RIGGED CONFIGURATIONS AND THE BETHE ANSATZ

ANNE SCHILLING Department of Mathematics, University of California, One Shields Avenue, Davis, CA 95616-8633, U.S.A. E-mail: anne@math.ucdavis.edu

This note is a review of rigged configurations and the Bethe Ansatz. In the first part, we focus on the algebraic Bethe Ansatz for the spin 1/2 XXX model and explain how rigged configurations label the solutions of the Bethe equations. This yields the bijection between rigged configurations and crystal paths/Young tableaux of Kerov, Kirillov and Reshetikhin. In the second part, we discuss a generalization of this bijection for the symmetry algebra D_{n}^{(1)}, based on work in collaboration with Okado and Shimozono.

1. Introduction

These notes arose from three lectures presented at the Summer School on Theoretical Physics ”Symmetry and Structural Properties of Condensed Matter” held in Myczkowce, Poland, on September 11-18, 2002. We review the algebraic Bethe Ansatz in the simple setting of the spin 1/2 XXX model, explain the physical meaning of rigged configurations and give a bijection between rigged configurations and crystal bases for type D_{n}^{(1)} ^{20} generalizing the bijection of Kerov, Kirillov and Reshetikhin ^{15,16} for type A_{n}^{(1)}.

The Bethe Ansatz originated in a paper by Bethe ^{2} in 1931 in which he studied the eigenvectors and eigenfunctions of the Hamiltonian of the Heisenberg antiferromagnet. The method he used is today often called the coordinate Bethe Ansatz, distinguishing it from the algebraic Bethe Ansatz that will be presented here. The algebraic Bethe Ansatz is a generalization of the coordinate Bethe Ansatz and is one of the most important outcomes of the quantum inverse scattering method introduced in ^{23,24,25}. The quantum inverse scattering method has unified the treatment of quantum integrable systems considering each model as a representation of the quantum monodromy matrix which satisfies certain commutation relations. The algebraic Bethe Ansatz is based on the idea of constructing eigenvectors of the Hamiltonian (resp. trace of the monodromy matrix) by creation and annihilation operators on a vacuum; the elements of the monodromy matrix play the role of these operators. The eigenvectors are parametrized by solutions of a system of algebraic equations, called the Bethe equations. The solutions in turn are labeled by combinatorial objects called rigged configurations.

We consider 𝔤-invariant models where 𝔤 is the symmetry algebra. The Hilbert space is the tensor product of irreducible representations of 𝔤 denoted H = h_{1} ⊗ ⋯ ⊗ h_{N}. The Bethe vectors are the highest weight vectors in the decomposition into irreducible components of H. It is known from representation theory that the highest weight vectors are also labeled by Young tableaux (see for example^{5}) or certain paths in crystal theory (see for example ^{7,8,18}). Assuming the completeness of the Bethe vectors, this suggests a bijection between rigged configurations and Young tableaux/crystal paths. For 𝔤 = 𝔤𝔩_{n} such a bijection was given by Kirillov and Reshetikhin 16 and generalized in ^{17}.

Analogous bijections for all 𝔤 of nonexceptional affine type were recently proven in ^{20} for tensor products of the fundamental representation. An important property of all these bijections is that they preserve statistics that can be defined on the set of rigged configurations and paths, respectively. As a corollary it follows that one-dimensional configuration sums defined in terms of crystal paths have fermionic formulas. Fermionic formulas reflect the quasiparticle structure of the underlying model and also reveal the statistics of the quasiparticles. For general affine Kac-Moody algebras fermionic formulas were conjectured by Hatayama, Kuniba, Okado, Takagi, Tsuboi and Yamada ^{7,8}. For type A_{n}^{(1)} they were proven in ^{17} and for nonexceptional types in special cases in ^{20}.

The paper is organized as follows. In section 2 we review the algebraic Bethe Ansatz for the spin 1/2 XXX model and derive the Bethe equations. In section 3 we present the solutions of the Bethe equations parametrized by rigged configurations and discuss the bijection between rigged configurations and paths in section 4. Sections 2 and 3 follow the presentation of Faddeev ^{4}. In sections 5-8 the bijection between paths and rigged configurations is generalized to types A_{n}^{(1)} and D_{n}^{(1)} based on work in collaboration with Okado and Shimozono ^{20}. Crystal bases are introduced in section 6 and section 7 states the fermionic formula and rigged configurations in the generalized set-up. The bijection is given explicitly in section 8.

2. Bethe Ansatz for the XXX model

In this section we discuss the algebraic Bethe Ansatz for the example of the spin 1/2 XXX Heisenberg chain. This is a one-dimensional quantum spin chain on N sites with periodic boundary conditions. It is defined on the Hilbert space H_{N} = ⨂_{n=1}^{N} h_{n} where in this case h_{n} = ℂ^{2} for all n. Associated to each site is a local spin variable s⃗ = \frac{1}{2} σ⃗ where

σ⃗ = (σ^{1}, σ^{2}, σ^{3}) = (( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , ( \begin{matrix}0 & −i \\ i & 0 \end{matrix} ) , ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ))

are the Pauli matrices. The spin variable acting on the n-th site is given by

s⃗_{n} = I ⊗ ⋯ ⊗ I ⊗ s⃗ ⊗ I ⊗ ⋯ ⊗ I

where I is the identity operator and s⃗ is in the n-th tensor factor. We impose periodic boundary conditions s⃗_{n} = s⃗_{n+N}.

The Hamiltonian of the spin 1/2 XXX model is

H_{N} = J ∑_{n=1}^{N} (s⃗_{n} · s⃗_{n+1} − \frac{1}{4}) .

Our goal is to determine the eigenvectors and eigenvalues of H_{N} in the antiferromagnetic regime J > 0 in the limit when N → ∞.

The main tool will be the Lax operator L_{n,a}(λ), also called the local transition matrix. It acts on h_{n} ⊗ ℂ^{2} where ℂ^{2} is an auxiliary space and is defined as

L_{n,a}(λ) = λI_{n} ⊗ I_{a} + is⃗_{n} ⊗ σ⃗_{a}.

Here I_{n} and I_{a} are unit operators acting on h_{n} and the auxiliary space ℂ^{2}, respectively; λ is a complex parameter, called the spectral parameter. Writing the action on the auxiliary space as a 2 × 2 matrix, we have

L_{n}(λ) = ( \begin{matrix}λ + is_{n}^{3} & is_{n}^{−} \\ is_{n}^{+} & λ − is_{n}^{3} \end{matrix} ) (5)

where s_{n}^{±} = s_{n}^{1} ± is_{n}^{2}.

