for quantum affine algebras of type D

DOI 10.1007/s10801-006-8347-9

Finite-dimensional crystals B

for quantum affine algebras of type D

Anne Schilling·Philip Sternberg

Received: August 23, 2004 / Revised: October 4, 2005 / Accepted: October 12, 2005

CSpringer Science+Business Media, LLC 2006

Abstract The Kirillov–Reshetikhin modules W^r,sare finite-dimensional representations of quantum affine algebras U_q(g), labeled by a Dynkin node r of the affine Kac–Moody algebra gand a positive integer s. In this paper we study the combinatorial structure of the crystal basis B^2,scorresponding to W^2,sfor the algebra of type D_n⁽¹⁾.

Keywords Quantum affine algebras . Crystal bases . Kirillov-Reshetikhin crystals 2000 Mathematics Subject Classification Primary—17B37; Secondary—81R10

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 quan- tum 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 finite-dimensional representations of quantum affine algebras. The irreducible finite- dimensional U_q(g)-modules for an affine Kac–Moody algebragwere 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

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

A Schilling ()

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

e-mail: anne@math.ucdavis.edu P. Sternberg

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

e-mail: sternberg@math.ucdavis.edu

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finite-dimensional modules. They naturally correspond to the weight sr, wherer is the r -th fundamental weight of the underlying finite algebrag.

Kashiwara [12, 13] showed that in the limit q→0 the highest-weight representations of the quantum algebra Uq(g) have very special bases, called crystal bases. This construc- tion 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,sexist. For type A⁽¹⁾_n , the crystals B^r,sare 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⁽²⁾_2n,A^(2)†_2n ,D⁽²⁾_n+1→ A⁽¹⁾_2n−1 A⁽²⁾_2n−1,B_n⁽¹⁾→D⁽¹⁾_n+1

E₆⁽²⁾,F₄⁽¹⁾→E⁽¹⁾₆ D⁽³⁾₄ ,G⁽¹⁾₂ →D⁽¹⁾₄ .

The main open problems in the theory of finite-dimensional affine crystals are therefore the proof of the existence of B^r,sand 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 irre- ducible 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].

In [5, 6], a conjecture is presented on the decomposition of B^r,sas a crystal for the underly- ing finite algebra of type Dn. Specifically, as a type Dnclassical crystal the crystals B^r,sof type D_n⁽¹⁾for r ≤n−2 decompose as

B^r,s∼=

B(),

where the direct sum is taken over all weights for the finite algebra corresponding to partitions obtained from an r×s rectangle by removing any number of 2×1 vertical domi- noes. Here B() is the Uq(Dn)-crystal associated with the highest weight representation of highest weight(see [17]). In the sequel, we consider the case r=2, for which the above direct sum specializes to

B^2,s∼= s

k=0

B(k2), (1)

where once again the summands in the right hand side of the equation are crystals for the finite algebra. Our approach to study the combinatorics of B^2,s is as follows. First, we in- troduce tableaux of shape (s,s) to define a U_q(Dn)-crystal whose vertices are in bijection with the classical tableaux from the direct sum decomposition (1). Using the automorphism of the D⁽¹⁾_n Dynkin diagram which interchanges nodes 0 and 1, we define the unique action of ˜f0and ˜e0which makes this crystal into a perfect crystal ˜B^2,s of level s with an energy function. (See sections 2.3 and 2.4 for definitions of these terms.)

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Assuming the existence of the crystal B^r,s, the main result of our paper states that our combinatorially constructed crystal ˜B^2,s is the unique perfect crystal of level s with the classical decomposition (1) with a given energy function. More precisely:

Theorem 1.1. If B^2,sexists with the properties as in Conjecture 3.4, then ˜B^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,sdecomposes as_s

k=0V (k2), where V () is the classical module with highest weight, 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, crys- tal bases and perfect crystals is reviewed. Section 3 is devoted to crystals and the plactic monoid of type Dn. The properties of B^2,sof type D_n⁽¹⁾as conjectured in [5] are given in Conjecture 3.4. In section 4 the set underlying ˜B^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). The branching component graph is introduced in section 5, which is used in section 6 to define ˜e0and ˜f₀on ˜B^2,s. This makes ˜B^2,sinto an affine crystal. It is shown in section 7 that ˜B^2,sis perfect and that ˜B^2,sis the unique perfect crystal having the classical decomposition (1) with the appropriate energy function. This proves in particular Theorem 1.1. Finally, we end in section 8 with some open problems.

