Existence of Crystal Bases for

Existence of Crystal Bases for

Kirillov-Reshetikhin Modules of Type D

Dedicated to Professor Noriaki Kawanaka on his sixtieth birthday

By

MasatoOkado^∗

Abstract

We show that a crystal base exists for any Kirillov-Reshetikhin module of type D⁽¹⁾n , generalizing the result of Kang et al. for the end nodes of the Dynkin diagram ofDn.

§1. Introduction

Let gbe an affine algebra and let U_q(g) be the corresponding quantum affine algebra without degree operator. Among irreducible finite-dimensional U_q(g)-modules there exists a distinguished family called Kirillov-Reshetikhin modules (KR modules for short). They were introduced in [17] in connec- tion with a certain conjectural formula of multiplicities of irreducible U_q(g₀)- modules in a tensor product of those modules. Here g₀ stands for the finite- dimensional simple Lie algebra whose Dynkin diagram is obtained by removing the 0-vertex, that is prescribed in [10], from that ofg. It is known [5, 4] that irreducible finite-dimensional U_q(g)-modules are classified byn-tuples of poly- nomials called Drinfeld polynomials, where nis the rank of g₀.

Let us define KR modules by the Drinfeld polynomials. Letk∈ {1,2, . . . , n}, l∈Z_>0andaan invertible element ofQ(q). A KR module ˜W_l,a^(k)is defined to be the unique irreducible finite-dimensional U_q(g)-module that has

P_i(u) =

(1−aq_i^1−lu)(1−aq_i^3−lu)· · ·(1−aq_i^l−1u) ifi=k,

1 otherwise

Communicated by M. Kashiwara. Received November 17, 2006.

2000 Mathematics Subject Classification(s): 17B37.

∗Department of Mathematical Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan.

as Drinfeld polynomials. (See Section 2.1 for the definition ofq_i.) Whenl= 1 it is also called a fundamental representation. Since a fundamental representation is known to have a crystal base [14], we choose a^† so that ˜W_l,a^(k)_† has a crystal base and redefineW_l,a^(k)= ˜W_l,a^(k)_†a. W_l,1^(k)is also denoted simply byW_l^(k).

The above-mentioned conjectural multiplicity formula was shown when g is non-twisted by combining two results. The first one is the proof of cer- tain algebraic relations among characters of KR modules, called Q-systems, presented in [20] for simply-laced cases and in [8] for all non-twisted cases.

The second one is a derivation of the multiplicity formula, called fermionic formula, in [16] for type A and in [7] for all non-twisted cases, by using the Q-systems. However, it deserves to emphasize that there is also a q-analogue of the conjecture, calledX =M conjecture [7, 6]. The definition ofX requires the existence of the crystal base of a KR module. Despite many efforts as in [12, 11, 25, 18, 9, 14, 19, 23, 2], this existence problem is yet to be settled. For type D for instance, the crystal base has been shown to exist forW_l^(k) where k= 1, n−1, n;l∈Z>0 in [12] and forW₁^(k)for arbitrarykin [18, 14].

In this paper we prove the following theorem, thereby settling the problem for typeD.

Theorem 1.1. For2≤k≤n−2andl≥1, theU_q(D⁽¹⁾_n )-moduleW_l^(k) has a crystal pseudobase.

Here (L, B) is said to be a crystal pseudobase if (L, B/{±1}) is a crystal base.

(See Definition 2.1 for the definition of a crystal base.) Let us give a short sketch of our proof. We follow the technique already developed in [12], namely, from a fundamental representationW₁^(k)we constructW_l^(k)for anylby fusion construction (Section 2.3), and we apply a criterion of the existence of a crystal pseudobase (Proposition 2.6) to the constructed moduleW_l^(k). Practically, we need to check two conditions ((2.27) and (2.28)). Checking the second one is not difficult, if once we find out the vectors {u_j} correctly, whereas checking the first one requires information on the image W and the kernel N of the R-matrix R(x, y) : W_1,x^(k)⊗W_1,y^(k) −→ W_1,y^(k)⊗W_1,x^(k) at x/y = q². Up to now such information was obtained by calculating the spectral decomposition of the R-matrix when dealing with W_l^(k) for higher l. It seems to be the reason why showing the existence of crystal bases of W_l^(k) for higher k has not been succeeded so far, since the calculation of the corresponding R-matrix is too much complicated. However, thanks to the result by Nakajima [20], we are now able to identify W and N with tensor products of KR modules (Lemma 3.4). Using the crystal base of W and a property of a bilinear form between