The crucial fact is that the Lax operator satisfies commutation relations in the auxiliary space V = ℂ^{2}. Altogether there are 16 relations which can be written compactly in tensor notation. Given two Lax operators L_{n,a_{1}}(λ) and L_{n,a_{2}}(μ) defined in the same quantum space h_{n}, but different auxiliary spaces V_{1} and V_{2}, the products L_{n,a_{1}}(λ)L_{n,a_{2}}(μ) and L_{n,a_{2}}(μ)L_{n,a_{1}}(λ) are defined on the triple tensor product h_{n} ⊗ V_{1} ⊗ V_{2}. There exists an operator R_{a_{1},a_{2}}(λ − μ) defined on V_{1} ⊗ V_{2} such that

R_{a_{1},a_{2}}(λ − μ)L_{n,a_{1}}(λ)L_{n,a_{2}}(μ) = L_{n,a_{2}}(μ)L_{n,a_{1}}(λ)R_{a_{1},a_{2}}(λ − μ). (6)

Explicitly, the R-matrix R_{a_{1},a_{2}}(λ) is given by

R_{a_{1},a_{2}}(λ) = (λ + \frac{i}{2}) I_{a_{1}} ⊗ I_{a_{2}} + \frac{i}{2}σ⃗_{a_{1}} ⊗ σ⃗_{a_{2}}.

To deduce the 16 relations explicitly, one may write (6) as matrices in the auxiliary space V_{1} ⊗ V_{2} using the convention (A ⊗ B)_{kℓ}^{ij} = A_{ij}B_{kℓ} where

M_{kℓ}^{ij} = ( \begin{matrix}M_{11}^{11} & M_{12}^{11} & M_{11}^{12} & M_{12}^{12} \\ M_{21}^{11} & M_{22}^{11} & M_{21}^{12} & M_{22}^{12} \\ M_{11}^{21} & M_{12}^{21} & M_{11}^{22} & M_{12}^{22} \\ M_{21}^{21} & M_{22}^{21} & M_{21}^{22} & M_{22}^{22} \end{matrix} ) .

In this notation the R-matrix reads

R(λ) = ( \begin{matrix}a(λ) & 0 & 0 & 0 \\ 0 & b(λ) & c(λ) & 0 \\ 0 & c(λ) & b(λ) & 0 \\ 0 & 0 & 0 & a(λ) \end{matrix} )

where a(λ) = λ + i, b(λ) = λ and c(λ) = i.

Geometrically, the Lax operator L_{n,a}(λ) can be interpreted as the transport between sites n and n + 1 of the quantum spin chain. Hence

T_{N,a}(λ) = L_{N,a}(λ) ⋯ L_{1,a}(λ)

is the monodromy around the circle (recall that we assume periodic boundary conditions). In the auxiliary space write

T_{N}(λ) = ( \begin{matrix}A(λ) & B(λ) \\ C(λ) & D(λ) \end{matrix} )

with entries in the full Hilbert space H_{N}. From (6) it is clear that the monodromy matrix satisfies the following commutation relation

R_{a_{1},a_{2}}(λ − μ)T_{N,a_{1}}(λ)T_{N,a_{2}}(μ) = T_{N,a_{2}}(μ)T_{N,a_{1}}(λ)R_{a_{1},a_{2}}(λ − μ). (12)

Explicitly, some of the relations contained in (12) are

[B(λ), B(μ)] = 0
A(λ)B(μ) = f(λ − μ)B(μ)A(λ) + g(λ − μ)B(λ)A(μ)  (13)
D(λ)B(μ) = h(λ − μ)B(μ)D(λ) + k(λ − μ)B(λ)D(μ)

where

\begin{matrix}f(λ) = \frac{λ−i}{λ} & g(λ) = \frac{i}{λ} \\ h(λ) = \frac{λ+i}{λ} & k(λ) = −\frac{i}{λ}. \end{matrix}

It is well-known ^{4,6,13} that the Hamiltonian is given in terms of the monodromy matrix as

H_{N} = \frac{i}{2} \frac{d}{dλ} ln t_{N}(λ)|_{λ=i/2} − \frac{N}{2}

where t_{N}(λ) = trT_{N}(λ) = A(λ) + D(λ).

Let ω_{n} = ( \begin{matrix}1 \\ 0 \end{matrix} ). In the auxiliary space the Lax operator is triangular on ω_{n}

L_{n}(λ)ω_{n} = ( \begin{matrix}λ + \frac{i}{2} & ∗ \\ 0 & λ − \frac{i}{2} \end{matrix} ) ω_{n}

where ∗ stands for an for us irrelevant quantity. This follows directly from (5). On the Hilbert space H_{N} we define Ω = ⨂_{n} ω_{n} so that

T_{N}(λ)Ω = ( \begin{matrix}α^{N}(λ) & ∗ \\ 0 & δ^{N}(λ) \end{matrix} ) Ω

where α(λ) = λ + \frac{i}{2} and δ(λ) = λ − \frac{i}{2}. Equivalently this means that

C(λ)Ω = 0
A(λ)Ω = α^{N}(λ)Ω
D(λ)Ω = δ^{N}(λ)Ω

so that Ω is an eigenstate of A(λ) and D(λ) and hence also of t_{N}(λ) = A(λ) + D(λ).

The claim is that the other eigenvectors of t_{N}(λ) are of the form

Φ(λ, Λ) = B(λ_{1}) ⋯ B(λ_{n})Ω

where the lambdas Λ = {λ_{1}, …, λ_{n}} satisfy a set of algebraic relations, called the Bethe equations. We will derive these now.

From the commutation relations (13) we find that

A(λ)B(λ_{1}) ⋯ B(λ_{n})Ω = ∏_{k=1}^{n} f(λ − λ_{k})α^{N}(λ)B(λ_{1}) ⋯ B(λ_{n})Ω

+ ∑_{k=1}^{n} M_{k}(λ, Λ)B(λ_{1}) ⋯ B̂(λ_{k}) ⋯ B(λ_{n})B(λ)Ω.

The first term on the right hand side is obtained by using only the first term on the right hand side of (13). The other terms come from a combination of the application of the first and second term when moving A past the B’s. In general the coefficients M_{k}(λ, Λ) are quite involved using the explicit formulas. However, M_{1}(λ, Λ) is obtained by using the second term in (13) moving A(λ) past B(λ_{1}) followed by applications of the first term in (13) only. This yields

M_{1}(λ, Λ) = g(λ − λ_{1}) ∏_{k=2}^{n} f(λ_{1} − λ_{k})α^{N}(λ_{1}).

