FINITE-DIMENSIONAL CRYSTALS B^{2,s} FOR QUANTUM AFFINE ALGEBRAS OF TYPE D_{n}^{(1)}

ANNE SCHILLING AND PHILIP STERNBERG

Abstract. The Kirillov–Reshetikhin modules W^{r,s} are finite-dimensional representations of quantum affine algebras U′_{q}(𝔤), labeled by a Dynkin node r of the affine Kac– Moody algebra 𝔤 and a positive integer s. In this paper we study the combinatorial structure of the crystal basis B^{2,s} corresponding to W^{2,s} for the algebra of type D_{n}^{(1)}.

1. Introduction

Quantum algebras were introduced independently by Drinfeld [4] and Jimbo [8] in their study of two dimensional solvable lattice models in statistical mechanics. Since then quantum algebras have surfaced in many areas of mathematics and mathematical physics, such as the theory of knots and links, representation theory, and topological quantum field theory. Of special interest, in particular for lattice models and representation theory, are finitedimensional representations of quantum affine algebras. The irreducible finite-dimensional U′_{q}(𝔤)-modules were classified by Chari and Pressley [2, 3] in terms of Drinfeld polynomials. The Kirillov–Reshetikhin modules W^{r,s}, labeled by a Dynkin node r and a positive integer s, form a special class of these finite-dimensional modules. They naturally correspond to the weight sΛ_{r}, where Λ_{r} is the r-th fundamental weight of 𝔤.

Kashiwara [12, 13] showed that in the limit q → 0 the representations of the quantum algebra U_{q}(𝔤) have very special bases, called crystal bases. This construction makes it possible to study modules over quantum algebras in terms of crystals graphs, which are purely combinatorial objects. However, in general it is not yet known which finite-dimensional representations of affine quantum algebras have crystal bases and what their combinatorial structure is. Recently, Hatayama et al. [5, 6] conjectured that crystal bases B^{r,s} for the Kirillov–Reshetikhin modules W^{r,s} exist. For type A_{n}^{(1)}, the crystals B^{r,s} are known to exist [10], and the explicit combinatorial crystal structure is also well-understood [28]. Assuming that the crystals B^{r,s} exist, their structure for non-simply laced algebras can be described in terms of virtual crystals introduced in [26, 27]. The virtual crystal construction is based on the following well-known algebra embeddings of non-simply laced into simply laced types:

C_{n}^{(1)}, A_{2n}^{(2)}, A_{2n}^{(2)†}, D_{n+1}^{(2)} ↪ A_{2n−1}^{(1)}
A_{2n−1}^{(2)}, B_{n}^{(1)} ↪ D_{n+1}^{(1)}
E_{6}^{(2)}, F_{4}^{(1)} ↪ E_{6}^{(1)}
D_{4}^{(3)}, G_{2}^{(1)} ↪ D_{4}^{(1)}.

2000 Mathematics Subject Classification. Primary 17B37; Secondary 81R10.

Date: August 2004.

Supported in part by the NSF grants DMS-0135345 and DMS-0200774.

The main open problems in the theory of finite-dimensional affine crystals are therefore the proof of the existence of B^{r,s} and the combinatorial structure of these crystals for types D_{n}^{(1)} (n ≥ 4) and E_{n}^{(1)} (n = 6, 7, 8). In this paper, we concentrate on type D_{n}^{(1)}. For irreducible representations corresponding to multiples of the first fundamental weight (indexed by a one-row Young diagram) or any single fundamental weight (indexed by a one-column Young diagram) the crystals have been proven to exist and the structure is known [10, 18]. It is conjectured in [5, 6] that as classical crystals, the crystals B^{r,s} of type D_{n}^{(1)} for r ≤ n − 2 decompose as

B^{r,s} ≅ ⨁_{Λ} B(Λ),

where the direct sum is taken over all weights or partitions Λ obtained from an r × s rectangle by removing any number of 2 × 1 vertical dominoes. Here B(Λ) is the U_{q}(D_{n})- crystal associated with the highest weight representation of highest weight Λ (see [17]). In the sequel, we consider the case r = 2, for which the above direct sum specializes to

(1.1)  B^{2,s} ≅ ⨁_{k=0}^{s} B(kΛ_{2}).

Our approach to study the combinatorics of B^{2,s} is as follows. First, we introduce tableaux of shape (s, s) to define a U_{q}(D_{n})-crystal whose vertices are in bijection with the classical tableaux from the direct sum decomposition (1.1). Using the automorphism of the D_{n}^{(1)} Dynkin diagram which interchanges nodes 0 and 1, we define the unique action of f̃_{0} and ẽ_{0} which makes this crystal into a perfect crystal B̃^{2,s} of level s with an energy function.

We may now state the main result of this paper.

Theorem 1.1. If B^{2,s} exists with the properties as in Conjecture 3.4, then B̃^{2,s} ≅ B^{2,s}.

This is the first step in confirming Conjecture 2.1 of [5], which states that as modules over the embedded classical quantum group, W^{2,s} decomposes as ⨁_{k=0}^{s} V (kΛ_{2}), where V (Λ) is the classical module with highest weight Λ, W^{2,s} has a crystal basis, and this crystal is a perfect crystal of level s.

The paper is structured as follows. In section 2 the definition of quantum algebras, crystal bases and perfect crystals is reviewed. Section 3 is devoted to crystals and the plactic monoid of type D_{n}. The properties of B^{2,s} of type D_{n}^{(1)} as conjectured in [5] are given in Conjecture 3.4. In section 4 the set underlying B̃^{2,s} is constructed in terms of tableaux of shape (s, s) obeying certain conditions. It is shown that this set is in bijection with the union of sets appearing on the right hand side of (1.1). The branching component graph is introduced in section 5, which is used in section 6 to define ẽ_{0} and f̃_{0} on B̃^{2,s}. This makes B̃^{2,s} into an affine crystal. It is shown in section 7 that B̃^{2,s} is perfect and that B̃^{2,s} is the unique perfect crystal having the classical decomposition (1.1) with the approporiate energy function. This proves in particular Theorem 1.1. Finally, we end in section 8 with some open problems.

Acknowledgments. We would like to thank the Max-Planck-Institut für Mathematik in Bonn and the Research Institute for Mathematical Sciences in Kyoto where part of this work was carried out. We would also like to thank Mark Shimozono, Masato Okado, and Mark Haiman for helpful conversations. AS would like to thank the University of British Columbia for their hospitality during her forced exile from the US.

2. Review of quantum groups and crystal bases

2.1. Quantum groups. For n ∈ ℤ and a formal parameter q, we use the notation

[n]_{q} = \frac{q^{n} − q^{−n}}{q − q^{−1}} , [n]_{q}! = ∏_{k=1}^{n}[k]_{q}, and \left[{m \atop n}\right]_{q} = \frac{[m]_{q}!}{[n]_{q}![m − n]_{q}!}.

These are all elements of ℚ(q), called the q-integers, q-factorials, and q-binomial coefficients, respectively.

Let 𝔤 be a Lie algebra with Cartan datum (A, Π, Π^{∨}, P, P^{∨}) and a Dynkin diagram indexed by I. Here A = (a_{ij})_{i,j∈I} is the Cartan matrix, P and P^{∨} are the weight lattice and dual weight lattice, respectively, Π = {α_{i} | i ∈ I} is the set of simple roots and Π^{∨} = {h_{i} | i ∈ I} is the set of simple coroots. Furthermore, let {s_{i} | i ∈ I} be the entries of the diagonal symmetrizing matrix of A and define q_{i} = q^{s_{i}} and K_{i} = q^{s_{i}h_{i}}. Then the quantum enveloping algebra U_{q}(𝔤) is the associative ℚ(q)-algebra generated by e_{i} and f_{i} for i ∈ I, and q^{h} for h ∈ P^{∨}, with the following relations (see e.g. [7, Def. 3.1.1]):

(1) q^{0} = 1, q^{h}q^{h′} = q^{h+h′} for all h, h′ ∈ P^{∨},

(2) q^{h}e_{i}q^{−h} = q^{α_{i}(h)}e_{i} for all h ∈ P^{∨},

(3) q^{h}f_{i}q^{−h} = q^{α_{i}(h)}f_{i} for all h ∈ P^{∨},

(4) e_{i}f_{j} − f_{j}e_{i} = δ_{ij} \frac{K_{i}−K_{i}^{−1}}{q_{i}−q_{i}^{−1}} for i, j ∈ I,

(5) ∑_{k=0}^{1−a_{ij}}(−1)^{k}\left[{1−a_{ij} \atop k}\right]_{q_{i}}e_{i}^{1−a_{ij}−k}e_{j}e_{i}^{k} = 0 for all i ≠ j,

(6) ∑_{k=0}^{1−a_{ij}}(−1)^{k}\left[{1−a_{ij} \atop k}\right]_{q_{i}}f_{i}^{1−a_{ij}−k}f_{j}f_{i}^{k} = 0 for all i ≠ j.

2.2. Crystal bases. The quantum algebra U_{q}(𝔤) can be viewed as a q-deformation of the universal enveloping algebra U(𝔤) of 𝔤. Lusztig [23] showed that the integrable highest weight representations of U(𝔤) can be deformed to U_{q}(𝔤) representations in such a way that the dimension of the weight spaces are invariant under the deformation, provided q ≠ 0 and q^{k} ≠ 1 for all k ∈ ℤ (see also [7]). Let M be a U_{q}(𝔤)-module and A the subset of all elements in ℚ(q) which are regular at q = 0. Kashiwara [12, 13] introduced Kashiwara operators ẽ_{i} and f̃_{i} as certain linear combinations of powers of e_{i} and f_{i}. A crystal lattice L is a free A-submodule of M that generates M over ℚ(q), has the same weight decomposition and has the property that ẽ_{i}L ⊂ L and f̃_{i}L ⊂ L for all i ∈ I. The passage from L to the quotient L/qL is referred to as taking the crystal limit. A crystal basis is a ℚ-basis of L/qL with certain properties.

Axiomatically, we may define a U_{q}(𝔤)-crystal as a nonempty set B equipped with maps wt : B → P and ẽ_{i}, f̃_{i} : B → B ∪ {∅} for all i ∈ I, satisfying

(2.1)  f̃_{i}(b) = b′ ⇔ ẽ_{i}(b′) = b if b, b′ ∈ B
(2.2)  wt(f̃_{i}(b)) = wt(b) − α_{i} if f̃_{i}(b) ∈ B
(2.3)  ⟨h_{i} , wt(b)⟩ = φ_{i}(b) − ϵ_{i}(b).

Here for b ∈ B

\begin{aligned}ϵ_{i}(b) & = max\{n ≥ 0 | ẽ_{i}^{n}(b) ≠ ∅\} \\ φ_{i}(b) & = max\{n ≥ 0 | f̃_{i}^{n}(b) ≠ ∅\}. \end{aligned}

(It is assumed that φ_{i}(b), ϵ_{i}(b) < ∞ for all i ∈ I and b ∈ B.) A U_{q}(𝔤)-crystal B can be viewed as a directed edge-colored graph (the crystal graph) whose vertices are the elements of B, with a directed edge from b to b′ labeled i ∈ I, if and only if f̃_{i}(b) = b′.

Let B_{1} and B_{2} be U_{q}(𝔤)-crystals. The Cartesian product B_{2} × B_{1} can also be endowed with the structure of a U_{q}(𝔤)-crystal. The resulting crystal is denoted by B_{2} ⊗ B_{1} and its elements (b_{2}, b_{1}) are written b_{2} ⊗ b_{1}. (The reader is warned that our convention is opposite to that of Kashiwara [14]). For i ∈ I and b = b_{2} ⊗ b_{1} ∈ B_{2} ⊗ B_{1}, we have wt(b) = wt(b_{1}) + wt(b_{2}),

(2.4)  f̃_{i}(b_{2} ⊗ b_{1}) = \begin{cases}f̃_{i}(b_{2}) ⊗ b_{1} & if ϵ_{i}(b_{2}) ≥ φ_{i}(b_{1}) \\ b_{2} ⊗ f̃_{i}(b_{1}) & if ϵ_{i}(b_{2}) < φ_{i}(b_{1}) \end{cases}

and

(2.5)  ẽ_{i}(b_{2} ⊗ b_{1}) = \begin{cases}ẽ_{i}(b_{2}) ⊗ b_{1} & if ϵ_{i}(b_{2}) > φ_{i}(b_{1}) \\ b_{2} ⊗ ẽ_{i}(b_{1}) & if ϵ_{i}(b_{2}) ≤ φ_{i}(b_{1}). \end{cases}

Combinatorially, this action of f̃_{i} and ẽ_{i} on tensor products can be described by the signature rule. The i-signature of b is the word consisting of the symbols + and − given by

\underbrace{− ⋯ −}_{φ_{i}(b_{2}) times} \underbrace{+ ⋯ +}_{ϵ_{i}(b_{2}) times} \underbrace{− ⋯ −}_{φ_{i}(b_{1}) times} \underbrace{+ ⋯ +}_{ϵ_{i}(b_{1}) times}.

The reduced i-signature of b is the subword of the i-signature of b, given by the repeated removal of adjacent symbols +− (in that order); it has the form

\underbrace{− ⋯ −}_{φ times} \underbrace{+ ⋯ +}_{ϵ times}.

If φ = 0 then f̃_{i}(b) = ∅; otherwise f̃_{i} acts on the tensor factor corresponding to the rightmost symbol − in the reduced i-signature of b. Similarly, if ϵ = 0 then ẽ_{i}(b) = ∅; otherwise ẽ_{i} acts on the leftmost symbol + in the reduced i-signature of b. From this it is clear that

\begin{matrix}φ_{i}(b_{2} ⊗ b_{1}) = φ_{i}(b_{2}) + max(0, φ_{i}(b_{1}) − ε_{i}(b_{2})), \\ ε_{i}(b_{2} ⊗ b_{1}) = ε_{i}(b_{1}) + max(0, −φ_{i}(b_{1}) + ε_{i}(b_{2})). \end{matrix}

2.3. Perfect crystals. Of particular interest is a class of crystals called perfect crystals. Perfect crystals can be used to construct the path realization of highest weight modules [11]. To define them, we need a few preliminary definitions.

Recall that P denotes the weight lattice of an affine Kac-Moody algebra 𝔤. The center of 𝔤 is one-dimensional and is generated by the canonical central element c = ∑_{i∈I} c_{i}h_{i}. Moreover, the imaginary roots of 𝔤 are nonzero integral multiples of the null root δ = ∑_{i∈I} d_{i}α_{i}, where c_{i} and d_{i} are nonnegative integers given in [9]. Define P_{cl} = P/ℤδ, P_{cl}^{+} = {λ ∈ P_{cl}|⟨h_{i}, λ⟩ ≥ 0 for all i ∈ I}, and U′_{q}(𝔤) to be the quantum enveloping algebra with the Cartan datum (A, Π, Π^{∨}, P_{cl}, P_{cl}^{∨}).

Define the set of level ℓ weights to be (P_{cl}^{+})_{ℓ} = {λ ∈ P_{cl}^{+}|⟨c, λ⟩ = ℓ}. For a crystal basis element b ∈ B, define

ε(b) = ∑_{i∈I} ε_{i}(b)Λ_{i} and φ(b) = ∑_{i∈I} φ_{i}(b)Λ_{i}.

Finally, for a crystal basis B, we define B_{min} to be the set of crystal basis elements b such that ⟨c, ε(b)⟩ is minimal over b ∈ B.

Definition 2.1. A crystal B is a perfect crystal of level ℓ if:

(1) B ⊗ B is connected;

(2) there exists λ ∈ P_{cl} such that wt(B) ⊂ λ + ∑_{i≠0} ℤ_{≤0}α_{i} and #(B_{λ}) = 1; (3) there is a finite-dimensional irreducible U′_{q}(𝔤)-module V with a crystal base whose

crystal graph is isomorphic to B;

(4) for any b ∈ B, we have ⟨c, ε(b)⟩ ≥ ℓ;

(5) the maps ε and φ from B_{min} to (P_{cl}^{+})_{ℓ} are bijective.

We use the notation lev(B) to indicate the level of the perfect crystal B.

2.4. Energy function. The existence of an affine crystal structure usually provides an energy function. Let B_{1} and B_{2} be finite U′_{q}(𝔤)-crystals. Then following [11, Section 4] we have:

(1) There is a unique isomorphism of U′_{q}(𝔤)-crystals R = R_{B_{2},B_{1}} : B_{2} ⊗ B_{1} → B_{1} ⊗ B_{2}.