2. Review of quantum groups and crystal bases 2.1. Quantum groups

For n∈Zand a formal parameter q, we use the notation [n]q =qⁿ−q⁻ⁿ

q−q⁻¹ , [n]q!=ⁿ

k=1

[k]q, and m

n

q

= [m]q! [n]q![m−n]q!.

These are all elements ofQ(q), called the q-integers, q-factorials, and q-binomial coef- ficients, respectively.

Letgbe an arbitrary Kac-Moody Lie algebra with Cartan datum ( A, , ^∨,P,P^∨) and a Dynkin diagram indexed by I . Here A=(ai j)_i,j∈I is the Cartan matrix, P and P^∨ are the weight lattice and dual weight lattice, respectively,= {αi |i ∈I}is the set of simple roots and^∨= {hi |i∈I}is the set of simple coroots. Furthermore, let{si|i∈I}be the entries of the diagonal symmetrizing matrix of A and define qi =q^si and Ki =q^sihi. Then the quantum enveloping algebra Uq(g) is the associativeQ(q)-algebra generated by ei and

f_ifor i∈I , and q^hfor h∈P^∨, with the following relations (see e.g. [7, Def. 3.1.1]):

(1) q⁰=1, q^hq^h =q^h+hfor all h,h∈ P^∨, (2) q^heiq^−h=q^αi(h)eifor all h∈P^∨, (3) q^hfiq^−h =q^αi(h)fifor all h∈P^∨, (4) eifj− fjei=δi jKi−Ki⁻¹

qi−qi⁻¹ for i,j∈I , (5)1−ai j

k=0 (−1)^k_{1−ai j}

k qie^1−a_i ^{i j−k}eje^k_i =0 for all i= j, (6)1−ai j

k=0 (−1)^k_{1−ai j}

k qif_i^{1−ai j−k}fjf_i^k =0 for all i = j.

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2.2. Crystal bases

The quantum algebra Uq(g) can be viewed as a q-deformation of the universal enveloping algebra U (g) ofg. Lusztig [23] showed that the integrable highest weight representations of U (g) can be deformed to Uq(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 Uq(g)-module and R the subset of all elements inQ(q) which are regular at q=0. Kashiwara [12, 13] introduced Kashiwara operators ˜ei and ˜f_i as certain linear combinations of powers of eiand fi. A crystal latticeLis a free R-submodule of M that generates M overQ(q), has the same weight decomposition and has the property that

˜

eiL⊂Land ˜fiL⊂Lfor all i∈I . The passage fromLto the quotientL/qLis referred to as taking the crystal limit. A crystal basis is aQ-basis ofL/qLwith certain properties.

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

f˜i(b)=b⇔e˜i(b)=b if b,b∈B (2) wt( ˜fi(b)) = wt(b)−αiif ˜fi(b)∈B (3)

hi,wt(b) = ϕi(b)−εi(b). (4)

Here for b∈B

εi(b)=max{n≥0|e˜_iⁿ(b)=∅} ϕi(b)=max{n≥0| f˜_iⁿ(b)=∅}.

(It is assumed thatϕi(b), εi(b)<∞for all i∈I and b∈B.) A Uq(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 blabeled i ∈I , if and only if ˜f_i(b)=b.

