^{r,s}, it is zero if λ ∉ Ω^{r,s} and 1 otherwise. On the other hand, for λ form Ω^{r,s}, tr(C^{⊗n}E_{λ}) = 0, for it is the quantum rank of M_{λ}.

We see that the scalar product above cannot be used to define the orthogonal

relations. Following [1] we compute the integral ∫ (Z_{λ}Z_{−μ}), where Z_{λ} is a coefficient

matrix of the simple comodule M_{λ}, i.e., Z_{λ} is the multiplicative matrix corresponding to certain basis of M_{λ}.

Let us fix a primitive idempotent E_{λ}, λ ⊢ n, and set M_{λ} = Imρ(E_{λ}). To choose a basis of M_{λ} we proceed as follows. First notice that there exists a one-one correspondence between the set of bases of M_{λ} and the set of maximal operators P_{b}^{a} in End_{𝕂}(M_{λ} with property P_{b}^{a}P_{d}^{c} = δ_{b}^{c}P_{d}^{a}. Further, since E_{λ} is an idempotent, End_{𝕂}(M_{λ}) = E_{λ}End_{𝕂}(V^{⊗n})E_{λ}. Therefore, instead of fixing a basis of M_{λ} we can equivalently choose a maximum set of operators P_{b}^{a} = E_{λ}Q_{b}^{a}E_{λ}, P_{b}^{a} ∈ End(V^{⊗n}), satisfying P_{b}^{a}P_{d}^{c} = δ_{b}^{c}P_{d}^{a}.

Using standard argument we deduce that the corresponding multiplicative matrix is given by Z_{λ}_{n}^{a} = tr(P_{b}^{a}Z^{⊗n}). Analogously, Z_{−λ}_{n}^{a} = tr(P_{b}^{a}\overline{T^{⊗n}}). Now we can find the integral

∫ (Z_{λ}_{b}^{a}Z_{−λ}_{d}^{c}) = ∫ tr(P_{b}^{a}Z^{⊗n})tr(P_{d}^{c}\overline{T^{⊗n}})
= k_{λ}^{−1}q_{λ}tr(C^{⊗n}E_{λ}^{ij}P_{b}^{a}E_{λ}^{ji}P_{d}^{c})
= k_{λ}^{−1}q_{λ}δ_{b}^{c}tr(C^{⊗n}P_{d}^{a}),

for we can choose E_{λ}^{ij} such that E_{λ} = E_{λ}^{ii} for some i.

Appendix

A partition of a positive integer n is a sequence of non-increasing positive integers λ = (λ_{1}, λ_{2}, ..., λ_{r}), whose sum |λ| is equal to n. We shall write λ ⊢ n or λ ∈ P_{n}. r is the length of λ. The set of partition of length r is denoted by P^{r}. P_{n}^{r} := P^{r} ∩ P_{n}. To shorten notations, we shall, for example, use (3^{3}, 2, 1^{2}) to denote the partition (3, 3, 3, 2, 1, 1).

The diagram [λ] of λ is like a matrix, its first row contains λ_{1} nodes, second row contains λ_{2} notes,... The conjugated to λ partition, denoted by λ′ is the one, whose diagram is obtained from [λ] by rotating along its diagonal 180^{∘}. A standard λtableau it the diagram [λ] filled with the numbers 1, 2, ..., |λ| in such a way that they are increasing along rows and columns. The content of the node x in the i^{th}-row and j^{th}-column is j − i, it is denoted by c_{λ}(x). For a tableau t(λ) we also speak of the content c_{t(λ)}(m) of a number m which is the content of the node containing m. The hook-length of the node x is h_{λ}(x) := λ_{i} +λ′_{j} −i−j +1. The number d_{λ} of all standard λ-tableau is equal to [5] d_{λ} = n!∏_{x∈[λ]} h_{λ}(x)^{−1}.

A (r, s)-hook-partition is a partition, whose (r + 1)^{th}-term is less than or equal to s. The set of all (r, s)-hook-partition is denoted by Γ^{r,s}, its subset of all partitions, of which the r^{th}-term is greater then or equal to s is denoted by Ω^{r,s}. Γ_{n}^{r,s} := Γ^{r,s} ∩ P_{n}, Ω_{n}^{r,s} := Ω^{r,s} ∩ P_{n}.