Note that the B’s commute with each other by (13). Hence M_{j}(λ, Λ) can be obtained from M_{1}(λ, Λ) by replacing λ_{1} by λ_{j} so that

M_{j}(λ, Λ) = g(λ − λ_{j}) ∏_{k≠j^{k=1}}^{n} f(λ_{j} − λ_{k})α^{N}(λ_{j}).

Similarly,

D(λ)B(λ_{1}) ⋯ B(λ_{n})Ω = ∏_{k=1}^{n} h(λ − λ_{k})δ^{N}(λ)B(λ_{1}) ⋯ B(λ_{n})Ω

+ ∑_{k=1}^{n} N_{k}(λ, Λ)B(λ_{1}) ⋯ B̂(λ_{k}) ⋯ B(λ_{n})B(λ)Ω

where

N_{j}(λ, Λ) = k(λ − λ_{j}) ∏_{k≠j^{k=1}}^{n} h(λ_{j} − λ_{k})δ^{N}(λ_{j}).

For Φ(λ, Λ) to be an eigenvector of t_{N}(λ) = A(λ) + D(λ) the terms

∑_{k=1}^{n} M_{k}(λ, Λ)B(λ_{1}) ⋯ B̂(λ_{k}) ⋯ B(λ_{n})B(λ)Ω

+ ∑_{k=1}^{n} N_{k}(λ, Λ)B(λ_{1}) ⋯ B̂(λ_{k}) ⋯ B(λ_{n})B(λ)Ω

need to cancel. Since g(λ − λ_{j}) = −k(λ − λ_{j}) this happens if the set of lambda’s Λ satisfy the following set of equations

∏_{k≠j^{k=1}}^{n} f(λ_{j} − λ_{k})α^{N}(λ_{j}) = ∏_{k≠j^{k=1}}^{n} h(λ_{j} − λ_{k})δ^{N}(λ_{j})

for all j = 1, 2, …, n. Explicitly this reads

(\frac{λ_{j} + \frac{i}{2}}{λ_{j} − \frac{i}{2}} )^{N} = ∏_{k≠j^{k=1}}^{n} \frac{λ_{j} − λ_{k} + i}{λ_{j} − λ_{k} − i} (23)

called the Bethe equations. In this case the eigenvalues of Φ(λ, Λ) are

α^{N}(λ) ∏_{k=1}^{n} f(λ − λ_{k}) + δ^{N}(λ) ∏_{k=1}^{n} h(λ − λ_{k}).

In the next section we will study solutions to (23) in the limit N → ∞.

3. Solutions to the Bethe equations

Let us rewrite (23) in the following way

(\frac{λ + \frac{i}{2}}{λ − \frac{i}{2}} )^{N} = ∏_{λ′≠λ^{λ′∈Λ}} \frac{λ − λ′ + i}{λ − λ′ − i} (25)

where λ ∈ Λ = {λ_{1}, …, λ_{n}}.

Suggested by numerical analysis, it is assumed that in the limit N → ∞ the λ’s form strings. This hypothesis is called the string hypothesis. A string of length ℓ = 2M +1, where M is an integer or half-integer depending on the parity of ℓ, is a set of λ’s of the form

λ_{jm}^{M} = λ_{j}^{M} + im

where λ_{j}^{M} ∈ ℝ and −M ≤ m ≤ M is integer or half-integer depending on M. The index j satisfies 1 ≤ j ≤ m_{ℓ} where m_{ℓ} is the number of strings of length ℓ. A decomposition of {λ_{1}, …, λ_{n}} into strings is called a configuration. Each configuration is parametrized by {m_{ℓ}}. It follows that

∑_{ℓ} ℓm_{ℓ} = n.

Now take (25) and multiply over a string

∏_{m=−M}^{M} (\frac{λ_{j}^{M} + i(m + \frac{1}{2})}{λ_{j}^{M} + i(m − \frac{1}{2})})^{N}

= ∏_{m=−M}^{M} ∏_{(M′,j′,m′)≠(M,j,m)^{M′,j′,m′}} \frac{λ_{j}^{M} − λ_{j′}^{M′} + i(m − m′ + 1)}{λ_{j}^{M} − λ_{j′}^{M′} + i(m − m′ − 1)}. (28)

Many of the terms on the left and right cancel so that this equation can be rewritten as

e^{iNp_{M} (λ_{j}^{M})} = ∏_{(M′,j′)≠(M,j)^{M′,j′}} e^{iS_{MM′} (λ_{j}^{M} −λ_{j′}^{M′})},  (29)

in terms of the momentum and scattering matrix

e^{ip_{M} (λ)} = \frac{λ + i(M + \frac{1}{2})}{λ − i(M + \frac{1}{2})}

e^{iS_{MM′} (λ)} = ∏_{m=|M−M′|}^{M+M′} \frac{λ + im}{λ − im} · \frac{λ + i(m + 1)}{λ − i(m + 1)}.

Taking the logarithm of (29) using the branch cut

\frac{1}{i} ln \frac{λ + ia}{λ − ia} = π − 2 arctan \frac{λ}{a}

we obtain

2N arctan \frac{λ_{j}^{M}}{M + \frac{1}{2}} = 2πQ_{j}^{M} + ∑_{(M′,j′)≠(M,j)^{M′,j′}} Φ_{MM′}(λ_{j}^{M} − λ_{j′}^{M′}),  (31)

where

Φ_{MM′}(λ) = 2 ∑_{m=|M−M′|}^{M+M′} (arctan \frac{λ}{m} + arctan \frac{λ}{m + 1}) .

The first term on the right is absent for m = 0. Here Q_{j}^{M} is an integer or half-integer depending on the configuration.

In addition to the string hypothesis, we assume that the Q_{j}^{M} classify the λ’s uniquely: λ_{j}^{M} increases if Q_{j}^{M} increases and in a given string no Q_{j}^{M} coincide. As we will see shortly with this assumption one obtains the correct number of solutions to the Bethe equations (25).

Using arctan ±∞ = ±\frac{π}{2} we obtain from (31) putting λ_{j}^{M} = ∞

Q_{∞}^{M} = \frac{N}{2} − (2M + \frac{1}{2})(m_{2M+1} − 1) − ∑_{M′≠M}(2 min(M, M′) + 1)m_{2M′+1}.