Let B1and B2be Uq(g)-crystals. The Cartesian product B2×B1 can also be endowed with the structure of a Uq(g)-crystal. The resulting crystal is denoted by B2⊗B1 and its elements (b₂,b₁) are written b₂⊗b₁. (The reader is warned that our convention is opposite to that of Kashiwara [14]). For i∈I and b=b₂⊗b₁∈B₂⊗B₁, we have wt(b)=wt(b1)+ wt(b2),

f˜_i(b2⊗b₁)=

f˜i(b2)⊗b1 ifεi(b2)≥ϕi(b1)

b₂⊗ f˜i(b1) ifεi(b2)< ϕi(b1) (5) and

˜

ei(b2⊗b1)=

e˜i(b2)⊗b₁ ifεi(b2)> ϕi(b1)

b2⊗e˜i(b1) ifεi(b2)≤ϕi(b1). (6) Combinatorially, this action of ˜f_i and ˜ei 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

− · · · −

ϕi(b2) times

+ · · · +

εi(b2) times

− · · · −

ϕi(b1) times

+ · · · +

εi(b1) times

.

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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

− · · · −

ϕtimes

+ · · · +

εtimes

.

If ϕ=0 then ˜fi(b)=∅; otherwise ˜fi acts on the tensor factor corresponding to the rightmost symbol−in the reduced i -signature of b. Similarly, ifε=0 then ˜ei(b)=∅; oth- erwise ˜eiacts on the leftmost symbol+in the reduced i -signature of b. From this it is clear that

ϕi(b2⊗b1)=ϕi(b2)+max(0, ϕi(b1)−εi(b2)), εi(b₂⊗b₁)=εi(b₁)+max(0,−ϕi(b₁)+εi(b₂)).

2.3. Perfect crystals

Of particular interest is a class of crystals called perfect crystals, which are crystals for affine algebras satisfying a set of very special properties. These properties ensure that 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 a Kac-Moody algebrag; for the remainder of this section,gis of affine type. The center ofgis one-dimensional and is generated by the canonical central element c=

i∈Ia_i^∨h_i, where the a_i^∨are the numbers on the nodes of the Dynkin diagram of the algebra dual toggiven in Table Aff of [9, section 4.8]. Moreover, the imaginary roots ofgare nonzero integral multiples of the null rootδ=

i∈Iaiαi, where the ai are the numbers on the nodes of the Dynkin diagram ofggiven in Table Aff of [9].

Define Pcl=P/Zδ, P_cl⁺= {λ∈ P_cl|hi, λ ≥0 for all i ∈I}, and U_q(g) to be the quantum enveloping algebra with the Cartan datum ( A, , ^∨,P_cl,P_cl^∨).

Define the set of levelweights to be (P_cl⁺)= {λ∈ P_cl⁺|c, λ =}. For a crystal basis element b∈B, define

ε(b)=

i∈I

εi(b)i and ϕ(b)=

i∈I

ϕi(b)i,

whereiis the i -th fundamental weight ofg. Finally, for a crystal basis B, we define Bmin

to be the set of crystal basis elements b such thatc, ε(b)is minimal over b∈B.

Definition 2.1. A crystal B is a perfect crystal of levelif:

(1) B⊗B is connected;

(2) there existsλ∈ P_clsuch that wt(B)⊂λ+

i=0Z_≤0αiand #(B_λ)=1;

(3) there is a finite-dimensional irreducible U_q(g)-module V with a crystal base whose crys- tal graph is isomorphic to B;

(4) for any b∈B, we havec, ε(b) ≥;

(5) the mapsεandϕfrom Bminto (P_cl⁺)are bijective.

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

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2.4. Energy function

The existence of an affine crystal structure usually provides an energy function. Let B1and B2be finite U_q(g)-crystals. Then following [11, Section 4] we have:

(1) There is a unique isomorphism of U_q(g)-crystals R=RB2,B1: B2⊗B1→B1⊗B2. (2) There is a function H=HB2,B1: B₂⊗B₁→Z, unique up to global additive constant,

such that H is constant on classical components and, for all b2∈B₂and b1∈B₁, if R(b₂⊗b₁)=b₁⊗b₂, then

H ( ˜e₀(b2⊗b₁))=H (b₂⊗b₁)+

⎧⎪

⎨

⎪⎩

−1 ifε0(b2)> ϕ0(b1) andε0(b₁)> ϕ0(b₂) 1 ifε0(b2)≤ϕ0(b1) andε0(b₁)≤ϕ0(b₂) 0 otherwise.