The Hecke algebra H_{n} = H_{q,n} is generated by the elements T_{i}, i = 1, 2, ..., n − 1, subject to the relations: T_{i}T_{i+1}T_{i} = T_{i+1}T_{i}T_{i+1}, and T_{i}^{2} = (q − 1)T_{i} + q. It is shown that H_{n} has a basis over 𝕂 {T_{w}|w ∈ 𝔖_{n}}, 𝔖_{n} is the symmetric group, for which T_{(i,i+1)} = T_{i}. If q is not a root of unity of order greater than 1, then H_{n} is semisimple. Thus it is decomposed into a direct sum of its minimal two-sided ideals, which are shown to be parameterized by partitions of n:

H_{n} = ⨁_{λ⊢n} A_{λ}.

A_{λ} itself is a matrix ring over 𝕂, its unit element is the minimal central idempotent F_{λ}. This element is uniquely determined by λ. Dipper and James showed in [4] that the set

E_{t_{i}(λ)} = ∏_{{1≤m≤n \atop 1−m≤k≤m−1} ; k≠c_{t_{i}(λ)}(m)} \frac{L_{m} − [k]_{q}}{[c_{t_{i}(λ)}(m)]_{q} − [k]_{q}} , 1 ≤ i ≤ d_{λ},

is a full set of primitive idempotent of A_{λ}. Their sum is F_{λ}.

References

[1] J. Alfaro, R. Medina, and L.F. Urrutia. The Itzykson-Zuber integral for U(m|n). J. Math. Phys., 36(6):3085–3093, 1995.

[2] J. Alfaro, R. Medina, and L.F. Urrutia. The orthogonality relations for the supergroup U(m|n). J. Phys. A, 28(16):4581–4588, 1995.

[3] C. C. Itzykson and J.B. Zuber. The planar approximation. II. J. Math. Phys, 21:411–421, 1980.

[4] R. Dipper and G. James. Block and Idempotents of Hecke Algebras of General Linear Groups. Proc. London Math. Soc., 54(3):57–82, 1987.

[5] J.A. Green. Polynomial Representations of GL_{n}. Lecture Note in Math., 830, 1980.

[6] Phung Ho Hai. Koszul Property and Poincaré Series of Matrix Bialgebra of Type A_{n}. Journal of Algebra, 192(2):734–748, 1997.

[7] Phung Ho Hai. On Matrix Quantum Groups of Type A_{n}. Preprint ICTP/97101, q-alg/9708007, 1997.

[8] Phung Ho Hai. Characters for Quantum Groups of Type A_{n}. Preprint ICTP,

xxx.lanl.gov/dvi/q-alg/98-7045, 1998.

[9] Phung Ho Hai. Poincaré Series of Quantum Spaces Associated to Hecke Operators. Acta Math. Vietnam., to appear, 1998. Available at xxx.lanl.gov/dvi/q-alg/9711020.

[10] Harish-Chandra. A formula for semisimple lie groups. Amer. J. Math, 79:733–760, 1957.

[11] M. Jimbo. A q-analogue of U(𝔤l(N + 1)), Hecke algebra, and the Yang-Baxter equation. Lett. Math. Phys., 11:247–252, 1986.

[12] Ch Kassel. Quantum Groups, volume 155 of Graduate Texts in Mathematics. Springer-Verlag, 1995. 531p.

[13] I.G. Macdonald. Symmetric functions and the Hall polynomials. Oxford University Press, 1979 (Second edition 1995). New York.

[14] Yu.I. Manin. Multiparametric Quantum Deformation of the General Linear Supergroups. Com. Math. Phys., 123:163–175, 1989.

[15] J.B. Sullivan. The Uniqueness of Integral for Hopf Algebras and Some Existence Theorems of Integrals for Commutative Hopf Algebras. Journal of Algebra, 19:426–440, 1971.

[16] M. Sweedler. Hopf Algebras. Benjamin, New York, 1969.

Hanoi Institute of Mathematics P.O. Box 631, 10000 Bo Ho, Hanoi, Vietnam

E-mail address: phung@ncst.ac.vn

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Bonn, Germany, phung@mpim-bonn.mpg.de