(2) There is a function H = H_{B_{2},B_{1}} : B_{2} ⊗ B_{1} → ℤ, unique up to global additive constant, such that H is constant on classical components and, for all b_{2} ∈ B_{2} and b_{1} ∈ B_{1}, if R(b_{2} ⊗ b_{1}) = b′_{1} ⊗ b′_{2}, then

(2.6) H(ẽ_{0}(b_{2} ⊗ b_{1})) = H(b_{2} ⊗ b_{1}) + \begin{cases}−1 & if ε_{0}(b_{2}) > φ_{0}(b_{1}) and ε_{0}(b′_{1}) > φ_{0}(b′_{2}) \\ 1 & if ε_{0}(b_{2}) ≤ φ_{0}(b_{1}) and ε_{0}(b′_{1}) ≤ φ_{0}(b′_{2}) \\ 0 & otherwise. \end{cases}

We shall call the maps R and H the local isomorphism and local energy function on B_{2} ⊗ B_{1}. The pair (R, H) is called the combinatorial R-matrix.

Let u(B_{1}) and u(B_{2}) be extremal vectors of B_{1} and B_{2}, respectively (see [15] for a definition of extremal vectors). Then

R(u(B_{2}) ⊗ u(B_{1})) = u(B_{1}) ⊗ u(B_{2}).

It is convenient to normalize the local energy function H by requiring that

H(u(B_{2}) ⊗ u(B_{1})) = 0.

With this convention it follows by definition that

H_{B_{1},B_{2}} ∘ R_{B_{2},B_{1}} = H_{B_{2},B_{1}}

as operators on B_{2} ⊗ B_{1}.

We wish to define an energy function D_{B} : B → ℤ for tensor products of perfect crystals of the form B^{r,s} [5, Section 3.3]. Let B = B^{r,s} be perfect. Then there exists a unique element b^{♮} ∈ B such that φ(b^{♮}) = lev(B)Λ_{0}. Define D_{B} : B → ℤ by

(2.7)  D_{B}(b) = H_{B,B}(b ⊗ b^{♮}) − H_{B,B}(u(B) ⊗ b^{♮}).

The intrinsic energy D_{B} for the L-fold tensor product B = B_{L} ⊗ ⋯ ⊗ B_{1} where B_{j} = B^{r_{j},s_{j}} is given by

D_{B} = ∑_{1≤i<j≤L} H_{i}R_{i+1}R_{i+2} ⋯ R_{j−1} + ∑_{j=1}^{L} D_{B_{j}}R_{1}R_{2} ⋯ R_{j−1},

where H_{i} and R_{i} are the local energy function and R-matrix on the i-th and i + 1-th tensor factor, respectively.

3. Crystals and plactic monoid of type D

From now on we restrict our attention to the classical Lie algebra of type D_{n} and the affine Kac-Moody algebra of type D_{n}^{(1)}. Denote by I = {0, 1, …, n} the index set of the Dynkin diagram for D_{n}^{(1)} and by J = {1, 2, …, n} the Dynkin diagram for type D_{n}.

3.1. Dynkin data. For type D_{n}, the simple roots are

\begin{aligned}α_{i} & = ε_{i} − ε_{i+1} for 1 ≤ i < n \\ α_{n} & = ε_{n−1} + ε_{n} \end{aligned} ; (3.1)

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

where ε_{i} ∈ ℤ^{n} is the i-th unit standard vector. The central element for D_{n}^{(1)} is given by

c = h_{0} + h_{1} + 2h_{2} + ⋯ + 2h_{n−2} + h_{n−1} + h_{n}.

3.2. Classical crystals. Kashiwara and Nakashima [17] described the crystal structure of all classical highest weight crystals B(Λ) of highest weight Λ explicitly. For the special case B(kΛ_{2}) as occuring in (1.1) each crystal element can be represented by a tableau of shape λ = (k, k) on the partially ordered alphabet

1 < 2 < ⋯ < n − 1 < n ; n̄ < \overline{n − 1} < ⋯ ¯2 < ¯1

such that the following conditions hold [7, page 202]:

Criterion 3.1.

(1) If ab is in the filling, then a ≤ b;

(2) If {a \atop b} is in the filling, then b ≰ a;

(3) No configuration of the form ^{a} a ; ā or a ; ā _{ā} appears;

(4) No configuration of the form {n−1 \atop n} n ; \overline{n−1} or n−1 ; n̄ n̄ ; \overline{n−1} appears;

(5) No configuration of the form 1 ¯1 appears.

Note that for k ≥ 2, condition 5 follows from conditions 1 and 3.

The crystal B(Λ_{1}) is described pictorially by the crystal graph:

[figure: 1; 1; 2; ⋯; n − 2; n − 1; n − 1; n; n; n; \overline{n}; n − 1; \overline{n − 1}; n − 2; ⋯; \overline{2}; 1; \overline{1}]

For a tableau T = \begin{matrix}a_{1} \\ b_{1} \end{matrix} ⋯ \begin{matrix}a_{s} \\ b_{s} \end{matrix} ∈ B(kΛ_{2}), the action of the Kashiwara operators f̃_{i} and

ẽ_{i} is defined as follows. Consider the column word w_{T} = b_{1}a_{1} ⋯ b_{s}a_{s} and view this word as an element in B(Λ_{1})^{⊗2s}. Then f̃_{i} and ẽ_{i} act by the tensor product rule as defined in section 2.2.

Example 3.2. Let n = 4. Then the tableau

T = 1 2 4 ¯3 ¯3 3 ¯4 ¯4 ¯2 ¯1

has column word w_{T} = 31¯42¯44¯2¯3¯1¯3. The 2-signature of T is + − + − −, derived from the subword 32¯2¯3¯3, and the reduced 2-signature is a single −. Therefore,

f̃_{2}(T) = 1 2 4 ¯3 ¯2 3 ¯4 ¯4 ¯2 ¯1 ,

since the rightmost − in the reduced 2-signature of T comes from the northeastmost ¯3. The 4-signature of T is − + + − ++, derived from the subword 3¯4¯44¯3¯3, and the reduced 4-signature is − + ++, from the subword 3¯4¯3¯3. This tells us that

f̃_{4}(T) = 1 2 4 ¯3 ¯3 ¯4 ¯4 ¯4 ¯2 ¯1 and ẽ_{4}(T) = 1 2 4 ¯3 ¯3 3 3 ¯4 ¯2 ¯1 .

3.3. Dual crystals. Let ω_{0} be the longest element in the Weyl group of D_{n}. The action of ω_{0} on the weight lattice P of D_{n} is given by

\begin{aligned}ω_{0}(Λ_{i}) & = −Λ_{τ(i)} \\ ω_{0}(α_{i}) & = −α_{τ(i)} \end{aligned}

where τ : J → J is the identity if n is even and interchanges n − 1 and n and fixes all other Dynkin nodes if n is odd.

There is a unique involution ∗ : B → B, called the dual map, satisfying

\begin{aligned}wt(b^{∗}) & = ω_{0}wt(b) \\ ẽ_{i}(b)^{∗} & = f̃_{τ(i)}(b^{∗}) \\ f̃_{i}(b)^{∗} & = ẽ_{τ(i)}(b^{∗}). \end{aligned}

The involution ∗ sends the highest weight vector u ∈ B(Λ) to the lowest weight vector (the unique vector in B(Λ) of weight ω_{0}(Λ)). We have

(B_{1} ⊗ B_{2})^{∗} ≅ B_{2} ⊗ B_{1}

with (b_{1} ⊗ b_{2})^{∗} ↦ b_{2}^{∗} ⊗ b_{1}^{∗}.

Explicitly, on B(Λ_{1}) the involution ∗ is given by

i ⟷ \overline{i}

except for i = n with n odd in which case n ↔ n and \overline{n} ↔ \overline{n}. For T ∈ B(Λ) the dual T^{∗} is obtained by applying the ∗ map defined for B(Λ_{1}) to each of the letters of w_{T}^{rev} (the reverse column word of T), and then rectifying the resulting word.

Example 3.3. If

T = 1 1 2 \overline{3} ∈ B(2Λ_{1} + Λ_{2})

we have

T^{∗} = 3 \overline{1} \overline{1} \overline{2} .

3.4. Plactic monoid of type D. The plactic monoid for type D is the free monoid generated by {1, …, n, n̄, …, ¯1}, modulo certain relations introduced by Lecouvey [22]. Note that we write our words in the reverse order compared to [22]. A column word C = x_{L}x_{L−1} ⋯ x_{1} is a word such that x_{i+1} ≰ x_{i} for i = 1, …, L − 1. Note that the letters n and n̄ are the only letters that may appear more than once in C. Let z ≤ n be a letter in C. Then N(z) denotes the number of letters x in C such that x ≤ z or x ≥ z̄. A column C is called admissible if L ≤ n and for any pair (z, z̄) of letters in C with z ≤ n we have N(z) ≤ z. The Lecouvey D equivalence relations are given by:

(1) If x ≠ z̄, then

xzy ≡ zxy for x ≤ y < z and yzx ≡ yxz for x < y ≤ z.

(2) If 1 < x < n and x ≤ y ≤ x̄, then

(x − 1)(\overline{x − 1})y ≡ x̄xy and yx̄x ≡ y(x − 1)(\overline{x − 1}).

(3) If x ≤ n − 1, then

\{ \begin{matrix}nx̄n̄ ≡ nn̄x̄ \\ n̄x̄n ≡ n̄nx̄ \end{matrix} and \{ \begin{matrix}xnn̄ ≡ nxn̄ \\ xn̄n ≡ n̄xn \end{matrix} .

(4)

\{ \begin{matrix}n̄n̄n ≡ n̄(n − 1)(\overline{n − 1}) \\ nnn̄ ≡ n(n − 1)(\overline{n − 1}) \end{matrix} and \{ \begin{matrix}(n − 1)(\overline{n − 1})n̄ ≡ nn̄n̄ \\ (n − 1)(\overline{n − 1})n ≡ n̄nn \end{matrix} .

(5) Consider w a non-admissible column word each strict factor of which is admissible. Let z be the lowest unbarred letter such that the pair (z, z̄) occurs in w and N(z) > z. Then w ≡ w̃ is the column word obtained by erasing the pair (z, z̄) in w if z < n, by erasing a pair (n, n̄) of consecutive letters otherwise.

This monoid gives us a bumping algorithm similar to the Schensted bumping algorithm. It is noted in [22] that a general type D sliding algorithm, if one exists, would be very complicated. However, for tableaux with no more than two rows, these relations provide us with the following relations on subtableaux:

(1) If x ≠ z̄, then

\begin{aligned}& y \\ x & z \end{aligned} ≡ \begin{aligned}x & y \\ & z \end{aligned} ≡ \begin{matrix}x & y \\ z \end{matrix} for x ≤ y < z,

and \begin{aligned}& x \\ y & z \end{aligned} ≡ \begin{matrix}x \\ y & z \end{matrix} ≡ \begin{matrix}x & z \\ y \end{matrix} for x < y ≤ z.

(2) If 1 < x < n and x ≤ y ≤ x̄, then

\begin{aligned}& y \\ x − 1 & \overline{x − 1} \end{aligned} ≡ x − 1 y ; \overline{x − 1} ≡ x y ; x̄

and \begin{aligned}& x \\ y & \overline{x} \end{aligned} ≡ \begin{matrix}x − 1 \\ y & \overline{x − 1} \end{matrix} ≡ \begin{matrix}x − 1 & \overline{x − 1} \\ y \end{matrix} .

(3) If x ≤ n − 1, then

\{  \begin{matrix}n̄ ; n \overline{x} ≡ n̄ ; n x̄ ≡ \begin{matrix}n̄ & x̄ \\ n \end{matrix} \\ \begin{aligned}& n \\ n̄ & \overline{x} \end{aligned} ≡ \begin{matrix}n \\ n̄ & x̄ \end{matrix} ≡ n x̄ ; n̄ \end{matrix}

and \{ \begin{matrix}n̄ ; x n ≡ \begin{aligned}x & n̄ \\ & n \end{aligned} ≡ \begin{matrix}x & n̄ \\ n \end{matrix} \\ \begin{aligned}& n \\ x & \overline{n} \end{aligned} ≡ x n ; n̄ ≡ x n ; n̄ \end{matrix} .

(4)

\{  \begin{matrix}\begin{aligned}& n \\ n̄ & n̄ \end{aligned} ≡ \begin{matrix}n − 1 \\ n̄ & \overline{n − 1} \end{matrix} ≡ n − 1 \overline{n − 1} ; n̄ \\ n̄ ; n n ≡ \begin{matrix}n − 1 \\ n & \overline{n − 1} \end{matrix} ≡ \begin{matrix}n − 1 & \overline{n − 1} \\ n \end{matrix} \end{matrix}

and \{ \begin{matrix}n̄ ; n − 1 \overline{n − 1} ≡ n − 1 n̄ ; \overline{n − 1} ≡ \begin{matrix}n̄ & n̄ \\ n \end{matrix} \\ \begin{aligned}& n \\ n − 1 & \overline{n − 1} \end{aligned} ≡ n − 1 n ; \overline{n − 1} ≡ n n ; n̄ \end{matrix} .

If a word is composed entirely of barred letters or entirely of unbarred letters, only relation (1) (the Knuth relation) applies, and the type A jeu de taquin may be used.

3.5. Properties of B^{2,s}. As mentioned in the introduction, it was conjectured in [5, 6] that there are crystal bases B^{r,s} associated with Kirillov–Reshetikhin modules W^{r,s}. In addition to the existence, Hatayama et al. [5] conjectured certain properties of B^{r,s} which we state here in the specific case of B^{2,s} of type D_{n}^{(1)}.

Conjecture 3.4 ([5]). If the crystal B^{2,s} of type D_{n}^{(1)} exists, it has the following properties:

(1) As a classical crystal B^{2,s} decomposes as B^{2,s} ≅ ⨁_{k=0}^{s} B(kΛ_{2}).

(2) B^{2,s} is perfect of level s.

(3) B^{2,s} is equipped with an energy function D_{B^{2,s}} such that D_{B^{2,s}}(b) = k − s if b is in the component of B(kΛ_{2}) (in accordance with (2.7)).

4. Classical decomposition of B̃^{2,s}

In this section we begin our construction of the crystal B̃^{2,s} mentioned in Theorem 1.1. We do this by defining a U_{q}(D_{n})-crystal with vertices labelled by the set T (s) of tableaux of shape (s, s) which satisfy conditions 1, 2, and 4 of Criterion 3.1. We will construct a bijection between T (s) and the vertices of ⨁_{i=0}^{s} B(iΛ_{2}), so that T (s) may be viewed as a U_{q}(D_{n})-crystal with the classical decomposition (1.1). In section 5 we will define f̃_{0} and ẽ_{0} on T (s) to give it the structure of a perfect U′_{q}(D_{n}^{(1)})-crystal. This crystal will be B̃^{2,s}.

Proposition 4.1. Let T ∈ T (s)\B(sΛ_{2}) with T ≠ 1 ¯1 ⋯ 1 ¯1, and define¯ ī = i for 1 ≤ i ≤ n. Then there is a unique a ∈ {1, …, n, n̄} and m ∈ ℤ_{>0} such that T contains one of the following configurations (called an a-configuration):

{a \atop b_{1}} \underbrace{a ; ā ⋯ a ; ā}_{m} {c_{1} \atop d_{1}} ,  where b_{1} ≠ ā, and c_{1} ≠ a or d_{1} ≠ ā;

{b_{2} \atop c_{2}} \underbrace{a ; ā ⋯ a ; ā}_{m} d_{2} ; ā ,  where d_{2} ≠ a, and b_{2} ≠ a or c_{2} ≠ ā;

{b_{3} \atop c_{3}} \underbrace{a ; ā ⋯ a ; ā}_{m+1} {d_{3} \atop e_{3}} ,  where b_{3} ≠ a and e_{3} ≠ ā.

Proof. If s = 1, the set T (s) \ B(sΛ_{2}) contains only 1 ¯1, so that the statement of the proposition is empty. Hence assume that s ≥ 2. The existence of an a-configuration for some a ∈ {1, …, n, n̄} follows from the fact that T violates condition 3 of Criterion 3.1. The conditions on b_{i}, c_{i}, d_{i} for i = 1, 2, 3 and e_{3} can be viewed as stating that m is chosen to maximize the size of the a-configuration. Condition 1 of Criterion 3.1 on the rows and the conditions on the parameters b_{i}, c_{i}, d_{i}, e_{3} imply that there can be no other a-configurations in T. □

The map D_{2,s} : T (s) → ⨁_{k=0}^{s} B(kΛ_{2}), called the height-two drop map, is defined as follows for T ∈ T (s). If T = 1 ¯1 ⋯ 1 ¯1, then D_{2,s}(T) = ∅ ∈ B(0). If T ∈ B(sΛ_{2}), D_{2,s}(T) = T. Otherwise by Proposition 4.1, T contains a unique a-configuration, and D_{2,s}(T) is obtained from T by removing \underbrace{a ; ā ⋯ a ; ā}_{m}. The result will satisfy Criterion 3.1, and

will therefore be a tableau in B((s − m)Λ_{2}).

Example 4.2. We have

T = 1 2 3 3 \overline{4} \overline{2} \overline{2} \overline{1} , D_{2,4}(T) = 1 3 3 \overline{4} \overline{2} \overline{1} .