(7)

We shall call the maps R and H the local isomorphism and local energy function on B2⊗B1, respectively. The pair (R,H ) is called the combinatorial R-matrix.

Let u(B1) and u(B2) be extremal vectors of B1and B2, respectively (see [15] for a defi- nition of extremal vectors). Then

R(u(B₂)⊗u(B₁))=u(B₁)⊗u(B₂).

It is convenient to normalize the local energy function H by requiring that H (u(B2)⊗u(B1))=0.

With this convention it follows by definition that H_B1,B2◦R_B2,B1=H_B2,B1 as operators on B2⊗B1.

We wish to define an energy function DB : B→Zfor tensor products of perfect crystals of the form B^r,s[5, Section 3.3]. Let B=B^r,sbe perfect. Then there exists a unique element b∈B such thatϕ(b)=lev (B)0. Define DB: B→Zby

DB(b)=H_B,B(b⊗b)−H_B,B(u(B)⊗b). (8) The intrinsic energy DB for the L-fold tensor product B=BL⊗ · · · ⊗B1where Bj= B^rj,sj is given by

D_B=

1≤i<j≤L

H_iR_i+1R_i+2· · ·R_j−1+^L

j=1

D_BjR₁R₂· · ·R_j−1,

where Hiand Ri 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 finite Lie algebra of type Dn and the affine Kac-Moody algebra of type D⁽¹⁾_n . 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 Dn.

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3.1. Dynkin data

For type Dn, the simple roots are

αi =εi−εi+1 for 1≤i <n αn =εn−1+εn

(9) and the fundamental weights are

i =ε1+ · · · +εi for 1≤i≤n−2 n−1=(ε1+ · · · +εn−1−εn)/2

n =(ε1+ · · · +εn−1+εn)/2

whereεi∈Zⁿis the i -th unit standard vector. The central element for D⁽¹⁾_n is given by c=h0+h1+2h2+ · · · +2h_n−2+h_n−1+hn.

3.2. Classical crystals

Kashiwara and Nakashima [17] described the crystal structure of all classical highest weight crystals B() of highest weightexplicitly. For the special case B(k2) as occuring in (1) each crystal element can be represented by a tableau of shapeλ=(k,k) on the partially ordered alphabet

1<2<· · ·<n−1< n

¯

n <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_b is in the filling, then ba;

3. No configuration of the form ^{a a}_a¯ or^a_{a ¯a¯} appears;

4. No configuration of the form ⁿ⁻¹_n . . ._n−1ⁿ orⁿ⁻¹_n¯ . . ._n−1^n¯ appears;

5. No configuration of the form ¹_¯1appears.

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

Also, observe that the conditions given in [7] apply only to adjacent columns, not to non- adjacent columns as in condition 4 above. However, Criterion 3.1 is unchanged by replacing condition 4 with the following:

(4a) No configuration of the form ⁿ⁻¹_{n n−1}ⁿ orⁿ⁻¹_{n¯ n−1}^n¯ appears.

To see this equivalence, observe that by conditions 1 and 2 the only columns that can appear between ⁿ⁻¹_n and _n−1ⁿ are ⁿ⁻¹_n , ⁿ⁻¹_n−1, and _n−1ⁿ , and they must appear in that order from left to right. If a column of the form ⁿ⁻¹_n−1appears, we have a configuration of the form

n−1 n−1

n−1, which is forbidden by condition 3. On the other hand, if no column of the formⁿ⁻¹_n−1 appears, the columns ⁿ⁻¹_n and_n−1ⁿ are adjacent, which is disallowed by condition 4a.