Since there are 2M + 1 strings in a given string of length 2M + 1, the maximal admissible Q_{max}^{M} is

Q_{max}^{M} = Q_{∞}^{M} − (2M + 1)

where we assume that if Q_{j}^{M} is bigger than Q_{max}^{M} then at least one root in the string is infinite and hence all are infinite which would imply Q_{j}^{M} = Q_{∞}^{M}.

With the already mentioned assumption that each admissible set of quantum number Q_{j}^{M} corresponds uniquely to a solution of the Bethe equations we may now count the number of Bethe vectors. Since arctan is an odd function and by the assumption about the monotonicity we have

−Q_{max}^{M} ≤ Q_{1}^{M} < ⋯ < Q_{m_{2M+1}}^{M} ≤ Q_{max}^{M}.

Hence defining P_{ℓ} as

P_{ℓ} = N − 2 ∑_{ℓ′} min(ℓ, ℓ′)m_{ℓ′}

so that

P_{ℓ} + m_{ℓ} = 2Q_{max}^{M} + 1 with ℓ = 2M + 1.

With this the number of Bethe vectors with configuration {m_{ℓ}} is given by

Z(N, n|\{m_{ℓ}\}) = ∏_{ℓ≥1} ( \begin{matrix}P_{ℓ} + m_{ℓ} \\ m_{ℓ} \end{matrix} )

where ( \begin{matrix}p + m \\ m \end{matrix} ) = (p + m)!/p!m! is the binomial coefficient. The total

number of Bethe vectors is

Z(N, n) = ∑_{∑_{ℓ} ℓm_{ℓ}=n^{\{m_{ℓ}\}}} ∏_{ℓ≥1} ( \begin{matrix}P_{ℓ} + m_{ℓ} \\ m_{ℓ} \end{matrix} ) . (39)

It should be emphasized that the derivation of (39) given here is not mathematically rigorous. Besides the various assumptions that were made we also did not worry about possible singularities of (28). However, as we shall see in the next section, (39) indeed yields the correct number of Bethe vectors.

4. Rigged configurations

In the last section we parametrized the Bethe vectors by solutions to the Bethe equations. As we have seen in section 2 the state space is the tensor product of irreducible representations of the underlying algebra, in our case the tensor product of ℂ^{2} with underlying algebra being 𝔰𝔲(2). The Bethe vectors are the highest weight vectors in the irreducible components in this tensor product.

In this section we will interpret (39) combinatorially in terms of rigged configurations. Since the Bethe vectors are also the irreducible components of the underlying tensor product which can be labeled by Young tableaux or crystal elements, one may expect a bijection between the rigged configurations and crystal elements. For the case A_{n} such a bijection is indeed known to exist ^{15,16,17}. For other types it was recently given in special cases in ^{20}.

To interpret (39) combinatorially let us view the set {m_{ℓ}} as a partition ν. A partition is a set of numbers ν = (ν_{1}, ν_{2}, …) such that ν_{i} ≥ ν_{i+1} and only finitely many ν_{i} are nonzero. The partition has part i if ν_{k} = i for some k. The size of partition ν is |ν| := ν_{1} +ν_{2} +· · ·. In the correspondence between {m_{ℓ}} and ν, m_{ℓ} specifies the number of parts of size ℓ in ν. For example, if m_{1} = 1, m_{2} = 3, m_{4} = 1 and all other m_{ℓ} = 0 then ν = (4, 2, 2, 2, 1).

It is well-known (see e.g. ^{1}) that ( \begin{matrix}p + m \\ m \end{matrix} ) is the number of partitions

in a box of size p × m, meaning, that the partition cannot have more than m parts and no part exceeds p. Let RC(N, n) be the set of all rigged configurations (ν, J) defined as follows. ν is a partition of size |ν| = n and J is a set of partition where J_{ℓ} is a partition in a box of size P_{ℓ} × m_{ℓ}. Then (39) can be rewritten as

Z(N, n) = ∑_{(ν,J)∈RC(N,n)} 1.

Example 4.1. Let N = 5 and n = 2. Then the following is the set of rigged configuration RC(5, 2)

1 1  1 1 1
0 1  1 1 0

0 1 0

The underlying partition on the left is (2) and on the right (1,1). The partitions J_{ℓ} attached to part length ℓ is specified by the numbers in each part. For example, the partition J_{1} for the top rigged configuration on the right is (1,1) whereas for the one in the middle and bottom is J_{1} = (1) and J_{1} = ∅, respectively. The numbers to the right of part ℓ is P_{ℓ}.

There exists a statistics on RC(N, n), called cocharge. It is given by

cc(ν, J) = cc(ν) + ∑_{ℓ} |J_{ℓ}|

where

cc(ν) = ∑_{j,k} min(j, k)m_{j}m_{k}.

For example, the cocharge for the rigged configurations in Example 4.1 from top to bottom, left to right is 3, 2, 6, 5, 4, respectively.

As mentioned before, rigged configurations are in bijection with crystal elements. For our 𝔰𝔲(2) example these are all sequences of 1’s and 2’s of length N such that the number of 2’s never exceeds the number of 1’s reading the sequence from right to left. The last condition is that of Yamanouchi words. The number n fixes the number of 2’s in the sequence. Denote the set of all such sequences by P(N, n). For a path p = p_{N} ⋯ p_{1} ∈ P(N, n) define the energy as

E(p) = ∑_{j=1}^{N−1}(N − j)χ(p_{j+1} > p_{j}) (43)

where χ(True) = 1 and χ(False) = 0. The generating function of paths is given by

X(N, n) = ∑_{p∈P(N,n)} q^{E(p)}.

Example 4.2. The set P(5, 2) is given by

P(5, 2) = {22111, 21211, 12211, 21121, 12121}.

The energies are 2, 4, 3, 5 and 6, respectively. Hence X(5, 2) = q^{2} + q^{3} + q^{4} + q^{5} + q^{6}.

The bijection between P(N, n) and RC(N, n) is defined recursively. A path p = p_{N} ⋯ p_{1} ∈ P(N, n) is built up successively from right to left. The empty path is mapped to the empty rigged configuration. Assume that p_{i−1} ⋯ p_{1} corresponds to (ν^{i−1}, J^{i−1}). If p_{i} = 1, (ν^{i}, J^{i}) = (ν^{i−1}, J^{i−1}). If p_{i} = 2, then add a box to the largest singular string in (ν^{i−1}, J^{i−1}) and make it singular again. A string is singular if its label is equal to the vacancy number, in other words, if J_{ℓ} has a part of size P_{ℓ}. In the final rigged configuration (ν^{N}, J^{N}) take the complement of the partitions J_{ℓ}^{N} in the box P_{ℓ}^{N} × m_{ℓ}^{N}. Let us call this map Ψ : P(N, n) → RC(N, n). We have the following theorem ^{15,16,17}.