The inverse of D_{2,s} is the height-two fill map F_{2,s} : ⨁_{k=0}^{s} B(kΛ_{2}) → T (s). Let t = {a_{1} \atop b_{1}} ⋯ {a_{k} \atop b_{k}} ∈ B(kΛ_{2}). If k = s, F_{2,s}(t) = t. If k < s, then F_{2,s}(t) is obtained by finding a subtableau {a_{i} \atop b_{i}} {a_{i+1} \atop b_{i+1}} in t such that

Criterion 4.3.

b_{i} ≤ ā_{i} ≤ b_{i+1} or a_{i} ≤ b̄_{i+1} ≤ a_{i+1}.

(Recall that ¯ ī = i for i ∈ {1, …, n}.) Note that the first pair of inequalities imply that a_{i} is unbarred, and the second pair of inequalities imply that b_{i+1} is barred. We may therefore insert between columns i and i + 1 of t either the configuration \underbrace{a_{i} ; ā_{i} ⋯ a_{i} ; ā_{i}}_{s−k} or

\underbrace{b̄_{i+1} ; b_{i+1} ⋯ b̄_{i+1} ; b_{i+1}}_{s−k}, depending on which part of Criterion 4.3 is satisfied. We say that i is the

filling location of t. If no such subtableau exists, then F_{2,s} will either append \underbrace{a_{k} ; ā_{k} ⋯ a_{k} ; ā_{k}}_{s−k}

to the end of t, or prepend \underbrace{b̄_{1} ; b_{1} ⋯ b̄_{1} ; b_{1}}_{s−k} to t. In these cases the filling locations are k and 0,

respectively.

Proposition 4.4. The map F_{2,s} is well-defined on ⨁_{i=0}^{s} B(iΛ_{2}).

The proof of this proposition follows from the next three lemmas.

Lemma 4.5. Suppose that t ∈ ⨁_{k=0}^{s−1} B(kΛ_{2}) has no subtableaux {a_{i} \atop b_{i}} {a_{i+1} \atop b_{i+1}} satisfying Cri-

terion 4.3. Then exactly one of either appending a_{k} ; ā_{k} ⋯ a_{k} ; ā_{k} or prepending b̄_{1} ; b_{1} ⋯ b̄_{1} ; b_{1} to t will produce a tableau in T (s) \ B(sΛ_{2}).

Proof. Suppose t = {a_{1} \atop b_{1}} ⋯ {a_{k} \atop b_{k}} ∈ B(kΛ_{2}) is as above for k < s. We will show that if

prepending b̄_{1} ; b_{1} ⋯ b̄_{1} ; b_{1} to t does not produce a tableau in T (s) \ B(sΛ_{2}), then appending a_{k} ; ā_{k} ⋯ a_{k} ; ā_{k} to t will produce a tableau in T (s) \ B(sΛ_{2}). There are two reasons we might

not be able to prepend b̄_{1} ; b_{1} ⋯ b̄_{1} ; b_{1} ; b_{1} may be unbarred, or we may have a_{1} < b̄_{1}.

First, suppose b_{1} is unbarred. If b_{k} is also unbarred, then b_{k} is certainly less than ā_{k}, so we may append a_{k} ; ā_{k} ⋯ a_{k} ; ā_{k} to t. Hence, suppose that b_{k} is barred. We will show that a_{k} is unbarred and ā_{k} > b_{k}.

We know that t has a subtableau of the form {a_{i} \atop b_{i}} {a_{i+1} \atop b_{i+1}} such that b_{i} is unbarred and b_{i+1} is barred. It follows that a_{i} is unbarred, and therefore ā_{i} > b_{i}. Since {a_{i} \atop b_{i}} {a_{i+1} \atop b_{i+1}} does not satisfy Criterion 4.3, this means that ā_{i} > b_{i+1}, which is equivalent to b̄_{i+1} > a_{i}. Once again observing that {a_{i} \atop b_{i}} {a_{i+1} \atop b_{i+1}} does not satisfy Criterion 4.3, this implies that b̄_{i+1} > a_{i+1}; i.e., a_{i+1} is unbarred, and ā_{i+1} > b_{i+1}.

We proceed with an inductive argument on i < j < k. Suppose that {a_{j} \atop b_{j}} {a_{j+1} \atop b_{j+1}} is a subtableau of t such that b_{j} and b_{j+1} are barred, a_{j} is unbarred, and ā_{j} > b_{j}. By reasoning identical to the above, we conclude that

ā_{j} > b_{j+1} ⇒ b̄_{j+1} > a_{j} ⇒ b̄_{j+1} > a_{j+1} ⇒ ā_{j+1} > b_{j+1}, (4.1)

which once again means that a_{j+1} is unbarred.

This inductively shows that a_{k} is unbarred and ā_{k} > b_{k}, so we may append a_{k} ; ā_{k} ⋯ a_{k} ; ā_{k} to t to get a tableau in T (s) \ B(sΛ_{2}). By a symmetrical argument, we conclude that if a_{k} is barred, then we may prepend b̄_{1} ; b_{1} ⋯ b̄_{1} ; b_{1} to t.

Now, suppose that b_{1} is barred and b̄_{1} > a_{1}. This means that a_{1} is unbarred and ā_{1} > b_{1}, so the induction carried out in equation 4.1 applies. It follows that a_{k} is unbarred and ā_{k} > b_{k}, so once again we may append a_{k} ; ā_{k} ⋯ a_{k} ; ā_{k} to t. Also, by a symmetrical argument, when

a_{k} is unbarred and b_{k} > ā_{k}, we may prepend b̄_{1} ; b_{1} ⋯ b̄_{1} ; b_{1} to t. Thus, when no subtableau of t

satisfy Criterion 4.3, either appending a_{k} ; ā_{k} ⋯ a_{k} ; ā_{k} or prepending b̄_{1} ; b_{1} ⋯ b̄_{1} ; b_{1} to t will produce a tableau in T (s) \ B(sΛ_{2}). □

Lemma 4.6. Any tableau t = {a_{1} \atop b_{1}} ⋯ {a_{k} \atop b_{k}} ∈ ⨁_{k=0}^{s−1} B(kΛ_{2}) has no more than two filling locations. If it has two, they are consecutive integers, and this choice has no effect on F_{2,s}(t).

Proof. Let 0 ≤ i_{∗} ≤ k be minimal such that i_{∗} is a filling location of t. First assume that 0 < i_{∗} < k. This implies the existence of a subtableau {a_{i∗} \atop b_{i∗}} {a_{i∗+1} \atop b_{i∗+1}} which satisfies Criterion 4.3.

Suppose that the first condition b_{i_{∗}} ≤ ā_{i_{∗}} ≤ b_{i_{∗}+1} of Criterion 4.3 is satisfied, and consider whether i_{∗} + 1 can be a filling location. If b_{i_{∗}+1} ≤ ā_{i_{∗}+1} ≤ b_{i_{∗}+2}, we have

b_{i_{∗}+1} ≤ ā_{i_{∗}+1} ≤ ā_{i_{∗}} ≤ b_{i_{∗}+1},

which implies that ā_{i_{∗}} = ā_{i_{∗}+1} = b_{i_{∗}+1}, so that t violates part 3 of Criterion 3.1. Similarly, if a_{i_{∗}+1} ≤ b̄_{i_{∗}+2} ≤ a_{i_{∗}+2}, then we have

ā_{i_{∗}+1} ≤ ā_{i_{∗}} ≤ b_{i_{∗}+1} ≤ b_{i_{∗}+2} ≤ ā_{i_{∗}+1},

which also implies that ā_{i_{∗}} = ā_{i_{∗}+1} = b_{i_{∗}+1}, once again violating part 3 of Criterion 3.1. We conclude that if i_{∗} is a filling location for which Criterion 4.3 is satisfied by b_{i_{∗}} ≤ ā_{i_{∗}} ≤ b_{i_{∗}+1}, then i_{∗} + 1 is not a filling location. Furthermore, this argument shows that a_{i_{∗}+1} > a_{i_{∗}} or b_{i_{∗}+1} > ā_{i_{∗}}. By the partial ordering on our alphabet, it follows that t has no other filling locations.

Now, suppose for the filling location i_{∗}, Criterion 4.3 is satisfied by a_{i_{∗}} ≤ b̄_{i_{∗}+1} ≤ a_{i_{∗}+1}. The condition a_{i_{∗}+1} ≤ b̄_{i_{∗}+2} ≤ a_{i_{∗}+2} for i_{∗} + 1 to be a filling location implies that

b̄_{i_{∗}+2} ≤ b̄_{i_{∗}+1} ≤ a_{i_{∗}+1} ≤ b̄_{i_{∗}+2},

which as above leads to a violation of part 3 of Criterion 3.1. However, i_{∗} + 1 may be a filling location if Criterion 4.3 is satisfied by b_{i_{∗}+1} ≤ ā_{i_{∗}+1} ≤ b_{i_{∗}+2}. Note that this inequality implies that a_{i_{∗}+1} ≤ b̄_{i_{∗}+1}, which tells us that a_{i_{∗}+1} = b̄_{i_{∗}+1}. Thus, choosing

to insert b̄_{i∗+1} ; b_{i∗+1} ⋯ b̄_{i∗+1} ; b_{i∗+1} between columns i_{∗} and i_{∗} +1 or to insert a_{i∗+1} ; ā_{i∗+1} ⋯ a_{i∗+1} ; ā_{i∗+1} between

columns i_{∗} + 1 and i_{∗} + 2 does not change F_{2,s}(t). Since i_{∗} + 1 is a filling location with Criterion 4.3 satisfied by b_{i_{∗}} ≤ ā_{i_{∗}} ≤ b_{i_{∗}+1}, the preceding paragraph implies that there are no other filling locations in t.

Finally, suppose that i_{∗} = 0 is a filling location for t; i.e., b_{1} is barred, a_{1} is unbarred, and b̄_{1} ≤ a_{1}. If 1 is a filling location, Criterion 4.3 is satisfied by b_{1} ≤ ā_{1} ≤ b_{2}; otherwise, part 3 of Criterion 3.1 is violated. Put together, this means that ā_{1} = b_{1}, so prepending b̄_{1} ; b_{1} ⋯ b̄_{1} ; b_{1} to t and inserting a_{1} ; ā_{1} ⋯ a_{1} ; ā_{1} between columns 1 and 2 results in the same tableau. As in the above cases, part 3 of Criterion 3.1 and the partial order on the alphabet prohibit any other filling locations. □

Example 4.7. Let s = 4. Then

t = 1 2 3 \overline{4} \overline{2} \overline{1} , F_{2,4}(t) = 1 2 2 3 \overline{4} \overline{2} \overline{2} \overline{1} .

While we could choose either column two or column three as the filling location, either choice results in the same tableau.

Lemma 4.8. If a filling location of t = {a_{1} \atop b_{1}} ⋯ {a_{k} \atop b_{k}} ∈ ⨁_{i=0}^{s−1} B(iΛ_{2}) satisfies Criterion 4.3 with both inequalities, then F_{2,s}(t) is independent of this choice.

Proof. Suppose that i_{∗} ≠ 0, k is a filling location for t where both parts of Criterion 4.3 are satisfied. This means that the subtableau {a_{i∗} \atop b_{i∗}} {a_{i∗+1} \atop b_{i∗+1}} satisfies both ā_{i_{∗}} ≤ b_{i_{∗}+1} and a_{i_{∗}} ≤ b̄_{i_{∗}+1}. The latter of these implies that b_{i_{∗}+1} ≤ ā_{i_{∗}}, so we have ā_{i_{∗}} = b_{i_{∗}+1} and

b̄_{i_{∗}+1} = a_{i_{∗}}. Thus, filling with either a_{i∗} ; ā_{i∗} ⋯ a_{i∗} ; ā_{i∗} or b̄_{i∗+1} ; b_{i∗+1} ⋯ b̄_{i∗+1} ; b_{i∗+1} between columns i_{∗}

and i_{∗} + 1 results in the same tableau F_{2,s}(t). □

Example 4.9. To illustrate, for

t = 2 3 3 \overline{4} \overline{2} \overline{1} we have F_{2,s}(t) = 2 2 3 3 \overline{4} \overline{2} \overline{2} \overline{1} .

By identifying T (s) with ⨁_{i=0}^{s} B(iΛ_{2}) via the maps D_{2,s} and F_{2,s}, we have defined a U_{q}(D_{n})-crystal with the decomposition (1.1), with vertices labelled by the 2 × s tableaux of T (s). The action of the Kashiwara operators ẽ_{i}, f̃_{i} for i ∈ {1, …, n} on this crystal is defined in terms of the above bijection, given explicitly by

\begin{aligned}ẽ_{i}(T) & = F_{2,s}(ẽ_{i}(D_{2,s}(T))) \\ f̃_{i}(T) & = F_{2,s}(f̃_{i}(D_{2,s}(T))), \end{aligned} ; (4.2)

for T ∈ T (s), where the ẽ_{i} and f̃_{i} on the right are the standard Kashiwara operators on U_{q}(D_{n})-crystals [17]. In section 6 we will discuss the action of ẽ_{0} and f̃_{0} on T (s), which will make T (s) into an affine crystal called B̃^{2,s}.

Using the filling and dropping map we obtain a natural inclusion of T (s′) into T (s) for s′ < s.

Definition 4.10. For s′ < s, the map Υ_{s′}^{s} : T (s′) ↪ T (s) is defined by Υ_{s′}^{s} = F_{2,s}∘D_{2,s′}.

5. The branching component graph

The Dynkin diagram of D_{n}^{(1)} has an automorphism interchanging the nodes 0 and 1, which induces a map σ : B^{2,s} → B^{2,s} on the crystals such that ẽ_{0} = σẽ_{1}σ and f̃_{0} = σf̃_{1}σ. With this in mind, suppose we have defined f̃_{0} on T (s) to produce B̃^{2,s}, and consider the following operations on B̃^{2,s}: Let K ⊂ I, and denote by B_{K} the graph which results from removing all k-colored edges from B̃^{2,s} for k ∈ K. Then as directed graphs, we expect B_{{0}} to be isomorphic to B_{{1}}; otherwise, B̃^{2,s} and B^{2,s} will not be isomorphic. We can gain some information about σ by considering B_{{0,1}}. The combinatorial structure of B_{{0,1}} is encoded in the branching component graph to be defined in this section.

5.1. Definitions and preliminary discussion. The connected components of B_{{0,1}} are U_{q}(D_{n−1})-crystals, indexed by partitions as described in this section. The decomposition of B̃^{2,s} into B_{{0,1}} produces a branching component graph for B̃^{2,s}, which we denote BC(B̃^{2,s}). The vertices of this graph correspond to the connected U_{q}(D_{n−1})-crystals; a vertex v_{λ} is labelled (non-uniquely) by the partition λ indicating the classical highest weight of the corresponding U_{q}(D_{n−1})-crystal. The edges of BC(B̃^{2,s}) are defined by placing an edge from v_{λ} to v_{μ} if there is a tableau b ∈ B(v_{λ}) such that f̃_{1}(b) ∈ B(v_{μ}), where B(v_{λ}) denotes the set of tableaux contained in the U_{q}(D_{n−1})-crystal indexed by v_{λ}.

Note that it suffices to describe the decomposition of the component of B̃^{2,s} with U_{q}(D_{n}) highest weight kΛ_{2} into U_{q}(D_{n−1})-crystals for any k ≥ 0, since

BC(⨁_{k=0}^{s} B(kΛ_{2})) = ⨁_{k=0}^{s} BC(B(kΛ_{2})).

Denote the branching component subgraph with classical highest weight kΛ_{2} by BC(kΛ_{2}). Since BC(kΛ_{2}) is determined by the action of ẽ_{i} and f̃_{i} on B(kΛ_{2}) for i = 1, …, n, which is in turn defined by composing the classical Kashiwara operators with D_{2,s} and F_{2,s} (see equation (4.2)), it in fact suffices to determine the structure of BC(sΛ_{2}) ⊂ BC(B̃^{2,s}).

The branching component graph BC(sΛ_{2}) is characterized by the following proposition. We denote by v_{s} the “highest weight” branching component vertex (that is to say the vertex v such that the highest weight vector u_{s} of B(sΛ_{2}) is in B(v)) of BC(sΛ_{2}).

Proposition 5.1. The graph distance from v_{s} defines a rank function on BC(sΛ_{2}). This graph has 2s + 1 ranks, and is symmetric as a non-directed graph over rank s. For j ≤ s, the j^{th} rank contains one of each partition λ = (λ_{1}, λ_{2}) ⊂ (s, j) such that |λ| = s−j+2m for some m ∈ ℤ_{≥0}. For all ranks 0 ≤ j ≤ 2s − 1, a vertex v_{λ} with rank j has an arrow to a vertex v_{μ} with rank j + 1 if and only if λ and μ are joined by an edge in Young’s lattice.

We begin by examining the first few ranks of BC(sΛ_{2}) in detail, then show that this proposition is true in general in sections 5.2 and 5.3.