W_1,q^(k)⊗W_1,q^(k)₋₁ and W_1,q^(k)₋₁ ⊗W_1,q^(k), we can check the first condition of the criterion. It is known that a U_q(g)-module with a connected crystal base is irreducible. Therefore, once W_l^(k) is shown to have a crystal pseudobase, it follows that W_l^(k) is an irreducible finite-dimensional module with the desired Drinfeld polynomials, since it is a simple quotient ofW_1,q^(k)_1−l⊗W_1,q^(k)_3−l⊗ · · · ⊗ W_1,q^(k)_l−1.

After the author finished the manuscript, he learned from Kashiwara that the moduleW_l^(k)can be shown to be irreducible by Theorem 9.2 of [14]. Once it is established, the character is known by [3]. Hence it turns out that there is no need to prove the inequality of the character in (i) just after Proposition 3.7. However, this does not seem to prove thatW_l^(k)is isomorphic to a module of the form ofV^⊗l/_l−2

i=0V^⊗i⊗N⊗V^{⊗(l−2−i)}. The author was also informed from Nakajima that the existence of a polarization on the fundamental rep- resentation W₁^(k)was shown in [24] (see also [21, 1] for more general results).

Hence similar calculations of the prepolarization as in Section 5 will give a proof of the existence of crystal bases for KR modules of other quantum affine algebras.

§2. Crystal Base and Fusion Construction

§2.1. Crystal base

In this subsection we briefly recall the definition of crystal bases. For more details along with the definition ofU_q(g), refer to [13].

Let gbe a symmetrizable Kac-Moody Lie algebra and letM be a U_q(g)- module. M is said to be integrable if M =⊕_λ∈PM_λ, dimM_λ<∞for any λ, and for anyi,M is a union of finite-dimensionalU_q(g_i)-modules. HereP is the weight lattice of g, M_λ is the weight space ofM of weight λand U_q(g_i) is the subalgebra generated by Chevalley generatorse_i andf_i. IfM is integrable, we have

(2.1) M =

0≤n≤hi,λ

f_i⁽ⁿ⁾(Kere_i∩M_λ).

Note that we use the following notations: [m]_i= (q^m_i −q_i^−m)/(q_i−q_i⁻¹),[n]_i! = _n

m=1[m]_i, f_i⁽ⁿ⁾=f_iⁿ/[n]_i! withq_i=q^(αi,αi), where (, ) is an invariant bilinear form on P. We define the endomorphisms ˜e_i,f˜_i ofM by

(2.2) f˜_i(f_i⁽ⁿ⁾u) =f_i⁽ⁿ⁺¹⁾u and e˜_i(f_i⁽ⁿ⁾u) =f_i⁽ⁿ⁻¹⁾u

foru∈Kere_i∩M_λ with 0≤n≤ h_i, λ. Similarly, we have

(2.3) M =

0≤n≤−hi,µ

e⁽ⁿ⁾_i (Kerf_i∩M_µ).

These two decompositions are related as follows:

if 0≤n≤ h_i, λand u∈Kere_i∩M_λ,

thenv=f_i^(hi,λ)ubelongs to Kerf_i∩M_si(λ)andf_i⁽ⁿ⁾u=e^(h_i ^i,λ−n)v.

Here s_i(λ) =λ− h_i, λα_i. Hence we obtain

(2.4) f˜_i(e⁽ⁿ⁾_i v) =e⁽ⁿ⁻¹⁾_i v and e˜_i(e⁽ⁿ⁾_i v) =e⁽ⁿ⁺¹⁾_i v forv∈Kerf_i∩M_µ with 0≤n≤ − h_i, µ.

Let us now look at the definition of a crystal base. Let Abe the subring of Q(q) consisting of rational functions without poles atq = 0. LetM be an integrable U_q(g)-module.