Theorem 4.1. The map Ψ : P(N, n) → RC(N, n) is a bijection and E(p) = cc(Ψ(p)) for all p ∈ P(N, n).

Example 4.3. Take p = 21121. We get successively

\begin{matrix}p & (ν, J) \\ ∅ & ∅ \\ 1 & ∅ \\ 21 & 0 0 \\ 121 & 0 1 \\ 1121 & 0 2 \\ 21121 & 1 1 0 \end{matrix}

Hence Ψ(21121) = 1 1 0 . Similarly,

Ψ(22111) = 0 1
Ψ(21211) = 0 1 0
Ψ(12211) = 1 1
Ψ(12121) = 1 1 1

Comparing with examples 4.1 and 4.2, the statistics match.

It follows immediately from Theorem 4.1 that

X(N, n) = ∑_{(ν,J)∈RC(N,n)} q^{cc(ν,J)}.

The q-binomial coefficient

[ \begin{matrix}p + m \\ m \end{matrix} ] = \frac{(q)_{p+m}}{(q)_{p}(q)_{m}} ,

where (q)_{m} = ∏_{i=1}^{m}(1−q^{i}), is the generating function of partitions in a box of size p × m ^{1}. Hence, defining C(N, n) to be the set of all partitions ν of n such that P_{ℓ} ≥ 0 for all ℓ the following corollary holds. The right-hand side is called fermionic formula.

Corollary 4.1.

X(N, n) = ∑_{ν∈C(N,n)} q^{cc(ν)} ∏_{ℓ} [ \begin{matrix}P_{ℓ} + m_{ℓ} \\ m_{ℓ} \end{matrix} ] .

5. Generalizations

So far we have only considered the spin 1/2 XXX model and its counting. This model is based on the fundamental representation of 𝔰𝔲(2). It turns out that the q-counting of Corollary 4.1 is associated with the Kac–Moody Lie algebra A_{1}^{(1)}. In the remainder of this note we will indicate how to generalize the q-counting that arises from the Bethe Ansatz.

The set of paths P(N, n), which is the set of Yamanouchi words in the letters 1 and 2 of length N with n twos, will be generalized to the set of highest weight elements in a tensor product of crystals of a given weight; the Yamanouchi condition is replaced by the highest weight condition and the condition on the number of twos becomes the requirement on the weight. Crystal bases were first introduced by Kashiwara ^{11} in connection with quantized universal enveloping algebras. The quantized universal enveloping algebra U_{q}(𝔤) associated with a symmetrizable Kac–Moody Lie algebra 𝔤 was discovered independently by Drinfeld ^{3} and Jimbo ^{9} in their study of two dimensional solvable lattice models in statistical mechanics. The parameter q corresponds to the temperature of the underlying model. Kashiwara ^{11} showed that at zero temperature or q = 0 the representations of U_{q}(𝔤) have bases, which he coined crystal bases, with a beautiful combinatorial structure and favorable properties such as uniqueness and stability under tensor products.

In the generalization from 𝔰𝔲(2) to other types, rigged configurations become sequences of partitions with riggings. The number of partitions depends on the rank of the underlying algebra.

The generalization of the bijection from paths to rigged configurations to type A_{n}^{(1)} is given in ^{16,17} and to other nonexceptional types in ^{20} in special cases. It was shown in ^{19,21} that all crystals can be realized as crystals of simply-laced type A, D, E. Hence the bijections for these types can be viewed as fundamental.

In the next section we will introduce crystal bases. The bijection algorithm for type A_{n}^{(1)} and D_{n}^{(1)} is presented in section 8.

6. Crystals

6.1. Axiomatic definition of crystals

Let 𝔤 be an affine Lie algebra and I the index set of its Dynkin diagram. Let α_{i}, h_{i}, Λ_{i} (i ∈ I) be the simple roots, simple coroots, and fundamental weights for 𝔤. Let δ = ∑_{i∈I} a_{i}α_{i} denote the standard null root and c = ∑_{i∈I} a_{i}^{∨}h_{i} the canonical central element, where a_{i}, a_{i}^{∨} are the positive integers given in ^{10}. Let P = ⨁_{i∈I} ℤΛ_{i} ⊕ ℤδ be the weight lattice and P^{+} = ∑_{i∈I} ℤ_{≥0}Λ_{i} ⊕ ℤδ the dominant weights.

A crystal B is a set B = ⊔_{λ∈P} B_{λ} (wt b = λ if b ∈ B_{λ}) with the maps

e_{i} : B_{λ} ⟶ B_{λ+α_{i}} ⊔ {0}, f_{i} : B_{λ} ⟶ B_{λ−α_{i}} ⊔ {0},
ε_{i} : B ⟶ ℤ ⊔ {−∞}, φ_{i} : B ⟶ ℤ ⊔ {−∞}

for all i ∈ I such that

for b ∈ B_{λ}, φ_{i}(b) = ⟨h_{i}, λ⟩ + ε_{i}(b),

for b ∈ B, we have

ε_{i}(b) = ε_{i}(e_{i}b) + 1 if e_{i}b ≠ 0,
= ε_{i}(f_{i}b) − 1 if f_{i}b ≠ 0,
φ_{i}(b) = φ_{i}(e_{i}b) − 1 if e_{i}b ≠ 0,
= φ_{i}(f_{i}b) + 1 if f_{i}b ≠ 0,

for b, b′ ∈ B, e_{i}b′ = b if and only if b′ = f_{i}b,

for b ∈ B, ε_{i}(b) = φ_{i}(b) = −∞ implies e_{i}b = f_{i}b = 0.

A crystal B can be regarded as a colored oriented graph by defining

b ⟶^{i} b′ ⟺ f_{i}b = b′.

If we want to emphasize I, B is called an I-crystal.

If B_{1} and B_{2} are crystals, then for b_{1} ⊗ b_{2} ∈ B_{1} ⊗ B_{2} the action of e_{i} is defined as

e_{i}(b_{1} ⊗ b_{2}) = \{ \begin{matrix}e_{i}b_{1} ⊗ b_{2} & if ε_{i}(b_{1}) > φ_{i}(b_{2}), \\ b_{1} ⊗ e_{i}b_{2} & else, \end{matrix}

where ε_{i}(b) = max{k | e_{i}^{k}b is defined} and φ_{i}(b) = max{k | f_{i}^{k}b is defined}. This is the opposite of the notation used by Kashiwara ^{11}.