The highest weight branching component vertex v_{s} is indexed by the one-part partition (s). To see that this is true, simply observe that the highest weight tableau of B(sΛ_{2}) is

\underbrace{{1 \atop 2} ⋯ {1 \atop 2}}_{s}, and acting by f̃_{2}, …, f̃_{n} in the most general possible way will affect only the

bottom row. When we map these bottom row subtableaux componentwise by a ↦ a − 1 and ā ↦ \overline{a − 1} to tableaux of shape (s), and apply the same map to the colors of the arrows, this is clearly isomorphic to the U_{q}(D_{n−1})-crystal with highest weight sΛ_{1}.

Now, consider what can result from acting on a tableau T = {a_{1} \atop b_{1}} ⋯ {a_{s} \atop b_{s}} in B(v_{s}) by f̃_{1}. Since a_{1} = ⋯ = a_{s} = 1, this will turn a_{s} into a 2. There are two cases to consider: if b_{s} = ¯2, this results in a tableau with a configuration 2 ¯2 at the right end (note that f̃_{i}, ẽ_{i} for i = 2, …, n do not act on this subtableau); otherwise, it is a tableau with a_{1} = ⋯ = a_{s−1} = 1 where some element of U_{q}(D_{n−1}) can act on the rightmost column. In either case, we can act with ẽ_{2}, …, ẽ_{n} to find a U_{q}(D_{n−1}) highest weight vector T′ = {a′_{1} \atop b′_{1}} ⋯ {a′_{s} \atop b′_{s}}, where we have b′_{1} = ⋯ = b′_{s−1} = 2; in the first case, we have b′_{s} = ¯2, in the other, we have b′_{s} = 3. Remove those parts of these tableaux on which ẽ_{i} and f̃_{i} for i = 2, …, n do not act; in both cases, we remove a′_{1}, …, a′_{s−1}, and in the first case we also remove the 2 ¯2 at the end. We then have a skew tableau, which when rectified by Lecouvey D equivalence (or, since there are no barred letters remaining, jeu de taquin), is either the tableau 2 ⋯ 2 of shape (s − 1), or the tableau of shape (s, 1) with 2’s in the first row and a 3 in the second. We conclude that there are two vertices of rank 1 in BC(sΛ_{2}), corresponding to the partitions (s − 1) and (s, 1).

Before we generalize this construction, we have a few technical remarks.

The number of 1-arrows in a minimal path in the crystal graph between the highest weight tableau and a tableau T is the “α_{1}-height” of T. Thus, the function

r_{s}(v) = d(v, v_{s}) = min_{P (v,v_{s})}{number of edges in P (v, v_{s})}

where P (v, v_{s}) is the set of all paths from v to v_{s} in BC(sΛ_{2}), is a rank function on BC(sΛ_{2}).

Definition 5.2. A null-configuration of size k is

\underbrace{1 ¯2 ⋯ 1 ¯2 2 ¯1 ⋯ 2 ¯1}_{k}  if k is even,

\underbrace{1 ¯2 ⋯ 1 ¯2 2 ¯2 2 ¯1 ⋯ 2 ¯1}_{k}  if k is odd,

where the number of 1’s equals the number of ¯1’s and the number of 2’s equals the number of ¯2’s.

Null-configurations are named thus because ẽ_{i} and f̃_{i} for i = 2, …, n send T to 0, where T is the 2×s tableau which is a null-configuration of size s. Therefore, T is the basis vector for the trivial representation of U_{q}(D_{n−1}) in BC(sΛ_{2}). Put another way, inserting a null-configuration into a tableau T has no effect on ε_{i}(T) or φ_{i}(T) for i = 2, …, n. This generalizes the phenomenon we observed in the case of 2 ¯2.

5.2. Content of rank j. We now characterize the partitions occuring in any rank 0 ≤ j ≤ s of the branching component graph. (Ranks greater than s will be defined by the ∗-duality of the crystal as defined in section 3.3.) We defer the discussion of the edges of the branching component graph to section 5.3.

Let T ∈ B(sΛ_{2}). We wish to determine the vertex v_{λ} of BC(sΛ_{2}) for which T ∈ B(v_{λ}), and also to determine r_{s}(v_{λ}). As demonstrated for ranks 0 and 1 above, determine the parts of T on which ẽ_{i} and f̃_{i} for i = 2, …, n do not act: this will be a null-configuration of size r_{2} (possibly of size 0), r_{1} many 1’s in the first row before the null-configuration, and r_{3} many ¯1’s in the second row after the null-configuration. We can extract from these data the pair

(5.1)  (t_{1}, t_{2}) = (r_{1} + r_{2}, r_{2} + r_{3}),

where t_{1}, t_{2} ≤ s. By observing the number of times 1 appears in a sequence i_{1}, …, i_{p} such that the highest weight vector of B(sΛ_{2}) is u_{s} = ẽ_{i_{1}} … ẽ_{i_{p}}T, it is easily seen that r_{s}(v_{λ}) = s − t_{1} + t_{2}.

Consider the set J of tableaux such that s − t_{1} + t_{2} = j ≤ s. We wish to determine the partitions λ such that T ∈ J are in a U_{q}(D_{n−1})-crystal with highest weight specified by λ. First, note that |λ| = 2s − t_{1} − t_{2}, since this is precisely the number of boxes where ẽ_{i} and f̃_{i} for i = 2, …, n act non-trivially. It follows that |λ| = s + j − 2t_{2}, so |λ| ≡ s + j (mod 2), and since t_{2} ranges from 0 to j, we have s − j ≤ |λ| ≤ s + j. Based on the definition of ẽ_{i} and f̃_{i} given in section 3.2, it is clear that other than the t_{1} + t_{2} boxes with 1’s, ¯1’s, and the null-configuration, a U_{q}(D_{n−1})-highest weight tableau must have only 2’s and 3’s. We may remove the irrelevant t_{1} + t_{2} boxes from T resulting in a skew tableau T^{#}. All the letters in T^{#} are unbarred, so the Lecouvey relations applied to w_{T^{#}} yield the column word of the rectification of T^{#} (we call this rectified tableau the completely reduced form of T), whose shape has no more than two parts. Let I ⊂ J be the set of U_{q}(D_{n−1})-highest weight tableaux with specified values for t_{1} and t_{2}. Then I includes tableaux where the number of 2’s ranges from s − t_{2} up to min(2s − t_{1} − t_{2}, s), and the number of 3’s ranges simultaneously from s − t_{1} down to max(0, s − t_{1} − t_{2}). The algorithm described above can therefore produce a tableau of any shape λ with two parts such that |λ| = 2s−t_{1} −t_{2}, λ_{1} ≤ s, and λ_{2} ≤ s−t_{1} = j −t_{2}. By properties of the plactic monoid, no two U_{q}(D_{n−1})-highest weight tableaux in J correspond to the same partition.

To summarize: In rank j ≤ s of BC(sΛ_{2}), the vertices correspond exactly to partitions λ = (λ_{1}, λ_{2}) ⊂ (s, j) such that |λ| = s − j + 2m for some m ∈ ℤ_{≥0}.

By the ∗-symmetry of B(sΛ_{2}) as described in section 3.3, it is clear that the U_{q}(D_{n−1})- crystals of rank j are the same as the U_{q}(D_{n−1})-crystals of rank 2s − j. This completely characterizes the vertices of BC(sΛ_{2}) by rank, and leads us to the following remark.

Remark 5.3. If we consider the embedding U_{q}(D_{n−1}) ↪ U_{q}(D_{n}) as implicitly described above, and think of the action of e_{1}, f_{1} ∈ U_{q}(D_{n}) as specifying a rank function on the embedded U_{q}(D_{n−1})-modules in a given U_{q}(D_{n})-module with highest weight sΛ_{2}, this provides a combinatorial proof that the ranks are multiplicity-free.

5.3. Edges of BC(sΛ_{2}). We must now confirm that the pairs of vertices which have an arrow between them are precisely those v_{λ} and v_{μ} such that r_{s}(v_{λ}) = j and r_{s}(v_{μ}) = j +1 for some 0 ≤ j ≤ 2s − 1, and for which λ and μ are adjacent in Young’s lattice, that is, μ is obtained from λ by either adding or removing a box. To do this, we will construct tableaux in B(v_{λ}) such that the shape of the completely reduced form of their image under f̃_{1} is the result of adding a box to λ. The question of removing boxes from λ then is simply a matter of appealing to the ∗-symmetry of the crystal graph as described in section 3.3.

Our analysis breaks into two cases, where our tableau T ∈ B(v_{λ}) may be of one of the following two forms:

(1)

T = \underbrace{{1 \atop b_{1}} · · · · · · {1 \atop b_{r_{1}}}}_{r_{1}} \underbrace{1 ¯2 ⋯ 1 ¯2(2 ¯2)2 ¯1 ⋯ 2 ¯1}_{r_{2}} \underbrace{a_{s−r_{3}+1} ; ¯1 ⋯ a_{s} ; ¯1}_{r_{3}},

(2)

T = \underbrace{{1 \atop b_{1}} ⋯ {1 \atop b_{r_{1}}}}_{r_{1}} \underbrace{{a_{r_{1}+1} \atop b_{r_{1}+1}} ⋯ {a_{s−r_{3}} \atop b_{s−r_{3}}}}_{u=s−r_{1}−r_{3}} \underbrace{a_{s−r_{3}+1} ; ¯1 ⋯ a_{s} ; ¯1}_{r_{3}},

where in case (1), the block of length r_{2} is a maximal null-configuration, and in case (2), a_{r_{1}+1} ≠ 1 and b_{s−r_{3}} ≠ ¯1 (we set r_{2} = 0 here). We now determine for which partitions μ we can have f̃_{1}(T) ∈ B(v_{μ}). Recall from the previous subsection that for T ∈ B(v_{λ}), we defined T^{#} to be the skew tableau which results from removing all 1’s, ¯1’s, and nullconfigurations from T. Observe that

w_{T^{#}} = \begin{cases}b_{1} ⋯ b_{r_{1}}a_{s−r_{3}+1} ⋯ a_{s} & for case (1) \\ b_{1} ⋯ b_{r_{1}}b_{r_{1}+1}a_{r_{1}+1} ⋯ b_{s−r_{3}}a_{s−r_{3}}a_{s−r_{3}+1} ⋯ a_{s} & for case (2). \end{cases}

In either case, if b_{r_{1}} = ¯2, the size of the null-configuration in f̃_{1}(T) is r_{2} + 1, since in case (1) f̃_{1} acts on the middle of the null-configuration, and in case (2) f̃_{1} acts on a_{r_{1}} = 1. It follows that w_{f̃_{1}(T)^{#}} is simply w_{T^{#}} with the ¯2 contributed by b_{r_{1}} removed. If b_{r_{1}} ≠ ¯2, we see that w_{f̃_{1}(T)^{#}} is w_{T^{#}} with a 2 inserted from a_{s−r_{3}} in case (1), and from a_{r_{1}} in case (2). Since we are currently concerned with adding boxes to λ, let us assume that b_{r_{1}} ≠ ¯2, and analyze how inserting a 2 as above affects the shape of the rectifications of these words.

Our augmented words are

(5.2) w_{f̃_{1}(T)^{#}} = \begin{cases}b_{1} ⋯ b_{r_{1}}2 a_{s−r_{3}+1} ⋯ a_{s} & for case (1) \\ b_{1} ⋯ b_{r_{1}}2 b_{r_{1}+1}a_{r_{1}+1} ⋯ b_{s−r_{3}}a_{s−r_{3}}a_{s−r_{3}+1} ⋯ a_{s} & for case (2). \end{cases}

Recall that we have assumed that b_{r_{1}} ≠ ¯2, which in turn implies that all letters b_{1}, …, b_{r_{1}} are strictly less than ¯2. Using relation (1) of Lecouvey type D equivalence, we may therefore move the 2 from position a_{r_{1}} to the second position in the word. This new word begins b_{1}2b_{2}, with b_{2} > 2. Since we may view all the plactic operations on this word as sliding moves, the subword b_{2} ⋯ a_{s} can be rectified to give a tableau with no more than two rows. Thus, all we have done is added one box to our shape.

We now show that this process can add a box to the top row of λ unless λ_{1} = s, and it can add a box to the bottom row unless λ_{2} = λ_{1}. In the U_{q}(D_{n−1})-crystal B(v_{λ}), we know that there is a U_{q}(D_{n−1}) highest weight tableau T_{λ} of the form

T_{λ} = \begin{cases}\underbrace{{1 \atop 2} ⋯ {1 \atop 2}}_{r_{1}−λ_{2}} \underbrace{{1 \atop 3} ⋯ {1 \atop 3}}_{λ_{2}} \underbrace{1 ¯2 ⋯ 1 ¯2(2 ¯2)2 ¯1 ⋯ 2 ¯1}_{r_{2}} \underbrace{2 ¯1 ⋯ 2 ¯1}_{r_{3}} & for case (1) \\ \underbrace{{1 \atop 2} ⋯ {1 \atop 2}}_{r_{1}−(λ_{2}−u)} \underbrace{{1 \atop 3} ⋯ {1 \atop 3}}_{λ_{2}−u} \underbrace{{2 \atop 3} ⋯ {2 \atop 3}}_{u} \underbrace{2 ¯1 ⋯ 2 ¯1}_{r_{3}} & for case (2). \end{cases}

Note that in case (1) we have λ_{2} ≤ r_{3} and in case (2) we have λ_{2} − u ≤ r_{3}; otherwise, acting by ẽ_{2} can turn another 3 into a 2.

These tableaux yield the words

w_{T_{λ}^{#}} = \begin{cases}\underbrace{2 ⋯ 2}_{r_{1}−λ_{2}} \underbrace{3 ⋯ 3}_{λ_{2}} \underbrace{2 ⋯ 2}_{r_{3}} & for case (1) \\ \underbrace{2 ⋯ 2}_{r_{1}−(λ_{2}−u)} \underbrace{3 ⋯ 3}_{λ_{2}−u} \underbrace{32 ⋯ 32}_{2u} \underbrace{2 ⋯ 2}_{r_{3}} & for case (2). \end{cases}

The completely reduced form of these tableaux is a two-row tableau with r_{1} + r_{3} − λ_{2} 2’s in the top row and λ_{2} 3’s in the bottom row, or 2u + r_{1} + r_{3} − λ_{2} 2’s in the top row and λ_{2} 3’s in the bottom row, respectively. It is easy to see that by adding a 2 to w_{T^{#}} as in (5.2), we simply add a box containing a 2 to the top row of the completely reduced form of T_{λ}. Note that this procedure fails precisely when T_{λ} can have no 2’s added to it, in which case there are s 2’s in T_{λ}, and thus λ_{1} = s.

Now suppose that λ_{1} − λ_{2} > 0, so that adding a box to the second row will produce a legal diagram. Consider T̃_{λ} = f̃_{2}^{λ_{1}−λ_{2}}(T_{λ}) (note that λ_{1} is the number of 2’s in T_{λ}). This tableau is in B(v_{λ}), so its completely reduced form has shape λ, and we see that

wT̃_{λ}^{#} = \begin{cases}\underbrace{3 ⋯ 3}_{|λ|−r_{3}} \underbrace{2 ⋯ 2}_{λ_{2}} \underbrace{3 ⋯ 3}_{r_{3}−λ_{2}} & for case (1) \\ \underbrace{3 ⋯ 3}_{|λ|−r_{3}−2u} \underbrace{32 ⋯ 32}_{2u} \underbrace{2 ⋯ 2}_{λ_{2}−u} \underbrace{3 ⋯ 3}_{r_{3}−λ_{2}+u} & for case (2). \end{cases}

The rectified tableau has λ_{2} 2’s followed by λ_{1} − λ_{2} 3’s in the top row, and λ_{2} 3’s in the bottom row. From this description, we see that adding a 2 to wT̃_{λ}^{#} as in (5.2) affects the completely reduced tableau by preventing one of the 3’s from the bottom row from being slid up to the top row; i.e., λ_{2} is increased by 1. Since we add only one box at a time and the only shape in rank 0 is (s, 0), we know that the number of boxes in the second row can never exceed the rank.

We now invoke the ∗-duality of the crystal graph to deal with how boxes can be removed from λ. If v_{λ} ∈ BC(sΛ_{2}) has rank p, there is a unique vertex v′_{λ}, called the complementary vertex of v_{λ}, with rank 2s−p for which the corresponding U_{q}(D_{n−1})-crystal is B(λ). This involution agrees with the ∗-crystal involution of section 3.3. We wish to show that there is an arrow from v_{λ} to v_{μ}, where λ/μ is a single box and r_{s}(v_{μ}) = r_{s}(v_{λ}) + 1. Recall that by definition, this is the case when for some T ∈ B(v_{λ}) we have f̃_{1}(T) ∈ B(v_{μ}). Observe that r_{s}(v′_{λ}) = r_{s}(v′_{μ}) + 1, and λ is the result of adding a box to μ; therefore, there is an arrow from v′_{μ} to v′_{λ}. It follows that we can find some T ∈ B(v′_{μ}) such that f̃_{1}(T) ∈ B(v′_{λ}). In turn, we have T^{∗} ∈ B(v_{μ}) and (f̃_{1}(T))^{∗} = ẽ_{1}(T^{∗}) ∈ B(v_{λ}). Since we know that f̃_{1}(ẽ_{1}(T^{∗})) = T^{∗} ∈ B(v_{μ}), we have shown that there is an arrow from v_{λ} to v_{μ}.