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

1 ¹- 2 · · · ⁿ⁻-² n−1

n−1

@@nR

n @@R

n

n

n−1

n−1 ⁿ⁻-² · · · 2 ¹- 1

For a tableau T= a1

b₁· · ·ak

bk ∈B(k2), the action of the Kashiwara operators ˜f_iand ˜ei is defined as follows. Consider the column wordwT =b₁a₁· · ·b_ka_kand view this word as an element in B(1)^⊗2k. Then ˜f_iand ˜eiact 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 wordwT =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,

f˜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

f˜₄(T )= 1 2 4 ¯3 ¯3

¯4 ¯4 ¯4 ¯2 ¯1 and e˜₄(T )= 1 2 4 ¯3 ¯3 3 3 ¯4 ¯2 ¯1 .

3.3. Dual crystals

Letω0 be the longest element in the Weyl group of Dn. The action ofω0 on the weight lattice P of Dnis given by

ω0(i)= −τ(i )

ω0(αi)= −ατ(i )

whereτ: 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.

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There is a unique involution∗: B→B, called the dual map, satisfying wt(b^∗)=ω0wt(b)

˜

ei(b)^∗= f˜_τ(i )(b^∗) f˜i(b)^∗=e˜_{τ(i )}(b^∗).

The involution∗sends the highest weight vector u∈B() to the lowest weight vector (the unique vector in B() of weightω0()). We have

(B1⊗B2)^∗∼= B2⊗B1

with (b1⊗b₂)^∗→b₂^∗⊗b₁^∗.

Explicitly, on B(1) the involution∗is given by i←→i

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

Example 3.3. If

T = 1 1 2

3 ∈B(21+2) we have

T^∗= 3 1 1

2 .

3.4. Plactic monoid of type D

The plactic monoid for type D is the free monoid generated by{1, . . . ,n,n, . . . ,¯ ¯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 =xLx_L−1· · ·x1is a word such that x_i+1≤xifor i =1, . . . ,L−1. Note that the letters n and ¯n 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≥z. A column C is called admissible if L¯ ≤n and for any pair (z,z) of let-¯ ters in C with z≤n we have N (z)≤z. The Lecouvey D equivalence relations are given by:

(1) If x =z, then¯

x zy≡zx y for x≤y<z and yzx≡yx z for x<y≤z.

(2) If 1<x<n and x≤y≤x, then¯

(x−1)(x−1)y≡x x y and y ¯x x¯ ≡y(x−1)(x−1).

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(3) If x≤n−1, then

n ¯x ¯n≡n ¯n ¯x n ¯xn¯ ≡nn ¯x¯ and

xn ¯n≡nx ¯n x ¯nn≡nxn¯ . (4)

n ¯nn¯ ≡n(n¯ −1)(n−1) nn ¯n≡n(n−1)(n−1) and

(n−1)(n−1) ¯n≡n ¯n ¯n (n−1)(n−1)n≡nnn¯ .

(5) Considerwa non-admissible column word each strict factor of which is admissible. Let z be the lowest unbarred letter such that the pair (z,¯z) occurs inwand N (z)>z. Then w≡w˜ is the column word obtained by erasing the pair (z,¯z) inwif z<n, by erasing a pair (n,n) 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=¯z, then

y

x z ≡ x y

z ≡ x y

z for x≤y<z,

and x

y z ≡ x

y z ≡ x z

y for x<y≤z.

(2) If 1<x<n and x ≤y≤x, then¯ y

x−1 x−1 ≡ x−1 y

x−1 ≡ x y x¯

and x

y x ≡ x−1

y x−1 ≡ x−1 x−1

y .

(3) If x≤n−1, then

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩ n¯ n x ≡ n¯

n ¯x ≡ n ¯x¯ n n

¯

n x ≡ n

¯

n ¯x ≡ n ¯x

¯ n

and

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎪⎪

⎩

¯ n

x n ≡ x ¯n

n ≡ x ¯n n n

x n ≡ x n

n¯ ≡ x n n¯

.