An element b ∈ B is classically highest weight if e_{i}b = 0 for all i = 1, 2, …, n. For B = B_{L} ⊗ ⋯ ⊗ B_{1} and Λ ∈ P^{+}, the set of paths is defined as follows

P(B, Λ) = {b ∈ B | e_{i}b = 0 for all i = 1, 2, …, n, wtb = Λ}.

In the following we will discuss the crystals of type A_{n}^{(1)} and D_{n}^{(1)} more explicitly.

6.2. Dynkin data of type A_{n} and D_{n}

Let ϵ_{i} be the i-th standard unit vector in ℤ^{n}. Then for type A_{n−1}, the simple roots are

α_{i} = ϵ_{i} − ϵ_{i+1} for 1 ≤ i < n

and the fundamental weights are

Λ_{i} = ϵ_{1} + ⋯ + ϵ_{i} for 1 ≤ i < n.

For type D_{n}, the simple roots are

α_{i} = ϵ_{i} − ϵ_{i+1}  for 1 ≤ i < n
α_{n} = ϵ_{n−1} + ϵ_{n}

and the fundamental weights are

Λ_{i} = ϵ_{1} + ⋯ + ϵ_{i}  for 1 ≤ i ≤ n − 2
Λ_{n−1} = (ϵ_{1} + ⋯ + ϵ_{n−1} − ϵ_{n})/2
Λ_{n} = (ϵ_{1} + ⋯ + ϵ_{n−1} + ϵ_{n})/2.

6.3. Affine crystals of type A_{n}^{(1)} and D_{n}^{(1)}

In ^{8} it is conjectured that there is a family of finite-dimensional irreducible U′_{q}(𝔤)-modules {W_{i}^{(a)} | a ∈ J, i ∈ ℤ_{≥0}} which, unlike most finitedimensional U′_{q}(𝔤)-modules, have crystal bases B^{a,i}. Here U′_{q}(𝔤) is the quantum universal enveloping algebra of the derived subalgebra of 𝔤, obtained by omitting the degree operator, and J = I\{0}.

Here we will restrict our attention to the simplest affine crystals B^{1,1} of type A_{n}^{(1)} and D_{n}^{(1)}. As a set B^{1,1} is {1 < 2 < ⋯ < n + 1} for type A_{n}^{(1)}

and \{1 < 2 < ⋯ < n − 1 < n ; \overline{n} < \overline{n − 1} < ⋯ < \overline{1}\} for type D_{n}^{(1)}. The

crystal graphs are given in Figure 1.

\begin{matrix}A_{n}^{(1)} & ⋯ \\ D_{n}^{(1)} & ⋯  ^{\overline{n}}  ^{\overline{n-1}}  ⋯  ^{\overline{2}}  ^{\overline{1}} \end{matrix}

Figure 1. Crystals B^{1,1}

6.4. One-dimensional sums

The energy function (43) can be generalized to the crystal setting. In the case B = (B^{1,1})^{⊗L} it takes a simple form. There is ^{14} a unique (up to global additive constant) function H : B^{1,1} ⊗ B^{1,1} → ℤ called the local energy function, such that

H(e_{i}(b ⊗ b′)) = H(b ⊗ b′) + \{ \begin{matrix}−1 & if i = 0 and e_{0}(b ⊗ b′) = b ⊗ e_{0}b′ \\ 1 & if i = 0 and e_{0}(b ⊗ b′) = e_{0}b ⊗ b′ \\ 0 & otherwise. \end{matrix}

We normalize H by the condition H(1 ⊗ 1) = 0.

Example 6.1. Let b⊗b′ ∈ B^{1,1}⊗B^{1,1}. Explicitly, the local energy function is given as follows. For type A_{n}^{(1)}, H(b ⊗ b′) = −χ(b > b′). For type D_{n}^{(1)}, H(b ⊗ b′) = 0 if b ≤ b′, H(b ⊗ b′) = −1 if b ⊗ b′ = n ⊗ \overline{n}, \overline{n} ⊗ n or b > b′ where b ⊗ b′ ≠ \overline{1} ⊗ 1, and H(\overline{1} ⊗ 1) = −2.

For b_{L} ⊗ ⋯ ⊗ b_{1} ∈ B = (B^{1,1})^{⊗L}

E(b_{L} ⊗ … ⊗ b_{1}) = ∑_{j=1}^{L−1}(L − j) H(b_{j+1} ⊗ b_{j}).

Define the one-dimensional sum X(B, λ; q) ∈ ℤ[q, q^{−1}] by

X(B, λ; q) = ∑_{b∈P(B,λ)} q^{E(b)}.

Example 6.2. In the crystal language the set of paths of Example 4.2 corresponds to B = (B^{1,1})^{⊗5} of type A_{1}^{(1)} of weight λ = Λ_{1} + 2Λ_{2}.

7. Fermionic formula and rigged configurations

Fermionic formulas associated to a Kac-Moody algebra 𝔤 were conjectured in ^{7,8}. We review the fermionic formulas for type A_{n}^{(1)} and D_{n}^{(1)}.

Let L_{i}^{(a)} with a ∈ J and i ∈ ℤ_{≥0} denote the number of tensor factors B^{a,i} in B and let λ be a dominant integral weight. Say that ν^{•} = (m_{i}^{(a)}) is a (B, λ)-configuration if

∑_{i∈ℤ_{≥0}^{a∈J}} i m_{i}^{(a)}α_{a} = ∑_{i∈ℤ_{≥0}^{a∈J}} i L_{i}^{(a)}Λ_{a} − λ. (58)

The configuration ν^{•} is admissible if all vacancy numbers are nonnegative

p_{i}^{(a)} ≥ 0 for all a ∈ J and i ∈ ℤ_{≥0},

p_{i}^{(a)} = ∑_{k∈ℤ_{≥0}} (L_{k}^{(a)} min(i, k) − ∑_{b∈J}(α_{a}|α_{b}) min(i, k) m_{k}^{(b)}) . (60)

Write C(B, λ) for the set of admissible (B, λ)-configurations. Define

cc(ν^{•}) = \frac{1}{2} ∑_{a,b∈J} ∑_{j,k∈ℤ_{≥0}}(α_{a}|α_{b}) min(j, k)m_{j}^{(a)}m_{k}^{(b)}.