The arguments of sections 5.2 and 5.3 prove Proposition 5.1.

5.4. Construction of BC(B̃^{2,s}). Observe that BC(B̃^{2,s}) = ⋃_{i=0}^{s} BC(iΛ_{2}). Let v_{λ} ∈ BC(iΛ_{2}) ⊂ BC(B̃^{2,s}). Define R(v_{λ}) = r_{i}(v_{λ}) + s − i. This defines a rank on all of BC(B̃^{2,s}). Note that BC(iΛ_{2}) ⊂ BC((i + 1)Λ_{2}), and this inclusion is compatible with R. Also note that if R(v_{λ}) = p, then v′_{λ}, the complementary vertex to v_{λ}, is now defined to be the vertex of rank 2s − p with the same shape and in the same component as v_{λ}.

To illustrate, BC(B̃^{2,2}) is given in Figure 1, with rank 0 in the first line, rank 1 in the second, etc.

6. Affine Kashiwara operators

Since we know that B_{{0}} and B_{{1}} are isomorphic as directed graphs, it is clear that we can put 0-colored edges in the branching component graph in such a way that interchanging the 1-edges and the 0-edges and applying some shape-preserving bijection σ̌ to the vertices of the branching component graph will produce an isomorphic colored directed graph. Such a bijection can be naturally extended to σ : B̃^{2,s} → B̃^{2,s} as follows. Let b ∈

[figure: ∅]

[figure: ∅; ∅]

Figure 1. Branching component graph BC(B̃^{2,2})

B(v_{λ}) ⊂ B̃^{2,s} for some branching component vertex v_{λ}, and let u_{λ} denote the U_{q}(D_{n−1})- highest weight vector of B(v_{λ}). Then for some finite sequence i_{1}, …, i_{k} of integers in {2, …, n}, we know that f̃_{i_{1}} ⋯ f̃_{i_{k}}u_{λ} = b. Let v_{λ}^{†} = σ̌(v_{λ}), and let u_{λ}^{†} be the highest weight vector of B(v_{λ}^{†}). We may define σ(b) = f̃_{i_{1}} ⋯ f̃_{i_{k}}u_{λ}^{†}. This involution of B̃^{2,s} allows us to define the affine structure of the crystal by the following equations:

(6.1)  f̃_{0} = σf̃_{1}σ  and  ẽ_{0} = σẽ_{1}σ.

Definition 6.1. The affine crystal B̃^{2,s} is given by the set T (s) as defined in section 4 with ẽ_{i}, f̃_{i} for 1 ≤ i ≤ n as in (4.2) and ẽ_{0}, f̃_{0} as in (6.1).

Using σ̌ as defined in section 6.1, it will be shown in section 7 that the resulting U′_{q}(D_{n}^{(1)})-crystal B̃^{2,s} is perfect.

6.1. Construction of σ̌. We will define σ̌(v_{λ}) for R(v_{λ}) ≤ s, and observe that σ̌(v′_{λ}) = σ̌(v_{λ})′, where v′ denotes the complementary vertex of v. Let v_{λ} ∈ BC(kΛ_{2}), R(v_{λ}) = p, and ℓ be minimal such that ι̌_{k}^{s}(v_{λ}) ∈ ι̌_{ℓ}^{s}(BC(ℓΛ_{2})), where ι̌_{i}^{j} is the embedding of BC(iΛ_{2}) in BC(jΛ_{2}) for i < j. Then by the inclusion BC(iΛ_{2}) ⊂ BC((i+1)Λ_{2}) for i = 0, …, s−1, there are s − ℓ + 1 vertices of the same shape as v_{λ} of rank p in BC(B̃^{2,s}), one in each BC(jΛ_{2}) for j = ℓ, …, s. We define σ̌(v_{λ}) to be the vertex of the same shape as v_{λ} of rank 2s − p in the component BC((s + ℓ − k)Λ_{2}).

The action of σ̌ on BC(B̃^{2,2}) is given in Figure 2.

6.2. Combinatorial construction of σ. We can also give a direct combinatorial description of σ(T) for any T ∈ B̃^{2,s}. As an auxilliary construction (which will also be useful in its own right later on), we combinatorially describe ι_{i}^{j} : B(iΛ_{2}) ↪ B(jΛ_{2}), the unique crystal embedding that agrees with ι̌_{i}^{j} : BC(iΛ_{2}) ↪ BC(jΛ_{2}).

Remark 6.2. It will often be useful to identify B(kΛ_{2}) with its image in B̃^{2,s}. We will use the notation T ∈ B(kΛ_{2}) ⊂ B̃^{2,s} to indicate this identification.

Let i ∈ {0, …, s − 1}, so ι_{i}^{i+1} denotes the embedding of B(iΛ_{2}) in B((i + 1)Λ_{2}). Let T ∈ B(iΛ_{2}) ⊂ B̃^{2,s}. This embedding can be combinatorially understood through the following observations:

∅

[figure: ∅; ∅]

Figure 2. Definition of σ̌ on BC(B̃^{2,2})

Remark 6.3.

• φ_{k}(T) = φ_{k}(ι_{i}^{i+1}(T)) and ε_{k}(T) = ε_{k}(ι_{i}^{i+1}(T)) for k = 2, …, n;

• D_{2,s}(ι_{i}^{i+1}(T)) has one more column than D_{2,s}(T) (recall D_{2,s} from section 4);

• Let v(T) be the branching component vertex containing T. Then R(v(T)) = R(v(ι_{i}^{i+1}(T))), so the rank of v(ι_{i}^{i+1}(T)) in BC((i+1)Λ_{2}) is one greater than the rank of v(T) in BC(iΛ_{2}).

In other words, we know that T has a maximal a-configuration of size s − i (section 4), and has completely reduced form T^{#} (section 5). Furthermore, let r_{1T} be the number of 1’s in the first row of D_{2,s}(T) to the left of a null-configuration, and similarly define r_{2T} , r_{3T} , t_{1T}, and t_{2T} as in (5.1). Then the rank of v(T) in BC(iΛ_{2}) is i − t_{1T} + t_{2T}. We wish to construct a tableau S with an a-configuration of size s − i − 1 such that S^{#} = T^{#} and (i + 1) − t_{1S} + t_{2S} = (i − t_{1T} + t_{2T}) + 1; i.e., t_{1S} − t_{2S} = t_{1T} − t_{2T}. Based on properties of the height 2 type D sliding algorithm of section 3.4, these conditions can only be satisfied when t_{jS} = t_{jT} + 1 for j = 1, 2.

We can calculate ι_{i}^{i+1}(T) by the following algorithm:

Algorithm 6.4.

(1) Remove the a-configuration of size s − i from T and slide it to get a 2 × i tableau.

(2) Remove the 1’s, ¯1’s and the null-configuration from the result to get a skew tableau of shape (i, i − t_{2T})/(t_{1T}).

(3) Using the type D sliding algorithm, produce a skew tableau of shape ((i + 1), (i + 1) − (t_{2T} + 1))/(t_{1T} + 1).

(4) Fill this tableau with 1’s, ¯1’s, and a null-configuration so that the result is a 2 × (i + 1) tableau.

(5) Use the height 2 fill map F_{2,s} (section 4) to insert s − i − 1 columns into the tableau.

This produces the unique tableau satisfying the three properties of Remark 6.3.

Example 6.5. Let

T = 1 1 2 2 2 ¯3 ¯2 2 2 3 ¯2 ¯2 ¯2 ¯1 ∈ B(5Λ_{2}) ⊂ B̃^{2,7}.

Running through the steps of our algorithm (using relation (2) of section 3.4 for step (3)) gives us

(1)  1 1 2 ¯3 ¯2 2 2 3 ¯2 ¯1

(2)  2 ¯3 ¯2 2 2 3 ¯2

(3)  3 ¯3 ¯2 2 2 3 ¯3

(4)  1 1 1 3 ¯3 ¯2 2 2 3 ¯3 ¯1 ¯1

(5) ι_{5}^{6}(T) = 1 1 1 3 3 ¯3 ¯2 2 2 3 ¯3 ¯3 ¯1 ¯1 ∈ B(6Λ_{2}) ⊂ B̃^{2,7}.

Composing these maps gives us the following algorithm for calculating ι_{i}^{j}(T), where T ∈ B(iΛ_{2}) and s ≥ j > i.

Algorithm 6.6.

(1) Remove the a-configuration of size s − i from T and slide it to get a 2 × i tableau.

(2) Remove the 1’s, ¯1’s and the null-configuration from the result to get a skew tableau of shape (i, i − t_{2T})/(t_{1T}).

(3) Using the type D sliding algorithm, produce a skew tableau of shape ((j), (j) − (t_{2T} + (j − i)))/(t_{1T} + (j − i)).

(4) Fill this tableau with 1’s, ¯1’s, and a null-configuration so that the result is a 2 × j tableau.

(5) Use the height 2 fill map F_{2,s} (section 4) to insert s − j columns into the tableau.

We can also define a map ι_{i}^{j} : B(iΛ_{2}) → B(jΛ_{2}) ∪ {0} for j < i by

ι_{i}^{j}(T) = \begin{cases}(ι_{j}^{i})^{−1}(T) & if T ∈ ι_{j}^{i}(B(jΛ_{2})), \\ 0 & otherwise . \end{cases}

Reversing the above algorithm makes this map explicit. Lastly, we define ι_{i}^{i} to be the identity map on B(iΛ_{2}), so ι_{i}^{j} is defined for all i, j ∈ {0, …, s}.

We have already observed that by the ∗-duality of B(kΛ_{2}) ⊂ B̃^{2,s}, each vertex v_{λ} ∈ B(kΛ_{2}) has a complementary vertex v′_{λ} ∈ B(kΛ_{2}) such that R(v_{λ}) + R(v′_{λ}) = 2s. We define the involution ∗_{BC} on B̃^{2,s} as follows: Let T ∈ B(v_{λ}) such that T = f̃_{i_{1}} ⋯ f̃_{i_{m}}u_{λ}, where u_{λ} is the U_{q}(D_{n−1})-highest weight tableau of B(v_{λ}). Then T^{∗_{BC}} = f̃_{i_{1}} ⋯ f̃_{i_{m}}u′_{λ}, where u′_{λ} is the U_{q}(D_{n−1})-highest weight tableau of B(v′_{λ}). Alternatively, this map is the composition of ∗ with the “local ∗” map, which applies only to the tableaux in B(v_{λ}) viewed as a U_{q}(D_{n−1})-crystal.

We now define σ(T) combinatorially. Suppose T ∈ B(kΛ_{2}) ⊂ B̃^{2,s}, and ℓ be minimal such that ι_{k}^{s}(T) ∈ ι_{ℓ}^{s}(B(ℓΛ_{2})). Then

(6.2)  σ(T) = ι_{k}^{s+ℓ−k}(T^{∗_{BC}}) = (ι_{k}^{s+ℓ−k}(T))^{∗_{BC}},

where it was used that ι_{i}^{j} commutes with ∗_{BC}.

6.3. Properties of f̃_{0} and ẽ_{0}. This combinatorial approach immediately gives us useful information about this crystal, such as the following lemma.

Lemma 6.7. For k = 0, 1, …, s, let u_{k} denote the highest weight vector of the classical component B(kΛ_{2}) ⊂ B̃^{2,s}. Then

f̃_{0}(u_{k}) = \begin{cases}u_{k+1} & ifk < s, \\ 0 & ifk = s. \end{cases}

Proof. Observe that

u_{k} = \underbrace{{1 \atop 2} ⋯ {1 \atop 2}}_{k} \underbrace{1 ¯1 ⋯ 1 ¯1}_{s−k};

We wish to calculate f̃_{0}(u_{k}) = σf̃_{1}σ(u_{k}).

Note that u_{k} ∉ ι_{k−1}^{k}(B((k − 1)Λ_{2})), so ℓ = k in the combinatorial definition of σ above. It follows that σ(u_{k}) = ι_{k}^{s}(u_{k}^{∗_{BC}}), which is

ι_{k}^{s}(u_{k}^{∗_{BC}}) = ι_{k}^{s}( \underbrace{1 ¯1 ⋯ 1 ¯1}_{s−k} \underbrace{2 ¯1 ⋯ 2 ¯1}_{k} ) = ∅_{s−k} \underbrace{2 ¯1 ⋯ 2 ¯1}_{k},

where ∅_{i} denotes a null-configuration of size i (see Definition 5.2). If k = s, f̃_{1} kills this tableau, as claimed in the second case of the Lemma. Otherwise, acting by f̃_{1} will decrease the size of the null-configuration by 1 and add another 2 ¯1 to the columns on the right. It follows that ι_{s}^{k} kills this tableau, but ι_{s}^{k+1} does not, so now ℓ = k + 1 in the combinatorial definition of σ. Thus,

σf̃_{1}σ(u_{k}) = ι_{s}^{k+1}( \underbrace{{1 \atop 2} ⋯ {1 \atop 2}}_{k+1} ∅_{s−k−1} ) = \underbrace{{1 \atop 2} ⋯ {1 \atop 2}}_{k+1} \underbrace{1 ¯1 ⋯ 1 ¯1}_{s−k−1} = u_{k+1}.

□

Corollary 6.8. Let u_{k} be as above for k > 0. Then

ẽ_{0}(u_{k}) = u_{k−1}.

A similar combinatorial analysis can be carried out on lowest weight tableaux to show that f̃_{0}(u_{k}^{∗}) = u_{k−1}^{∗} and ẽ_{0}(u_{k}^{∗}) = u_{k+1}^{∗} for appropriate values of k. Since u_{0} = u_{0}^{∗}, this gives us the following Corollary:

Corollary 6.9. For highest weight vectors u_{k} and lowest weight vectors u_{k}^{∗}, we have

φ_{0}(u_{k}) = ε_{0}(u_{k}^{∗}) = s − k  and  φ_{0}(u_{k}^{∗}) = ε_{0}(u_{k}) = s + k
7. Perfectness of B̃^{2,s}

7.1. Overview. To show that B̃^{2,s} is perfect, it must be shown that all criteria of Definition 2.1 are satisfied with ℓ = s. We have taken part 3 of Definition 2.1 as part of our hypothesis for Theorem 1.1, so we do not attempt to prove this here.

Part 2 of Definition 2.1 is satisfied by simply noting that λ = sΛ_{2} − 2sΛ_{0} is a weight in P_{cl} such that B_{λ} = {u_{s}} contains only one tableau and all other tableaux in B̃^{2,s} have “lower” weights.

In section 7.2, we show that B̃^{2,s} ⊗ B̃^{2,s} is connected proving part 1 of Definition 2.1. Parts 4 and 5 of Definition 2.1 will be dealt with simultaneously in sections 7.4 and 7.5 by examining the levels of tableaux combinatorially. We will see that the level of a generic tableau is at least s and the tableaux of level s are in bijection with the level s weights. In section 7.6 we show that B̃^{2,s} is the unique affine crystal satsifying the properties of Conjecture 3.4 thereby proving Theorem 1.1.

7.2. Connectedness of B̃^{2,s}.

Lemma 7.1 (Part 1 of Definition 2.1). The crystal B̃^{2,s} ⊗ B̃^{2,s} is connected.

Proof. (This proof is very similar to that in [18, Proposition 5.1].) For k = 0, 1, …, s, let u_{k} denote the highest weight vector of the classical component B(kΛ_{2}) ⊂ B̃^{2,s}, as in Lemma 6.7. We will show that an arbitrary vertex b ⊗ b′ ∈ B̃^{2,s} ⊗ B̃^{2,s} is connected to u_{0} ⊗ u_{0}.

We know that for some j ∈ {0, …, s}, we have b′ ∈ B(jΛ_{2}). Then for some pair of sequences i_{1}, i_{2}, …, i_{p} (with entries in {1, …, n}) and m_{1}, m_{2}, …, m_{p} (with entries in ℤ_{>0}) and some b^{1} ∈ B̃^{2,s}, we have ẽ_{i_{1}}^{m_{1}}ẽ_{i_{2}}^{m_{2}} ⋯ ẽ_{i_{p}}^{m_{p}}(b ⊗ b′) = b^{1} ⊗ u_{j}.

By Corollary 6.9, φ_{0}(u_{j}) = s − j, so if ε_{0}(b^{1}) ≤ s − j, Lemma 6.7 tells us that ẽ_{0}^{j}(b^{1} ⊗ u_{j}) = b^{1} ⊗ u_{0}. If ε_{0}(b^{1}) = r > s − j, then ẽ_{0}^{r−s+j}(b^{1} ⊗ u_{j}) = b^{2} ⊗ u_{j}, where ε_{0}(b^{2}) = r − (r − s + j) = s − j, so ẽ_{0}^{j}(b^{2} ⊗ u_{j}) = b^{2} ⊗ u_{0}. In either case, our arbitrary b ⊗ b′ is connected to an element of the form b′′ ⊗ u_{0}.