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(4)

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩ n

n ¯n¯ ≡ n−1

¯

n n−1 ≡ n−1 n−1

¯ n

¯ n

n n ≡ n−1

n n−1 ≡ n−1 n−1 n

and

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

n¯

n−1 n−1 ≡ n−1 n¯

n−1 ≡ n ¯n¯ n n

n−1 n−1 ≡ n−1 n

n−1 ≡ n n

¯ n

.

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,swhich we state here in the specific case of B^2,sof type D⁽¹⁾_n .

Conjecture 3.4 ([5]). If the crystal B^2,sof type D⁽¹⁾_n exists, it has the following properties:

(1) As a classical crystal B^2,sdecomposes as B^2,s∼=_s

k=0B(k2).

(2) B^2,sis perfect of level s.

(3) B^2,sis equipped with an energy function DB^2,s such that DB^2,s(b)=k−s if b is in the component of B(k2) (in accordance with (8)).

4. Classical decomposition of ˜B^2,s

In this section we begin our construction of the crystal ˜B^2,s mentioned in Theorem 1.1.

We do this by defining a Uq(Dn)-crystal with vertices labeled by the setT(s) of tableaux of shape (s,s) which satisfy conditions 1, 2, and 4 of Criterion 3.1. We will construct a bijection betweenT(s) and the vertices of_s

i=0B(i2), so thatT(s) may be viewed as a Uq(Dn)-crystal with the classical decomposition (1). In section 6 we will define ˜f0and ˜e0

onT(s) to give it the structure of a perfect U_q(D_n⁽¹⁾)-crystal. This crystal will be ˜B^2,s. The reader may note in later sections that the main result of the paper does not depend on this explicit labeling of the vertices of ˜B^2,s. We have included it here because a description of the crystal in terms of tableaux will be needed to obtain a bijection with rigged config- urations. It is through such a bijection that we anticipate being able to prove the X =M conjecture for type D, as has already been done for special cases in [25, 29, 30].

Proposition 4.1. Let T ∈T(s)\B(s2) with T = ¹_¯1· · ·¹_¯1, and define ¯¯i=i for 1≤i≤n.

Then there is a unique a∈ {1, . . . ,n,n}¯ and m∈Z>0 such that T contains one of the

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following configurations (called an a-configuration):

a b1

a

¯ a· · ·a

¯ a

m

c1

d1, where b₁=a, and c¯ ₁=a or d₁=a;¯ b₂

c₂ a a¯· · ·a

a¯

m

d₂

a¯, where d₂=a, and b₂=a or c₂=a;¯ b₃

c₃ a a¯· · ·a

a¯ m+1

d₃

e₃, where b₃=a and e₃=a.¯

Proof: If s=1, the setT(s)\B(s2) contains only ¹_¯1, so that the statement of the propo- sition is empty. Hence assume that s≥2. The existence of an a-configuration for some a∈ {1, . . . ,n,n}¯ follows from the fact that T violates condition 3 of Criterion 3.1. The conditions on bi,ci,di for i=1,2,3 and e3 can be viewed as stating that m is chosen to maximize the size of the a-configuration. Condition 1 of Criterion 3.1 and the conditions on the parameters bi,ci,di,e₃imply that there can be no other a-configurations in T .

The map D_2,s:T(s)→_s

k=0B(k2), called the height-two drop map, is defined as fol- lows for T∈T(s). If T = ¹_¯1· · ·¹_¯1, then D_2,s(T )=∅∈B(0). If T ∈B(s2), 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 removinga

a¯· · ·a a¯

m

.

Theorem 4.2. Let T ∈T(s). Then D_2,s(T ) satisfies Criterion 3.1, and is therefore a tableau in_s

k=0B(k2).

Proof: Condition 1 is satisfied since the relation ≤ on our alphabet is transitive.