The fermionic formula is defined by

M(B, λ; q) = ∑_{ν^{•}∈C(B,λ)} q^{cc(ν^{•})} ∏_{a∈J} ∏_{i∈ℤ_{≥0}} [ \begin{matrix}p_{i}^{(a)} + m_{i}^{(a)} \\ m_{i}^{(a)} \end{matrix} ] . (62)

The X = M conjecture of ^{7,8} states that

X(B, λ; q^{−1}) = M(B, λ; q).

The fermionic formula M(B, λ) can be interpreted using rigged configurations. Denote by (ν^{•}, J^{•}) a pair where ν^{•} = (m_{i}^{(a)}) is a matrix and J^{•} = (J^{(a,i)}) is a matrix of partitions with a ∈ J and i ∈ ℤ_{≥0}. Then a rigged configuration is a pair (ν^{•}, J^{•}) such that ν^{•} ∈ C(B, λ) and the partition J^{(a,i)} is contained in a m_{i}^{(a)} × p_{i}^{(a)} rectangle for all a, i. The set of rigged (B, λ)-configurations for fixed λ and B is denoted by RC(B, λ). Then (62) is equivalent to

M(B, λ) = ∑_{(ν^{•},J^{•})∈RC(B,λ)} q^{cc(ν^{•},J^{•})}

where cc(ν^{•}, J^{•}) = cc(ν^{•}) + |J^{•}| and |J^{•}| = ∑_{(a,i)} |J^{(a,i)}|. To emphasize the dependence on ν^{•} we also write m_{i}^{(a)}(ν^{•}) and P_{i}^{(a)}(ν^{•}) for m_{i}^{(a)} and p_{i}^{(a)}, respectively.

8. Bijection between rigged configurations and paths

In this section we give the description of the bijection Φ : RC(B, λ) → P(B, λ) for types A_{n}^{(1)} and D_{n}^{(1)} when B = (B^{1,1})^{⊗L}.

Let (ν^{•}, J^{•}) ∈ RC(B, λ). We shall define a map rk : RC(B, λ) → B^{1,1} which associates to (ν^{•}, J^{•}) an element of B^{1,1} called its rank. Denote by RC_{b}(B, λ) the elements of RC(B, λ) of rank b. We shall define a bijection δ : RC_{b}(B, λ) → RC(B̃, λ − wt(b)) where B̃ = (B^{1,1})^{⊗(L−1)}. The disjoint union of these bijections then defines a bijection δ : RC(B, λ) → ⋃_{b∈B^{1,1}} RC(B̃, λ − wt(b)).

The bijection Φ is defined recursively as follows. For b ∈ B^{1,1} let P_{b}(B, λ) be the set of paths in B that have b as leftmost tensor factor. For L = 0 the bijection Φ sends the empty rigged configuration (the only element of the set RC(B, λ)) to the empty path (the only element of P(B, λ)). Otherwise assume that Φ has been defined for B̃ and define it for B by the commutative diagram

RC_{b}(B, λ)  ⟶^{Φ}  P_{b}(B, λ)
δ↓  ↓
RC(B̃, λ − wt(b)) ⟶^{Φ} P(B̃, λ − wt(b))

where the right hand vertical map removes the leftmost tensor factor b. In short,

Φ(ν^{•}, J^{•}) = rk(ν^{•}, J^{•}) ⊗ Φ(δ(ν^{•}, J^{•})).

We also require the bijection Φ̃ : RC(B, λ) → P(B, λ) given by Φ̃ = Φ∘comp where comp : RC(B, λ) → RC(B, λ) with comp(ν^{•}, J^{•}) = (ν^{•}, J̃^{•}) is the function which complements the riggings, meaning that J̃^{•} is obtained from J^{•} by complementing all partitions J^{(a,i)} in the m_{i}^{(a)}(ν^{•}) × P_{i}^{(a)}(ν^{•}) rectangle.

Remark 8.1. The bijection Ψ of section 4 is the inverse of Φ̃ for type A_{1}^{(1)}.

Theorem 8.1. Φ : RC(B, λ) → P(B, λ) is a bijection such that

cc(ν^{•}, J^{•}) = −E(Φ̃(ν^{•}, J^{•})) for all (ν^{•}, J^{•}) ∈ RC(B, λ).

For type A_{n}^{(1)} a generalization of this theorem for B = B^{a_{L},i_{L}} ⊗ ⋯ ⊗ B^{a_{1},i_{1}} was proven in ^{17}. For other types Theorem 8.1 is proved in ^{20}.

To describe the bijection explicitly for types A_{n}^{(1)} and D_{n}^{(1)}, the following notation is needed. The matrix ν^{•} = (m_{i}^{(a)}) can be viewed as a sequence of partitions ν^{•} = (ν^{(1)}, ν^{(2)}, …, ν^{(n)}) where m_{i}^{(a)} is the number of parts of size i in the partition ν^{(a)}. Denote by Q_{i}(ρ) the number of boxes in the first i columns of the partition ρ. Finally the partition J^{(a,i)} is called singular if it has a part of size p_{i}^{(a)}.

8.1. Bijection for type A_{n}^{(1)}

Using the Dynkin data for type A_{n} the vacancy numbers (60) and the constraints (58) can be rewritten in the following explicit way

P_{i}^{(a)}(ν^{•}) = Q_{i}(ν^{(a−1)}) − 2Q_{i}(ν^{(a)}) + Q_{i}(ν^{(a+1)}) + Lδ_{a,1} for 1 ≤ a ≤ n and

|ν^{(a)}| = L − ∑_{b=1}^{a} λ_{b} for 1 ≤ a ≤ n.

The algorithm δ is given as follows. Set ℓ^{(0)} = 0 and repeat the following process for a = 1, 2, …, n or until stopped. Find the minimal index i ≥ ℓ^{(a−1)} such that J^{(a,i)} is singular. If no such i exists, set b = a and stop. Otherwise set ℓ^{(a)} = i and continue with a + 1. If the process did not stop, set b = n + 1. Set all undefined ℓ^{(a)} to ∞.

The new rigged configuration is defined by

m_{i}^{(a)}(ν̃^{•}) = m_{i}^{(a)}(ν^{•}) + \{ \begin{matrix}1 & if i = ℓ^{(a)} − 1 \\ −1 & if i = ℓ^{(a)} \\ 0 & otherwise. \end{matrix}

The partition J̃^{(a,i)} is obtained from J^{(a,i)} by removing a part of size P_{i}^{(a)}(ν^{•}) for i = ℓ^{(a)}, adding a part of size P_{i}^{(a)}(ν̃^{•}) for i = ℓ^{(a)} − 1, and leaving it unchanged otherwise.