Let j′ be such that b′′ ∈ B(j′Λ_{2}). Since u_{0} is the unique element of B(0), the crystal for the trivial representation of U_{q}(D_{n}), we know that B(j′Λ_{2})⊗B(0) ≃ B(j′Λ_{2}). Therefore, b′′ ⊗ u_{0} is connected to u_{j′} ⊗ u_{0}. Finally, we note that φ_{0}(u_{0}) = s < s + j′ = ε_{0}(u_{j′}) for j′ ≠ 0, so ẽ_{0}^{j′}(u_{j′} ⊗ u_{0}) = ẽ_{0}^{j′}(u_{j′}) ⊗ u_{0} = u_{0} ⊗ u_{0}, completing the proof. □

7.3. Preliminary observations. We first make a few observations.

Proposition 7.2. Let T ∈ B(kΛ_{2}) ⊂ B̃^{2,s}, and set T_{m} = ι_{k}^{m}(T) for m = s, s − 1, …, ℓ, where ℓ is minimal such that ι_{k}^{ℓ}(T) ≠ 0. If ℓ ≠ s, we have for s > m ≥ ℓ

ε_{1}(T_{m+1}) = ε_{1}(T_{m}) + 1 and ε_{0}(T_{m+1}) = ε_{0}(T_{m}) − 1,
φ_{1}(T_{m+1}) = φ_{1}(T_{m}) + 1 and φ_{0}(T_{m+1}) = φ_{0}(T_{m}) − 1.

Proof. Let ℓ ≤ m ≤ s−1, so ι_{m}^{m+1} is defined. We first consider the difference between the reduced 1-signatures of D_{2,s}(T_{m}) and D_{2,s}(T_{m+1}) = D_{2,s}(ι_{m}^{m+1}(T_{m})), since the action of ẽ_{1} on these tableaux is defined by the action of the classical ẽ_{1} on their image under D_{2,s}. Let −^{M}+^{P} be the reduced 1-signature of D_{2,s}(T_{m}). As in section 5.3, let r_{1} denote the number of 1’s in D_{2,s}(T_{m}), r_{3} the number of ¯1’s, r_{2} the size of the null-configuration, and t_{1} = r_{1} + r_{2}, t_{2} = r_{2} + r_{3}. Then there is a contribution −^{r_{2}}+^{r_{2}} to the 1-signature from the null-configuration, and the remaining −’s and +’s come from 1’s with a letter greater than 2 below them and ¯1’s with a letter less than ¯2 above them, respectively.

We now have two cases, as in section 5.3. If t_{1} + t_{2} ≥ s, ι_{m}^{m+1} simply increases the size of the null-configuration in D_{2,s}(T_{m}) by 1. It follows that the reduced 1-signature of D_{2,s}(T_{m+1}) is −^{M+1}+^{P +1}, as we wished to show. On the other hand, if t_{1} + t_{2} < s, after step (2) of Algorithm 6.4 for ι_{m}^{m+1} we have a tableau of shape (m, m − r_{3})/(r_{1}). In step (3), we slide this into shape (m + 1, m + 1 − (r_{3} + 1))/(r_{1} + 1). We claim that the rightmost “uncovered” letter in the second row of this tableau is greater than 2 and the leftmost “unsupported” letter in the first row is less than ¯2. As observed in the preceding paragraph, this implies that after refilling the empty spaces as in step (4) of Algorithm 6.4 the reduced 1-signature of our tableau is −^{M+1}+^{P +1} in this case as well.

Let us first consider the leftmost “unsupported” letter. After step (2), our tableau is of the form

\begin{aligned}& & & a_{r_{1}+1} & ⋯ & a_{m−r_{3}} & a_{m−r_{3}+1} & ⋯ & a_{s} \\ b_{1} & ⋯ & b_{r_{1}} & b_{r_{1}+1} & ⋯ & b_{m−r_{3}} \end{aligned}

and its column word is unchanged by the slide

\begin{aligned}& & & a_{r_{1}+1} & ⋯ & & a_{m−r_{3}} & a_{m−r_{3}+1} & ⋯ & a_{s} \\ b_{1} & ⋯ & b_{r_{1}} & b_{r_{1}+1} & ⋯ & b_{m−r_{3}} \end{aligned} ,

so we have a_{m−r_{3}} < b_{m−r_{3}} ≤ ¯2.

The second row of this tableau has m−r_{3} boxes just as it did before sliding, so the boxes in the bottom row will never be moved. It follows that this sliding procedure only changes

L-shaped subtableaux into Γ -shapes (i.e.,  into  ) and never involves

any Γ- or L -shapes. According to the Lecouvey D-equivalence relations from section 3.4, such moves can only be made when the letters in the bottom row are strictly greater than 2. Specifically, in relations (3) and (4), the letter which is “uncovered” is either n or n̄, while in relations (1) and (2) only the second case of each relation applies. This proves our claim, and thus the first half of the proposition.

Since ẽ_{0} = σ ∘ ẽ_{1} ∘ σ, we can derive the statements about ε_{0} and φ_{0} from the corresponding statements about ε_{1} and φ_{1}. More precisely, ε_{0}(T) = ε_{1}(σ(T)) and φ_{0}(T) = φ_{1}(σ(T)) and by (6.2) we have

σ(T_{m}) = (ι_{m}^{s+ℓ−m} ∘ ι_{k}^{m}(T))^{∗_{BC}} = ι_{k}^{s+ℓ−m}(T^{∗_{BC}}).

Hence

ε_{0}(T_{m+1}) = ε_{1}(σ(T_{m+1})) = ε_{1}(ι_{k}^{s+ℓ−m−1}(T^{∗_{BC}}))

= ε_{1}(ι_{k}^{s+ℓ−m}(T^{∗_{BC}})) − 1 = ε_{1}(σ(T_{m})) − 1 = ε_{0}(T_{m}) − 1.

A similar computation can be carried out for φ_{0}. □

Corollary 7.3. Given the above hypotheses, we have

⟨h_{0} + h_{1}, ε(T_{s})⟩ = ⟨h_{0} + h_{1}, ε(T_{s−1})⟩ = ⋯ = ⟨h_{0} + h_{1}, ε(T_{ℓ})⟩ ≠ 0.

The following observation is an immediate consequence of Remark 6.3:

Corollary 7.4. For i = 2, …, n,

⟨h_{i}, ε(T_{s})⟩ = ⟨h_{i}, ε(T_{s−1})⟩ = ⋯ = ⟨h_{i}, ε(T_{ℓ})⟩.

Lemma 7.5. The map Υ_{s−1}^{s} : T (s − 1) ↪ T (s) viewed as sending the set underlying B̃^{2,s−1} into the set underlying B̃^{2,s} increases the level of a tableau by exactly 1.

Proof. This map is defined by sending each summand B(kΛ_{2}) ⊂ B̃^{2,s−1} to B(kΛ_{2}) ⊂ B̃^{2,s} for k = 0, …, s − 1, so φ_{i}(Υ_{s−1}^{s}(T)) = φ_{i}(T) for i = 1, …, n. To calculate the change in φ_{0}(T), we must consider the difference between φ_{1}(σ_{s−1}(T)) and φ_{1}(σ_{s}(Υ_{s−1}^{s}(T))). By our descriptions of maps on crystals, we have σ_{s}(Υ_{s−1}^{s}(T)) = Υ_{s−1}^{s}(ι_{j}^{j+1}(σ_{s−1}(T))), where j is determined by σ_{s−1}(T) ∈ B(jΛ_{2}) ⊂ B̃^{2,s−1}. By Proposition 7.2, φ_{1}(ι_{j}^{j+1}(σ_{s−1}(T))) = φ_{1}(σ_{s−1}(T)) + 1. □

7.4. Surjectivity. Given a weight λ ∈ (P_{cl}^{+})_{s}, we construct a tableau T_{λ} ∈ B̃^{2,s} such that ε(T_{λ}) = φ(T_{λ}) = λ. This amounts to constructing T_{λ} so that its reduced i-signature is −^{ε_{i}(T_{λ})}+^{ε_{i}(T_{λ})}. Note that such a tableau is invariant under the ∗-involution, so its symmetry allows us to define it beginning with the middle, and proceeding outwards.

For i = 0, …, n, let k_{i} = ⟨h_{i}, λ⟩. We first construct a tableau T_{λ′} corresponding to the weight λ′ = ∑_{i=2}^{n} k_{i}Λ_{i}. We begin with the middle k_{n−1} + k_{n} columns of T_{λ′}. If k_{n−1} + k_{n} is even and k_{n} ≥ k_{n−1}, these columns of T_{λ′} are

\underbrace{{n − 2 \atop n − 1} ⋯ {n − 2 \atop n − 1}}_{k_{n−1}} \underbrace{{n − 1 \atop n} ⋯ {n − 1 \atop n}}_{(k_{n}−k_{n−1})/2} \underbrace{\overline{n} ; \overline{n − 1} ⋯ \overline{n} ; \overline{n − 1}}_{(k_{n}−k_{n−1})/2} \underbrace{\overline{n − 1} ; \overline{n − 2} ⋯ \overline{n − 1} ; \overline{n − 2}}_{k_{n−1}}

If k_{n−1} + k_{n} is odd and k_{n} ≥ k_{n−1}, we have

\underbrace{{n − 2 \atop n − 1} ⋯ {n − 2 \atop n − 1}}_{k_{n−1}} \underbrace{{n − 1 \atop n} ⋯ {n − 1 \atop n}}_{(k_{n}−k_{n−1}−1)/2} n̄ ; n \underbrace{\overline{n} ; \overline{n − 1} ⋯ \overline{n} ; \overline{n − 1}}_{(k_{n}−k_{n−1}−1)/2} \underbrace{\overline{n − 1} ; \overline{n − 2} ⋯ \overline{n − 1} ; \overline{n − 2}}_{k_{n−1}}

In either case, if k_{n} < k_{n−1}, interchange n and n̄, and k_{n} and k_{n−1} in the above configurations.

Next we put a configuration of the form

\underbrace{{1 \atop 2} ⋯ {1 \atop 2}}_{k_{2}} \underbrace{{2 \atop 3} ⋯ {2 \atop 3}}_{k_{3}}  ⋯  \underbrace{{n − 3 \atop n − 2} ⋯ {n − 3 \atop n − 2}}_{k_{n−2}}

on the left, and a configuration of the form

\underbrace{\overline{n − 2} ; \overline{n − 3} ⋯ \overline{n − 2} ; \overline{n − 3}}_{k_{n−2}} \underbrace{\overline{n − 3} ; \overline{n − 4} ⋯ \overline{n − 3} ; \overline{n − 4}}_{k_{n−3}}  ⋯  \underbrace{¯2 \overline{1} ⋯ ¯2 \overline{1}}_{k_{2}}

on the right. Denote the set of tableaux constructed by the procedure up to this point by M(s′).

Observe that the reduced 1-signature of T_{λ′} is empty, so ⟨h_{1}, φ(T_{λ′})⟩ = 0. Furthermore, since λ′ has the same number of 1’s as ¯1’s, it is fixed by σ, so ⟨h_{0}, φ(T_{λ′})⟩ = 0 as well. Thus Proposition 7.2 implies that T_{λ′} ∈ B̃_{min}^{2,s}∩B(s′Λ_{2})\ι_{s′−1}^{s′}(B((s′−1)Λ_{2})) as a subset of B̃^{2,s′}. Recall the embedding Υ_{s′}^{s} : B̃^{2,s′} ↪ B̃^{2,s} from Definition 4.10. Since s′ is the minimal m for which ι_{s′}^{m}(T_{λ′}) ≠ 0, Lemma 7.5 and its proof tell us that ε_{0}(F_{2,s}(T_{λ′})) = s − s′, where the fill map F_{2,s} inserts an a-configuration to increase the width of T_{λ′} to s. By the same proposition the desired tableau is T_{λ} = ι_{s′}^{s′+k_{1}} ∘ F_{2,s}(T_{λ′}). We denote by M_{min}(s) the set of tableaux constructed by this procedure.

7.5. Injectivity. In this subsection we show that the tableaux in M_{min}(s) are all the minimal tableaux in B̃^{2,s}.

We first introduce some useful notation. Observe that any tableau T can be written as T = T_{1}T_{2}T_{3}T_{4}T_{5}, where the block T_{i} has width k_{i}, and all letters in T_{1} (resp. T_{5}) are unbarred or n̄ in the second row (resp. barred or n in the first row), all columns in T_{2} (resp. T_{4}) are of the form a ; b̄ where a < b ≤ n − 1 (resp. b < a ≤ n − 1), and all columns in T_{3} are of the form a ; ā for some a. Note that for a tableau in B(sΛ_{2}) ⊂ B̃^{2,s} we have 0 ≤ k_{3} ≤ 1. Also note that T_{2} and T_{4} do not contain any n’s or n̄’s.

Theorem 7.6. We have M_{min}(s) = B̃_{min}^{2,s}.

Proof. We prove the theorem by induction on s. For the base case, we checked explicitly that the statement of the theorem is true for s = 0, 1, 2.

By our induction hypothesis, M_{min}(s − 1) = B̃_{min}^{2,s−1}. By Lemma 7.5, Υ_{s−1}^{s} increases the level of a tableau by 1, and therefore Υ_{s−1}^{s}(B̃_{min}^{2,s−1}) = (B̃^{2,s} \ B(sΛ_{2}))_{min}. By Corollaries 7.3 and 7.4, ι_{s−1}^{s} does not change the level of a tableau, so it suffices to show that

(7.1)  M(s) = B̃_{min}^{2,s} ∩ (B(sΛ_{2}) \ ι_{s−1}^{s}(B((s − 1)Λ_{2}))).

By Lemmas 7.8 and 7.9 below, if T is a minimal tableau not in the image of ι_{s−1}^{s}, it has k_{2} = k_{4} = 0, and if k_{3} = 1, then T_{3} = n ; n̄ or T_{3} = n̄ ; n. By Lemma 7.10, equation (7.1) follows. □

Here we state the lemmas used in the proof of Theorem 7.6. The proofs are given in Appendices A-D and together with Theorem 7.6 all rely on induction on s. The base cases s = 0, 1, 2 have been checked explicitly.

Lemma 7.7. For all t ∈ B̃^{2,s} we have ⟨c, ε(t)⟩ ≥ s and ⟨c, φ(t)⟩ ≥ s.

Lemma 7.8. If T ∈ (B̃^{2,s} ∩ B(sΛ_{2}))_{min}, we have k_{1} + k_{2} = k_{4} + k_{5} = ⌊s/2⌋ with k_{1}, k_{2}, k_{4} and k_{5} as defined in the beginning of this subsection.

Lemma 7.9. Suppose T ∈ (B̃^{2,s} ∩ B(sΛ_{2}))_{min} has both an unbarred letter and a barred letter in a single column a ; b̄ other than n ; n̄, n̄ ; n, n−1 ; n̄, or n ; \overline{n−1}. Then T ∈ ι_{s−1}^{s}(B((s − 1)Λ_{2}).

Lemma 7.10. Let T ∈ (B̃^{2,s} ∩ B(sΛ_{2}))_{min} such that T does not contain any column a ; b̄ for 1 ≤ a, b ≤ n except possibly n−1 ; n̄, n ; n̄, or n ; \overline{n−1}. Then T ∈ M(s).

7.6. Uniqueness. Theorem 1.1 follows as a corollary from the next Proposition.

Proposition 7.11. B̃^{2,s} is the unique affine finite-dimensional crystal structure satisfying the properties of Conjecture 3.4.

For the proof of Proposition 7.11 we must show that our choice of σ is the only choice satisfying the properties of Conjecture 3.4. Recall from the beginning of section 6 the relationship between σ and σ̌. Let T ∈ B̃^{2,s}. We know that

(7.2)  wt(T) = ∑_{i=0}^{n} k_{i}Λ_{i} ⇔ wt(σ(T)) = k_{1}Λ_{0} + k_{0}Λ_{1} + ∑_{i=2}^{n} k_{i}Λ_{i}.

Once σ̌ is determined, the given definition of σ (sending D_{n−1} highest weight vectors to D_{n−1} highest weight vectors, etc.) is the only involution of the set of tableaux in B̃^{2,s} satisfying (7.2) and agreeing with σ̌.

As we observed in section 6.1, σ̌(v) and v must be associated with the same partition, and if v′ is the complementary vertex of v, σ̌(v′) is the complementary vertex of σ̌(v). We now prove a few lemmas that uniquely determine σ̌.

Please note that in this section we often use the phrase “the tableau b is in the branching component vertex v” to mean b ∈ B(v).

Lemma 7.12. Suppose b ∈ B̃^{2,s} is in a branching component vertex of rank p and f̃_{0}(b) ≠ 0. Then the branching component vertex containing f̃_{0}(b) has rank p − 1.