Conditions 2 and 5 are automatically satisfied, since the columns that remain are not changed. Condition 3 is satisfied since by Proposition 4.1, there can be no more than one a-configuration in T . Condition 4 is satisfied since D_2,sdoes not remove any columns of the

formⁿ⁻¹_n ,ⁿ⁻¹_n¯ ,_n−1ⁿ , or_n−1^n¯ .

In Proposition 4.5, we will show that D_2,sis a bijection by constructing its inverse.

Example 4.3. We have

T= 1 2 3 3

4 2 2 1 , D_2,4(T )= 1 3 3 4 2 1 . The inverse of D_2,s is the height-two fill map F_2,s:_s

k=0B(k2)→T(s). Let t=

a1

b1· · ·^a_b^k_k ∈B(k2). If k=s, F_2,s(t)=t. If k<s, then F_2,s(t) is obtained by finding a sub- tableau^a_bⁱ

i a_i+1

bi+1 in t such that Criterion 4.4.

bi≤a¯i≤b_i+1or ai ≤¯b_i+1≤a_i+1.

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(Recall that ¯¯i=i for i∈ {1, . . . ,n}.) Note that the first pair of inequalities imply that ai

is unbarred, and the second pair of inequalities imply that b_i+1is barred. We may therefore insert between columns i and i+1 of t either the configurationai

¯ ai · · ·ai

¯ ai

s−k

or ¯b_i+1 b_i+1· · · ¯b_i+1

b_i+1

s−k

, depending on which part of Criterion 4.4 is satisfied. We say that i is the filling location of t. If no such subtableau exists, then F_2,swill either append ak

¯ ak · · ·ak

¯ ak

s−k

to the end of t, or

prepend ¯b1

b₁· · · ¯b1

b₁

s−k

to t. In these cases the filling locations are k and 0, respectively.

Proposition 4.5. The map F_2,sis well-defined on_s

i=0B(i2).

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

Lemma 4.6. Suppose that t ∈_s−1

k=0B(k2) has no subtableaux ^a_bⁱ_i^a_bⁱ⁺¹_i+1 satisfying Criterion 4.4. Then exactly one of either appending ^a_a¯^k

k· · ·^a_a¯^k_k or prepending _b^¯b1

1· · ·_b^¯b1₁ to t will produce a tableau inT(s)\B(s2).

Proof: Suppose t = ^a_b¹₁· · ·^a_b^k_k ∈B(k2) is as above for k<s. We will show that if prepend- ing _b^¯b1

1· · ·_b^¯b1₁ to t does not produce a tableau inT(s)\B(s2), then appending ^a_a¯^k

k· · ·^a_a¯^k_k to t will produce a tableau inT(s)\B(s2). There are two reasons we might not be able to prepend_b^¯b1₁· · ·_b^¯b1₁; b1may be unbarred, or we may have a1< ¯b1.

First, suppose b1is unbarred. If bkis also unbarred, then bkis certainly less than ¯ak, so we may append^a_a¯^k_k· · ·^a_a¯^k_k to t. Hence, suppose that bkis barred. We will show that akis unbarred and ¯ak>bk.

We know that t has a subtableau of the form ^a_bⁱ

i ai+1

bi+1 such that bi is unbarred and b_i+1is barred. It follows that ai is unbarred, and therefore ¯ai >bi. Since ^a_bⁱ_i^a_bⁱ⁺¹_i+1 does not satisfy Criterion 4.4, this means that ¯ai>b_i+1, which is equivalent to ¯b_i+1>ai. Once again ob- serving that ^a_bⁱ

i ai+1

bi+1 does not satisfy Criterion 4.4, this implies that ¯b_i+1>a_i+1; i.e., a_i+1is unbarred, and ¯a_i+1>b_i+1.

We proceed with an inductive argument on i< j<k. Suppose that^a_b^j

j aj+1

b_j+1is a subtableau of t such that bjand b_j+1are barred, ajis unbarred, and ¯aj >b_j. By reasoning identical to the above, we conclude that

a¯_j >b_j+1⇒ ¯b_j+1>a_j⇒ ¯b_j+1>a_j+1⇒a¯_j+1>b_j+1, (10) which once again means that a_j+1is unbarred.