Example 8.1. Take B = (B^{1,1})^{⊗7}, λ = Λ_{3} + Λ_{4} and (ν^{•}, J^{•}) ∈ RC(B, λ) as

( 0 0 0 3 3 , 0 0 0 0 , ^{0 0} ).

The algorithm for Φ on comp(ν^{•}, J^{•}) yields

(ν^{•}, J^{•})^{(1)} (ν^{•}, J^{•})^{(2)} (ν^{•}, J^{•})^{(3)} rk
0 0 0 0 3  0 0 0 0  0 0
0 0 2 2 0  0 0 0  0 0  3
0 0 0 2  _{0 0}  ∅  4
1 1 0  _{0 0}  ∅  2
0 1  ∅  ∅  3
0 0  ∅  ∅  1
∅  ∅  ∅  2
∅  ∅  ∅  1

Hence Φ̃(ν^{•}, J^{•}) = b = 3 ⊗ 4 ⊗ 2 ⊗ 3 ⊗ 1 ⊗ 2 ⊗ 1 and E(b) = cc(ν^{•}, J^{•}) = 12. 8.2. Bijection for type D_{n}^{(1)} Using the Dynkin data for type D_{n} the vacancy numbers (60) and the constraints (58) can be rewritten in the following explicit way

P_{i}^{(a)}(ν^{•}) = Q_{i}(ν^{(a−1)}) − 2Q_{i}(ν^{(a)}) + Q_{i}(ν^{(a+1)}) + Lδ_{a,1}  for 1 ≤ a < n − 2
P_{i}^{(n−2)}(ν^{•}) = Q_{i}(ν^{(n−3)}) − 2Q_{i}(ν^{(n−2)}) + Q_{i}(ν^{(n−1)}) + Q_{i}(ν^{(n)})
P_{i}^{(n−1)}(ν^{•}) = Q_{i}(ν^{(n−2)}) − 2Q_{i}(ν^{(n−1)})
P_{i}^{(n)}(ν^{•}) = Q_{i}(ν^{(n−2)}) − 2Q_{i}(ν^{(n)})

and

|ν^{(a)}| = L − ∑_{b=1}^{a} λ_{b}  for 1 ≤ a ≤ n − 2
|ν^{(n−1)}| = \frac{1}{2}(L − ∑_{b=1}^{n−1} λ_{b} + λ_{n})

|ν^{(n)}| = \frac{1}{2}(L − ∑_{b=1}^{n} λ_{b}).

The algorithm δ is given as follows. Set ℓ^{(0)} = 0 and repeat the following process for a = 1, 2, …, n − 2 or until stopped. Find the minimal index i ≥ ℓ^{(a−1)} such that J^{(a,i)} is singular. If no such i exists, set b = a and stop. Otherwise set ℓ^{(a)} = i and continue with a + 1.

If the process has not stopped at a = n − 2 continue as follows. Find the minimal indices i, j ≥ ℓ^{(n−2)} such that J^{(n−1,i)} and J^{(n,j)} are singular. If neither i nor j exist, set b = n − 1 and stop. If i exists, but not j, set ℓ^{(n−1)} = i, b = n and stop. If j exists, but not i, set ℓ^{(n)} = j, b = \overline{n} and stop. If both i and j exist, set ℓ^{(n−1)} = i, ℓ^{(n)} = j and continue with a = n − 2.

Now continue for a = n − 2, n − 3, …, 1 or until stopped. Find the minimal index i ≥ \overline{ℓ}^{(a+1)} where \overline{ℓ}^{(n−1)} = max(ℓ^{(n−1)}, ℓ^{(n)}) such that J^{(a,i)} is singular (if i = ℓ^{(a)} then there need to be two parts of size P_{i}^{(a)}(ν^{•}) in J^{(a,i)}). If no such i exists, set b = \overline{a + 1} and stop. If the process did not stop, set b = \overline{1}.

Set all yet undefined ℓ^{(a)} and \overline{ℓ}^{(a)} to ∞.

The new rigged configuration is defined by

m_{i}^{(a)}(ν̃^{•}) = m_{i}^{(a)}(ν^{•}) + \{ \begin{matrix}1 & if i = ℓ^{(a)} − 1 \\ −1 & if i = ℓ^{(a)} \\ 1 & if i = \overline{ℓ}^{(a)} − 1 and 1 ≤ a ≤ n − 2 \\ −1 & if i = \overline{ℓ}^{(a)} and 1 ≤ a ≤ n − 2 \\ 0 & otherwise \end{matrix}

The partition J̃^{(a,i)} is obtained from J^{(a,i)} by removing a part of size P_{i}^{(a)}(ν^{•}) for i = ℓ^{(a)} and i = \overline{ℓ}^{(a)}, adding a part of size P_{i}^{(a)}(ν̃^{•}) for i = ℓ^{(a)} − 1 and i = \overline{ℓ}^{(a)} − 1, and leaving it unchanged otherwise.

Example 8.2. Take B = (B^{1,1})^{⊗6}, λ = 2Λ_{3} and (ν^{•}, J^{•}) ∈ RC(B, λ) as

( 0 0 0 0 2 , 0 0 0 , ^{0 0} , ^{0 0} ).

Then the algorithm for Φ on comp(ν^{•}, J^{•}) gives the following intermediate steps

(ν^{•}, J^{•})^{(1)} (ν^{•}, J^{•})^{(2)} (ν^{•}, J^{•})^{(3)} (ν^{•}, J^{•})^{(4)} rk
0 0 0 2 2  0 0 0  _{0 0}  0 0

0 0 0  0 0 0 0  _{0 0}  0 0  \overline{4}

0 0 2 2  0 0 0  _{0 0}  0 0  3

1 1  ∅  ∅  ∅  \overline{1}
∅  ∅  ∅  ∅  2
∅  ∅  ∅  ∅  1
∅  ∅  ∅  ∅  1

so that Φ̃(ν^{•}, J^{•}) = b = \overline{4} ⊗ 3 ⊗ \overline{1} ⊗ 2 ⊗ 1 ⊗ 1. The statistics in this case are E(b) = cc(ν^{•}, J^{•}) = 8.

Acknowledgements

Many thanks to Professor T. Lulek and the Organizing Committee for the invitation to the Summer School at Myczkowce and for providing excellent conditions for the meeting. I would also like to thank the Max-Planck-Institut für Mathematik in Bonn and the University of Wuppertal for hosting me while this work was completed. This work was partially supported by the Humboldt foundation and NSF grant DMS-0200774.

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