Proof. Recall from the weight structure of type D algebras that α_{0} = 2Λ_{0} − Λ_{2} and wt(f̃_{0}(b)) = wt(b) − α_{0}. Define cw(b) = wt(b) − (φ_{0}(b) − ε_{0}(b))Λ_{0}. The above implies that cw(f̃_{0}(b)) = cw(b) + Λ_{2} = cw(b) + ε_{1} + ε_{2}. Similarly, cw(f̃_{i}(b)) = cw(b) − α_{i}, so that by (3.1) only f̃_{1} changes the ε_{1} component by −1. Since f̃_{1} increases the rank by one and f̃_{0} changes the ε_{1} component by +1, it follows that f̃_{0} decreases the rank by one. □

Since f̃_{1} increases the rank by one, f̃_{0} decreases the rank by one and f̃_{0} = σf̃_{1}σ, the following corollary holds.

Corollary 7.13. Suppose b is in a branching component vertex v of rank p. Then σ̌(v), the branching component vertex containing σ(b), has rank 2s − p.

Note that this determines σ̌ on B(sΛ_{2}) \ ι_{s−1}^{s}(B((s − 1)Λ_{2})), and is in agreement with our definition of σ̌ restricted to that domain.

Lemma 7.14. Let v ∈ BC(kΛ_{2}) be a branching component vertex of rank s associated with a rectangular partition, and let ℓ be minimal such that v ∈ ι̌_{ℓ}^{k}(BC(ℓΛ_{2})). Then the hypothesis that B̃^{2,s} is perfect of level s can only be satisfied if σ̌(v) is the vertex associated with the same shape as v with rank s in BC((s + ℓ − k)Λ_{2}).

Proof. We have already shown that σ̌(v) has the same shape as v and has rank s, so it only remains to show that σ̌(v) ∈ BC((s + ℓ − k)Λ_{2}).

First, observe that v must contain a minimal tableau as constructed in section 7.4, according to the following table.

shape associated with v  weight of tableau in v
(2m)  mΛ_{2} + (k − 2m)Λ_{1} + (s − k)Λ_{0}
(m, m)  mΛ_{3} + (k − 2m)Λ_{1} + (s − k)Λ_{0}

Let T be the tableau constructed by this prescription, so that ⟨c, cw(T)⟩ = k. The criterion that ⟨c, φ(T)⟩ ≥ s forces us to have φ_{0}(T) = φ_{1}(σ(T)) ≥ s − k. We denote T_{i} = ι_{k}^{i}(T), and thus have φ_{1}(T_{i}) = i − ℓ.

We show inductively that σ(T_{i}) = T_{s+ℓ−i} for ℓ ≤ i ≤ s. As a base case, we see that ⟨c, cw(T_{ℓ})⟩ = ℓ, so we must have φ_{1}(σ(T_{ℓ})) ≥ s−ℓ. The only T_{i} for which this inequality holds is T_{s}, where we have φ_{1}(T_{s}) = s − ℓ.

For the induction step, assume that σ sends T_{ℓ}, T_{ℓ+1}, …, T_{k−1} to T_{s}, …, T_{s+ℓ−k+1}, respectively. By the above inequality this implies φ_{1}(σ(T_{k})) ≥ s − k, which specifies that σ(T_{k}) = T_{s+ℓ−k}. □

Definition 7.15. Recall the definition of Υ_{s′}^{s} from Definition 4.10. We define Υ̌_{s′}^{s} : BC(B̃^{2,s′}) ↪ BC(B̃^{2,s}) by Υ̌_{s′}^{s}(v) = v′ if for some T ∈ B(v), we have Υ_{s′}^{s}(T) ∈ B(v′).

Lemma 7.16. Let v ∈ BC(kΛ_{2}) be a branching component vertex of rank 1 ≤ p ≤ s − 1 associated to the shape (λ_{1}, λ_{2}). Suppose that for the branching component vertex w ∈ BC(kΛ_{2}) of rank p + 1 with shape (λ_{1} − 1, λ_{2}), B̃^{2,s} has the correct energy function and is perfect only if σ̌(w) is as described in section 6.1. Then B̃^{2,s} has the correct energy function only if σ̌(v) is as described in section 6.1.

Proof. First, recall that the partitions associated to vertices of rank p in BC(B̃^{2,s}) are produced by adding or removing one box from the partitions associated to vertices of rank p−1. Since the vertex of rank 0 is associated with a rectangle of shape (s), the lowest rank for which we can have a two-row rectangle is s. It follows that removing a box from the first row of v results in a partition of rank p + 1, so there is in fact a vertex w as described in the statement of the lemma.

Let ℓ be minimal such that v ∈ ι̌_{ℓ}^{k}(BC(ℓΛ_{2})). We may assume v ∉ BC(ℓΛ_{2}), and let σ̌(v) be determined by the involutive property of σ̌ in the case v ∈ BC(ℓΛ_{2}). Specifically, we will show that the vertex v′ ∈ BC(sΛ_{2}) with the same shape and rank as v has the property that σ̌(v′) is the complementary vertex of v, and therefore σ̌(v) is the complementary vertex of v′.

The top row of w is one box shorter than the top row of v, so that w ∈ ι̌_{ℓ−1}^{k}(BC((ℓ − 1)Λ_{2})), and ℓ−1 is minimal with this property. By our hypothesis, σ̌(w) is the vertex with shape (λ_{1} − 1, λ_{2}) of rank 2s − p − 1 in BC((s + (ℓ − 1) − k)Λ_{2}).

We now use induction on s. Suppose that the only choice of σ̌, for which B̃^{2,s−1} is perfect and has an energy function, is the choice of section 6.1. Part 3 of Conjecture 3.4 states that in B̃^{2,s}, the energy on the component B((s − k)Λ_{2}) is −k, and so the difference in energy between B((s − k)Λ_{2}) and B((s − j)Λ_{2}) is j − k. In order for this to be true for all 1 ≤ k, j ≤ s − 1, the action of f̃_{0} and ẽ_{0} on B̃^{2,s} must agree with the action on B̃^{2,s−1}. More precisely, if v and w are in different classical components of B̃^{2,s−1} and f̃_{0}(v) = w in B̃^{2,s−1}, then f̃_{0}(Υ_{s−1}^{s}(v)) = Υ_{s−1}^{s}(w); this statement extends naturally to BC(B̃^{2,s−1}) and BC(B̃^{2,s}).

Let v^{†} denote the vertex with shape (λ_{1}, λ_{2}) of rank 2s−p in BC((s+ℓ−k)Λ_{2}). Since we assumed k ≠ ℓ, we know that v^{†} ∉ BC(sΛ_{2}), and therefore v^{†} has a preimage under Υ̌_{s−1}^{s}. From our construction of σ̌ we know that in BC(B̃^{2,s−1}), (Υ̌_{s−1}^{s})^{−1}(v^{†}) has a 0 arrow to (Υ̌_{s−1}^{s})^{−1}(σ̌(w)). Our induction argument tells us that in BC(B̃^{2,s}), v^{†} has a 0 arrow to σ̌(w). Since v has a 1 arrow to w and we must have f̃_{0} = σf̃_{1}σ, we conclude that in fact σ̌(v) = v^{†}. □

Lemma 7.17. Let v ∈ BC(kΛ_{2}) be a branching component vertex of rank s associated to the non-rectangular shape (λ_{1}, λ_{2}), and suppose that for the branching component vertex w ∈ BC(kΛ_{2}) of rank s − 1 with shape (λ_{1}, λ_{2} + 1), B̃^{2,s} has the correct energy function and is perfect only if σ̌(w) is as described in section 6.1. Then B̃^{2,s} has the correct energy function only if σ̌(v) is as described in section 6.1.

Proof. (This proof is very similar to the proof of Lemma 7.16.)

Let ℓ be minimal such that v ∈ ι̌_{ℓ}^{k}(BC(ℓΛ_{2})), assuming k ≠ ℓ. Note that ℓ is also minimal for w ∈ ι̌_{ℓ}^{k}(BC(ℓΛ_{2})), since the shapes for v and w have the same number of boxes in the first row. By our hypothesis, σ̌(w) is the vertex with shape (λ_{1}, λ_{2}+1) of rank s+1 in BC((s+ℓ−k)Λ_{2}). Let v^{†} be the vertex with shape (λ_{1}, λ_{2}) of rank s in BC((s+ℓ−k)Λ_{2}). From our construction of σ̌, we know that in BC(B̃^{2,s−1}), (Υ̌_{s−1}^{s})^{−1}(w) has a 0 arrow to (Υ̌_{s−1}^{s})^{−1}(σ̌(v^{†})). It follows from our induction argument that in BC(B̃^{2,s}), w has a 0 arrow to σ̌(v^{†}). Since w has a 1 arrow to v and we must have f̃_{0} = σf̃_{1}σ, we conclude that in fact σ̌(v) = v^{†}. □

Corollary 7.18. Corollary 7.13 and Lemmas 7.14, 7.16 and 7.17 determine σ̌ on BC(B̃^{2,s}) uniquely.

Proof. For any vertex v associated with shape (λ_{1}, λ_{2}) with rank p ≤ s, σ̌(v) is fixed by the image under σ̌ of a vertex with shape (λ_{1} − s + p, λ_{2}) and rank s by Lemma 7.16. If λ_{1} − s + p ≠ λ_{2}, Lemmas 7.16 and 7.17 may be used together to reduce determining σ̌(v) to determining the action of σ̌ on a rectangular vertex of rank s, which is given by Lemma 7.14. □

8. Discussion

In this section we discuss some applications and open problems regarding the crystals B̃^{2,s} introduced in this paper.

The major open question regarding B̃^{2,s} is of course its existence, which was assumed throughout this paper. A possible method of proof is a generalization of the fusion construction of [10].

In [16], Kashiwara conjectures that for any quantum affine algebra, B^{r,s} is isomorphic as a classical crystal to a Demazure crystal in an irreducible affine highest weight module of weight s max(1, 2/(α_{r}, α_{r}))ϖ_{r} − sΛ_{0}, where ϖ_{r} = Λ_{r} − c_{r}Λ_{0} (except for type A_{2n}^{(2)}). The f̃_{0} edges that stay within the Demazure crystal should be among the f̃_{0} edges of B^{r,s}. The combinatorial structure of B̃^{2,s} as constructed in this paper might give a hint on how to make this correspondence more precise.

For a tensor product of affine finite crystals B = B_{L} ⊗⋯⊗B_{1} and a dominant integral weight λ define the set of classically restricted paths as

P(B, λ) = {b ∈ B | wt(b) = λ, ẽ_{i}(b) = 0 for all i ∈ J}

where J = {1, 2, …, n}. The classically restricted one dimensional sum is defined to be

X(B, λ; q) = ∑_{b∈P(B,λ)} q^{D_{B}(b)}.

In [5, Section 4] fermionic formulas M(B, λ; q) are defined which are sums of products of q-binomial coefficients, and it is conjectured that X(B, λ; q) = M(B, λ; q). This conjecture has been proven for type A_{n}^{(1)} [19, 20, 21] and various other cases [1, 24, 25, 26, 27, 29, 30]. It is expected that the X = M conjecture can also be proven in the case of tensor products of crystals B̃^{2,s} as constructed in this paper by using the splitting map (see [30]) and the single column bijection of type D_{n}^{(1)} (see [29]).

Appendix A. Proof of Lemma 7.7

By induction hypothesis, B̃_{min}^{2,s−1} = M_{min}(s − 1).

Observe that by Corollaries 7.3 and 7.4 ι_{i}^{j} is level preserving, and by Lemma 7.5, the map Υ_{s−1}^{s} increases the level of a tableau by one. Our induction hypothesis therefore allows us to assume that t ∈ B(sΛ_{2}) \ ι_{s−1}^{s}(B((s − 1)Λ_{2}). Combinatorially, we may characterize such tableaux as being those which are legal in the classical sense and for which removing all 1’s, ¯1’s, and null configurations produces a tableau which is Lecouvey D-equivalent to a tableau whose first row has width s. This characterization follows from the combinatorial description of ι_{s−1}^{s} in Algorithm 6.4.

We may further restrict our attention by the observation that if T is minimal, then so is T^{∗}. We may therefore assume T to be in the top half (inclusive of the middle row) of the branching component graph. This means that T has no more ¯1’s than 1’s.

Our approach is to consider the tableau T′ that results from removing the leftmost column from T. We will show that if T′ is minimal, the level of T exceeds the level of T′ by at least 2, and if T′ is not minimal, the level of T is at least as great as the level of T′.

First consider the case when T′ is minimal. Since T is assumed to be such that removing all 1’s and ¯1’s produces a tableau which is Lecouvey D-equivalent to a tableau whose first row has width s, it is the case that removing all 1’s and ¯1’s from T′ produces a tableau which is Lecouvey D-equivalent to a tableau whose first row has width s−1. The minimal tableaux of B̃^{2,s−1} with this property are precisely M(s − 1). By properties of M(s − 1), we know that φ_{0}(T′) = 0, so φ_{0}(T) ≥ φ_{0}(T′). Since our base case is s = 2, we know that the first column of T is {a \atop b}, where a and b are both unbarred. Observe that φ(T) = φ(T′) + 2Λ_{b} + non-negative weight if b ≠ n − 1 or φ(T) = φ(T′) + Λ_{n−1} + Λ_{n} + non-negative weight if b = n − 1. Hence, the level is increased by at least 2.

Now suppose T′ is not minimal. The level of the i-signatures (that is to say, the level of the sum of the weights φ_{i}(T′) which depend on i-signatures) cannot have a net decrease for i = 1, …, n, but there is now a possibility that φ_{0}(T) < φ_{0}(T′). We will show that when φ_{0}(T) < φ_{0}(T′), the level of the i-signatures goes up by at least φ_{0}(T′) − φ_{0}(T).

First, suppose T has no 1’s. Then by one of our hypotheses, it also has no ¯1’s, and is therefore fixed by σ: it follows that φ_{0}(T) = φ_{1}(T), so we may assume the upper-left entry of T to be 1.

We know that φ_{0}(T) is equal to the number of −’s in the reduced 1-signature of σ(T). Consider the following tableaux:

T′ = \underbrace{{1 \atop b_{1}} ⋯ {1 \atop b_{m_{1}}}}_{m_{1}} {a_{m_{1}+1} \atop b_{m_{1}+1}} ⋯ {a_{s−m_{2}} \atop b_{s−m_{2}}} \underbrace{a_{s−m_{2}+1} ; ¯1 ⋯ a_{s} ; ¯1}_{m_{2}}

T = {1 \atop b} \underbrace{{1 \atop b_{1}} ⋯ {1 \atop b_{m_{1}}}}_{m_{1}} {a_{m_{1}+1} \atop b_{m_{1}+1}} ⋯ {a_{s−m_{2}} \atop b_{s−m_{2}}} \underbrace{a_{s−m_{2}+1} ; ¯1 ⋯ a_{s} ; ¯1}_{m_{2}}

σ(T′) = \underbrace{{1 \atop b_{1}} ⋯ {1 \atop b_{m_{2}−1}} {1 \atop b′_{m_{2}}}}_{m_{2}} {a′_{m_{2}+1} \atop b′_{m_{2}+1}} ⋯ {a′_{s−m_{1}} \atop b′_{s−m_{1}}} \underbrace{a′_{s−m_{1}+1} ; ¯1 ⋯ a′_{s} ; ¯1}_{m_{1}}

σ(T) = \underbrace{{1 \atop b} {1 \atop b_{1}} ⋯ {1 \atop b_{m_{2}−1}}}_{m_{2}} {a′′_{m_{2}} \atop b′′_{m_{2}}} ⋯ {a′′_{s−m_{1}−1} \atop b′′_{s−m_{1}−1}} \underbrace{a′′_{s−m_{1}} ; ¯1 ⋯ a′′_{s} ; ¯1}_{m_{1}+1} .

Note that our assumption that m_{2} ≤ m_{1} + 1 ensures that the absence of primes on b, b_{1}, …, b_{m_{2}−1} is accurate.

Let us consider all possible ways for the number of −’s in the 1-signature to be smaller for σ(T) than for σ(T′). The number of 1’s is the same, so the only way this contribution could be decreased is by having more 2’s in the first m_{2} letters of the bottom row. This can only come about by having b = 2, and only one − may be removed in this way.

The other possibility is for the number of −’s contributed by ¯2’s to be decreased. The only Lecouvey relation which removes a ¯2 assumes the presence of a column 2 ¯2, which we disallow (null-configuration). To decrease this contribution therefore requires an additional + in the 1-signature of σ(T) compared to that of σ(T′), which will bracket one of the −’s from a ¯2. The additional + may come from one of the additional ¯1’s, or from a 2 that is “pushed out” from under the 1’s at the beginning in the case b = 2. Note that this second possibility is mutually exclusive with having more 2’s bracketing 1’s at the beginning.

In any case, we see that φ_{0}(T) − φ_{0}(T′) ≤ 2, and that when this value is 2, the first column of T is {1 \atop 2}. This column adds no +’s to the i-signatures, but does provide a new − in the 2-signature. Since Λ_{2} is a level 2 weight, the level stays the same in this case.