This inductively shows that ak is unbarred and ¯ak >b_k, so we may append ^a_a¯^k_k · · ·^a_a¯^k_k to t to get a tableau inT(s)\B(s2). By a symmetrical argument, we conclude that if ak is barred, then we may prepend _b^¯b1

1· · ·_b^¯b1₁ to t.

Now, suppose that b₁is barred and ¯b₁>a₁. This means that a₁is unbarred and ¯a₁>b₁, so the induction carried out in equation 10 applies. It follows that akis unbarred and ¯ak >b_k, so once again we may append ^a_a¯^k_k · · ·^a_a¯^k_k to t. Also, by a symmetrical argument, when ak is

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unbarred and bk >a¯k, we may prepend_b^¯b1

1· · ·_b^¯b1₁ to t. Thus, when no subtableau of t satisfy Criterion 4.4, either appending ^a_a¯^k

k· · ·^a_a¯^k_k or prepending _b^¯b1

1· · ·_b^¯b1₁ to t will produce a tableau

inT(s)\B(s2).

Lemma 4.7. Any tableau t= ^a_b¹₁· · ·^a_b^k_k ∈_s−1

k=0B(k2) has no more than two filling loca- tions. 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_b^i∗

i∗

ai∗+1

b_i∗+1which satisfies Criterion 4.4.

Suppose that the first condition bi∗≤a¯i∗≤bi∗+1of Criterion 4.4 is satisfied, and consider whether i_∗+1 can be a filling location. If bi∗+1≤a¯i∗+1≤bi∗+2, we have

b_i∗+1≤a¯_i∗+1≤a¯_i∗≤b_i∗+1,

which implies that ¯ai∗=a¯i∗+1=bi∗+1, so that t violates part 3 of Criterion 3.1. Similarly, if a_i∗+1≤¯bi∗+2≤a_i∗+2, then we have

¯

ai∗+1≤a¯i∗≤bi∗+1≤bi∗+2≤a¯i∗+1,

which also implies that ¯ai∗=a¯_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 4.4 is satisfied by bi∗≤a¯i∗≤bi∗+1, then i_∗+1 is not a filling location. Furthermore, this argument shows that ai∗+1>ai∗ or bi∗+1>a¯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 4.4 is satisfied by ai∗≤ ¯bi∗+1≤a_i∗+1. The condition ai∗+1≤ ¯bi∗+2≤ai∗+2for i_∗+1 to be a filling location implies that

¯b_i∗+2≤ ¯b_i∗+1≤a_i∗+1≤ ¯b_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 4.4 is satisfied by bi∗+1≤a¯i∗+1≤bi∗+2. Note that this inequal- ity implies that ai∗+1≤ ¯b_i∗+1, which tells us that ai∗+1= ¯b_i∗+1. Thus, choosing to insert

¯bi∗+1

b_i∗+1· · ·_b^¯bi_i∗+1^∗+1 between columns i_∗ and i_∗+1 or to insert ^a_a¯^i∗+1

i∗+1· · ·^a_a¯ⁱ_i∗+1^∗+1 between columns i_∗+1 and i_∗+2 does not change F_2,s(t). Since i_∗+1 is a filling location with Criterion 4.4 satisfied by bi∗≤a¯i∗≤bi∗+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., b1is barred, a1is unbarred, and

¯b1≤a₁. If 1 is a filling location, Criterion 4.4 is satisfied by b1≤a¯₁≤b₂; otherwise, part 3 of Criterion 3.1 is violated. Put together, this means that ¯a1=b₁, so prepending _b^¯b1₁· · ·_b^¯b1₁ to t and inserting ^a_a¯¹₁· · ·^a_a¯¹₁ 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 4.8. Let s=4. Then t= 1 2 3

4 2 1 , F_2,4(t)= 1 2 2 3 4 2 2 1 .

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