If φ_{0}(T) − φ_{0}(T′) = 1 and the first column of T is {1 \atop 2}, we in fact have a net increase in level. If b ≠ 2, the i-signature levels go up by at least 1, so still the total level cannot decrease.

Appendix B. Proof of Lemma 7.8

We first establish that ⟨c, φ(T)⟩ ≥ 2k_{1} + 2k_{2} + k_{3}, and thus by ∗-duality, ⟨c, ε(T)⟩ ≥ k_{3} + 2k_{4} + 2k_{5} as well. Recall that 0 ≤ k_{3} ≤ 1.

First, observe that every letter in the bottom row of T_{1} contributes:

• a − to the reduced a-signature if 2 ≤ a ≤ n − 2 is in the bottom row;

• a − to both the (n − 1)-signature and the n-signature if n − 1 is in the bottom row;

• a − to the n-signature (resp. (n − 1)-signature) if n (resp. n̄) is the bottom row. Suppose T_{1} has a column of the form {a \atop b} with b ≠ a + 1, or b = n̄ and a ≠ n − 1. For the − in the a-signature of T contributed by this a to be bracketed, we must have a column of the form {a′ \atop a+1} to the left of this column in T_{1}, with a′ < a. Applying this observation recursively, we see that to bracket as many −’s as possible we must eventually have a column of the form {1 \atop c} for some c ≠ 2. Note that in the case of columns of the form {n−1 \atop n} (resp. n−1 ; n̄) the unbracketed − in the n-signature (resp. (n − 1)-signature) from n − 1 cannot be bracketed, since n and n̄ may not appear in the same row.

Now, consider a column a ; b̄ in T_{2}, so we have a < b, and thus also ā > b̄. Recall that T_{2} has no n’s or n̄’s, so b ≤ n − 1. This column contributes −’s to the a-signature and the (b − 1)-signature of T. In this case, these −’s may be bracketed. Due to the conditions that the rows and columns of T are increasing, the − from the a can only be bracketed by an a + 1 in the bottom row of T_{1} and the − from the b̄ can only be bracketed by a b in the bottom row of T_{1}. Furthermore, the letter above these must be strictly less than a and b−1, respectively. By the reasoning in the previous paragraph, we see that to bracket every − engendered by the column a ; b̄ we must have two columns of the form {1 \atop a′}, with each a′ ≠ 2.

If k_{3} = 1, T has a column of the form a ; ā. We have two cases; 2 ≤ a ≤ n − 1, and a = n (resp. a = n̄). In the first case, we have a − in the (a − 1)-signature from the ā in this column. Because of the prohibition against configurations of the form ^{a} a ; ā, this − can only be bracketed by a + from an a in the bottom row of T_{1}. Therefore, this column engenders another column of the form {1 \atop a′}. In the case of a = n (resp. a = n̄), we have a − in the (n − 1)-signature (resp. n-signature) which cannot be bracketed.

To bracket the maximal number of −’s (i.e., to minimize ⟨c, φ(T)⟩) we see that unless T_{3} = n ; n̄ or T_{3} = n̄ ; n, we must have

(B.1)  T_{1}T_{2}T_{3} = \underbrace{{1 \atop b_{1}} ⋯ {1 \atop b_{2k_{2}+k_{3}}}}_{2k_{2}+k_{3}} \underbrace{{a_{2k_{2}+k_{3}+1} \atop b_{2k_{2}+k_{3}+1}} ⋯ {a_{k_{1}} \atop b_{k_{1}}}}_{k_{1}−(2k_{2}+k_{3})} \underbrace{a_{k_{1}+1} ; b̄_{k_{1}+1} ⋯ a_{k_{1}+k_{2}+k_{3}} ; b̄_{k_{1}+k_{2}+k_{3}}}_{k_{2}+k_{3}},

where each column in the first block contributes 3 to ⟨c, φ(T)⟩, each column in the second block contributes 2 to ⟨c, φ(T)⟩, and the third block contributes nothing. In the case T_{3} = n ; n̄ or T_{3} = n̄ ; n, we have k_{2} = 0, so we simply have T_{1}T_{2}T_{3} = T_{1}T_{3}, where each column in T_{1} increases ⟨c, φ(T)⟩ by at least 2 and T_{3} increases ⟨c, φ(T)⟩ by 1. We therefore have in the first case ⟨c, φ(T)⟩ ≥ 3(2k_{2} + k_{3}) + 2(k_{1} − 2k_{2} − k_{3}) = 2k_{1} + 2k_{2} + k_{3}, and in the second case ⟨c, φ(T)⟩ ≥ 2k_{1} + k_{3} = 2k_{1} + 2k_{2} + k_{3} , as we wished to show.

Since by Lemma 7.7 elements in B̃^{2,s} have level at least s, it follows that when T ∈ (B̃^{2,s} ∩ B(sΛ_{2}))_{min}, we have k_{1} + k_{2} ≤ ⌊s/2⌋, and by ∗-duality that k_{4} + k_{5} ≤ ⌊s/2⌋. Furthermore, since s = k_{1} + k_{2} + k_{3} + k_{4} + k_{5}, it follows that k_{1} + k_{2} = k_{4} + k_{5} = ⌊\frac{s}{2}⌋ and k_{3} = 0 if s is even and k_{3} = 1 if s is odd.

Appendix C. Proof of Lemma 7.9

By using the reverse of Algorithm 6.4, it suffices to show the following:

(1) T has a 1;

(2) T has a ¯1;

(3) after removing all 1’s and ¯1’s, applying the Lecouvey D relations will reduce the width of T.

The proof of Lemma 7.8 shows that if k_{2} ≠ 0, or k_{3} = 1 and T_{3} ≠ n ; n̄, n̄ ; n, then T has a 1. By ∗-duality, if k_{4} ≠ 0, or the same condition is placed on k_{3} and T_{3}, then T has a ¯1. We will show that if k_{2} + k_{3} ≠ 0, then k_{3} + k_{4} ≠ 0, which will prove statements (1) and (2) above.

If k_{3} = 1 this statement is trivial, so we assume k_{3} = 0. We show that the assumptions k_{2} ≠ 0 and k_{4} = 0 lead to a contradiction. From the proof of Lemma 7.8, we know that for T to be minimal, every − from T_{5} must be bracketed. Because of the increasing conditions on the rows and columns of T, the −’s from the bottom row of T_{5} cannot be bracketed by +’s from T_{5}, so there must be at least k_{5} +’s from T_{1}T_{2}. Inspection of (B.1) shows us that the first block contributes no +’s, the second block contributes k_{1} − 2k_{2} many +’s, and the third block contributes 2k_{2} many +’s. We thus have k_{1} ≥ k_{5}; but Lemma 7.8 tells us that k_{1} + k_{2} = k_{4} + k_{5}, contradicting our assumption that k_{2} ≠ 0 and k_{4} = 0.

For the proof of statement (3), we must show that every configuration _{a} {c \atop b} in T avoids the following patterns (recall the Lecouvey D sliding algorithm from section 3.4): _{x} n ; n̄ and x n̄ ; n with x ≤ n − 1; _{n−1} n̄ ; \overline{n−1} ; _{n−1} n ; \overline{n−1} ; and c ≥ a, unless c = a = b̄. If T has any of these patterns, the top row will not slide over.

First, simply observe that the first four specified configurations exclude the possibility of having a column of the form a ; b̄ other than n ; n̄ or n̄ ; n. It therefore suffices to show that the presence of a column d ; ē, 2 ≤ d, e ≤ n − 1 implies that T avoids c ≥ a, unless c = a = b̄. We break our analysis of this criterion into several special cases:

Case 1: a and b are barred, c is unbarred: trivial.

Case 2: a is unbarred, b and c are barred: This excludes the possibility of having d ; ē ∈ T. Case 3: a and c are unbarred, b is barred: We know the − in the c-signature from c

must be bracketed; if it is by b, we have b = c̄. As we saw in the proof of Lemma 7.8, we must have the − in the (c − 1)-signature from c̄ bracketed by a c in the bottom row. This forces a ≥ c. If the − in the c-signature from c is bracketed by a c + 1, it also must be in the bottom row, forcing a > c.

Case 4: a, b, c all unbarred: Suppose c ≥ a. Let d be the leftmost unbarred letter weakly to the right of c which does not have its − bracketed by the letter immediately below it. (Such a letter exists, since we assume the occurence of d ; ē ∈ T, except when d = e; this case will be treated below.) This letter d must be bracketed by a d + 1 in the bottom row, and it must be weakly to the left of a. But we have d+1 > d ≥ c ≥ a ≥ d+1; contradiction.

If instead we have a d ; d̄ column, we must have the − from the d̄bracketed by a d in the bottom row weakly to the left of a. In this case d ≥ c ≥ a ≥ d so c = d, and we have a d d ; d̄ configuration, contradicting our assumption that T ∈ B(sΛ_{2}).

Case 5: a, b, c all barred: Similarly to case 4, suppose c ≥ a and let d be the rightmost barred letter weakly to the left of a which does not have its + bracketed by the letter immediately above it. (If none exists, we have a d̄ ; d case, see below.) It must be bracketed by a \overline{d̄+ 1} in the top row to the right of c. We then have d ≤ a ≤ c ≤ \overline{d̄+ 1}; contradiction.

If we have a d̄ ; d column (note that d is barred), the + from the d̄must be bracketed by a d in the top row to the right of c. This implies that d ≥ c ≥ a ≥ d, so a = d and we have a d̄ ; d_{d} configuration, again contradicting our assumption that T ∈ B(sΛ_{2}).

Appendix D. Proof of Lemma 7.10

In the notation of section 7.5, we have k_{2} = k_{4} = 0, and if k_{3} = 1, T_{3} = n ; n̄ or T_{3} = n̄ ; n. Lemma 7.8 thus tells us that k_{1} = k_{5}.

Next we show that a column {j \atop i} must be of the form {i−1 \atop i} for T to be in B̃_{min}^{2,s}. For i = 2 we have j = 1 by columnstrictness. Now suppose that {j \atop i} is the leftmost column such that j < i − 1. Then j contributes a Λ_{j} to φ(T) and hence ⟨c, φ(T)⟩ ≥ 2k_{1} + k_{3} + 1 = s + 1, so that T is not minimal. By a similar argument ⟨c, ε(T)⟩ > s unless all columns of the form ī ; j̄ must obey j = i − 1.

A column \overline{i} ; \overline{i−1} for i > 2 (resp. n ; \overline{n−1}) contributes a − to the (i − 2)-signature (resp. (n − 2)-signature) of T. This − can only be compensated by a + in the (i − 2)-signature (resp. (n − 2)-signature) from a column {i−1 \atop i} (resp. n−1 ; \overline{n}). Hence for T to be minimal the number of columns of the form {i−1 \atop i} (resp. n−1 ; \overline{n}) needs to be the same as the number of columns of the form \overline{i} ; \overline{i−1} (resp. n ; \overline{n−1}). This proves that T ∈ M(s).

References

[1] V. Chari, On the fermionc formula and the Kirillov-Reshetikhin conjecture, Internat. Math. Res. Notices 2001, no. 12, 629–65.

[2] V. Chari and A. Pressley, Quantum affine algebras and their representations, Representations of groups (Banff, AB, 1994), 59–78, CMS Conf. Proc., 16, Amer. Math. Soc., Providence, RI, 1995.

[3] V. Chari and A. Pressley, Twisted quantum affine algebras, Comm. Math. Phys. 196 (1998) 461–476.

[4] V. G. Drinfeld, Hopf algebra and the Yang–Baxter equation, Soviet. Math. Dokl. 32 (1985) 254–258.

[5] G. Hatayama, A. Kuniba, M. Okado, T. Takagi and Z. Tsuboi, Paths, crystals and fermionic formulae, MathPhys odyssey, 2001, 205–272, Prog. Math. Phys., 23, Birkhäuser Boston, Boston, MA, 2002.

[6] G. Hatayama, A. Kuniba, M. Okado, T. Takagi and Y. Yamada, Remarks on fermionic formula, Recent developments in quantum affine algebras and related topics (Raleigh, NC, 1998), 243–291, Contemp. Math., 248, Amer. Math. Soc., Providence, RI, 1999.

[7] J. Hong and S.-J. Kang, Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, 42. American Mathematical Society, Providence, RI, 2002. xviii+307 pp. ISBN: 0-8218-2874-6.

[8] M. Jimbo, A q-difference analogue of U(G) and the Yang–Baxter equation, Lett. Math. Phys. 10 (1985) 63–69.

[9] V.G. Kac, Infinite-dimensional Lie algebras. Third edition. Cambridge University Press, Cambridge, 1990. xxii+400 pp. ISBN: 0-521-37215-1.

[10] S.-J. Kang, M. Kashiwara, K.C. Misra, T. Miwa, T. Nakashima and A. Nakayashiki, Perfect crystals of quantum affine Lie algebras, Duke Math. J. 68 (1992), no. 3, 499–607.

[11] S.-J. Kang, M. Kashiwara, K.C. Misra, T. Miwa, T. Nakashima and A. Nakayashiki, Affine crystals and vertex models, Infinite analysis, Part A, B (Kyoto, 1991), 449–484, Adv. Ser. Math. Phys., 16, World Sci. Publishing, River Edge, NJ, 1992.

[12] M. Kashiwara, Crystalizing the q-analogue of universal enveloping algebras, Comm. Math. Phys. 133 (1990), no. 2, 249–260.

[13] M. Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63 (1991), no. 2, 465–516.

[14] M. Kashiwara, On crystal bases, in: Representations of groups (Banff, AB, 1994), 155–197, CMS Conf. Proc., 16, Amer. Math. Soc., Providence, RI, 1995.

[15] M. Kashiwara, On level-zero representation of quantized affine algebras, Duke Math. J. 112 (2002) no. 1, 117–195.

[16] M. Kashiwara, Level zero fundamental representations over quantized affine algebras and Demazure modules, preprint math.QA/0309142.

[17] M. Kashiwara and T. Nakashima, Crystal graphs for representations of the q-analogue of classical Lie algebras, J. Algebra 165 (1994), no. 2, 295–345.

[18] Y. Koga, Level one perfect crystals for B_{n}^{(1)}, C_{n}^{(1)}, and D_{n}^{(1)}, J. Algebra 217 (1999), no. 1, 312–334.

[19] S. V. Kerov, A. N. Kirillov and N. Y. Reshetikhin, Combinatorics, the Bethe ansatz and representations of the symmetric group, J. Soviet Math. 41 (1988), no. 2, 916–924.

[20] A. N. Kirillov and N. Y. Reshetikhin, The Bethe ansatz and the combinatorics of Young tableaux, J. Soviet Math. 41 (1988), no. 2, 925–955.

[21] A. N. Kirillov, A. Schilling and M. Shimozono, A bijection between Littlewood-Richardson tableaux and rigged configurations, Selecta Mathematica (N.S.) 8 (2002) 67–135.

[22] C. Lecouvey, Schensted-type correspondences and plactic monoid for types B_{n} and D_{n}, J. Algebraic Combin. 18 (2003), no. 2, 99–133.

[23] G. Lusztig, Quantum deformation of certain simple modules over enveloping algebras, Adv. Math. 70 (1988), no. 2, 237–249.

[24] H. Nakajima, t-analogue of the q-characters of finite dimensional representations of quantum affine algebras, Physics and combinatorics, 2000 (Nagoya), 196–219, World Sci. Publishing, River Edge, NJ, 2001.

[25] M. Okado, A. Schilling and M. Shimozono, A crystal to rigged configuration bijection for nonexceptional affine algebras, ”Algebraic Combinatorics and Quantum Groups”, Edited by N. Jing, World Scientific (2003), 85–124.

[26] M. Okado, A. Schilling and M. Shimozono, Virtual crystals and fermionic formulas of type D_{n+1}^{(2)}, A_{2n}^{(2)},

and C_{n}^{(1)}, Represent. Theory 7 (2003) 101–163.

[27] M. Okado, A. Schilling and M. Shimozono, Virtual crystals and Kleber’s algorithm Commun. Math. Phys. 238 (2003) 187–209.

[28] M. Shimozono, Affine type A crystal structure on tensor products of rectangles, Demazure characters, and nilpotent varieties, J. Algebraic Combin. 15 (2002), no. 2, 151–187.

[29] A. Schilling, A bijection between type D_{n}^{(1)} crystals and rigged configurations, preprint math.QA/0406248.

[30] A. Schilling and M. Shimozono, X = M for symmetric powers, in preparation.

Department of Mathematics, University of California, One Shields Avenue, Davis, CA 95616-8633, U.S.A.

   E-mail address: anne@math.ucdavis.edu
   URL: http://www.math.ucdavis.edu/∼anne

Department of Mathematics, University of California, One Shields Avenue, Davis, CA 95616-8633, U.S.A.

   E-mail address: sternberg@math.ucdavis.edu
   URL: http://www.math.ucdavis.edu/∼sternberg