Definition 2.1. A pair (L, B) is called a crystal base ofM if it satisfies the following 6 conditions:

Lis a free sub-A-module ofM such that M Q(q)⊗_AL, (2.5)

B is a base of theQ-vector spaceL/qL, (2.6)

˜

e_iL⊂Land ˜f_iL⊂Lfor anyi.

(2.7)

By (2.7) ˜e_i and ˜f_i act onL/qL.

˜

e_iB⊂B∪ {0} and ˜f_iB⊂B∪ {0}. (2.8)

L=⊕_λ∈PL_λandB =_λ∈PB_λ (2.9)

where L_λ=L∩M_λ andB_λ=B∩(L_λ/qL_λ).

(2.10) Forb, b ∈B, b= ˜f_ib if and only if ˜e_ib =b.

Standard notations are in order. Forb∈B we set

ε_i(b) = max{m∈Z≥0|e˜^m_i b= 0}, ϕ_i(b) = max{m∈Z≥0|f˜_i^mb= 0}, (2.11)

ε(b) =

i

ε_i(b)Λ_i, ϕ(b) =

i

ϕ_i(b)Λ_i, (2.12)

wtb=ϕ(b)−ε(b).

(2.13)

Here {Λ_i}stands for the set of fundamental weights ofg.

The crystal base behaves nicely under the tensor product. Let (L_j, B_j) be the crystal base of an integrableU_q(g)-moduleM_j (j= 1,2). SetL=L₁⊗AL₂ andB ={b₁⊗b₂|b_j∈B_j(j= 1,2)}. Then (L, B) is a crystal base ofM₁⊗M₂. Moreover, the action of ˜e_i and ˜f_i becomes very simple as

˜

e_i(b₁⊗b₂) =

˜e_ib₁⊗b₂ ifϕ_i(b₁)≥ε_i(b₂), b₁⊗e˜_ib₂ ifϕ_i(b₁)< ε_i(b₂), (2.14)

f˜_i(b₁⊗b₂) =

f˜_ib₁⊗b₂ ifϕ_i(b₁)> ε_i(b₂), b₁⊗f˜_ib₂ ifϕ_i(b₁)≤ε_i(b₂).

(2.15)

Here 0⊗b and b⊗0 are understood to be 0. We denote this B byB₁⊗B₂. ε_i, ϕ_i andwt are given by

ε_i(b₁⊗b₂) = max(ε_i(b₁), ε_i(b₁) +ε_i(b₂)−ϕ_i(b₁)), (2.16)

ϕ_i(b₁⊗b₂) = max(ϕ_i(b₂), ϕ_i(b₁) +ϕ_i(b₂)−ε_i(b₂)), (2.17)

wt(b₁⊗b₂) =wtb₁+wtb₂. (2.18)

Next lemma is used later in Section 4.

Lemma 2.2. Let (L, B)be a crystal base. Assume that ˜e³_ib = ˜f_i³b= 0 for any b∈B. Letv∈L be such thatv≡b modqL. Then we have

e_iv≡q^−ϕ_i ^i(b)˜e_iv modqq^−ϕ_i ^i(b)L, f_iv≡q^−ε_i ^i(b)f˜_iv modqq^−ε_i ^i(b)L.

In particular, e_iv≡0 (resp. f_iv≡0)if ε_i(b) = 0 (resp. ϕ_i(b) = 0).

Proof. We prove the second relation. Letλbe the weight ofv. From the assumption it suffices to check the relation for the following cases, since the other cases are trivial.

(i)ε_i(b) = 0, h_i, λ= 1 or 2, (ii)ε_i(b) = 0 or 1, h_i, λ= 0.

In case (i) we have f_iv = ˜f_iv by (2.1) and (2.2). In case (ii) let us write v = f_iv₁+v₂withv_j∈Kere_i∩L(j = 1,2) such that h_i,wtv₁= 2, h_i,wtv₂= 0.

Then we have f_iv=f_i²v₁= [2]_if˜_iv≡q⁻¹_i f˜_ivmodqq⁻¹_i L.

The first relation can be checked similarly by using (2.3) and (2.4).

§2.2. Polarization

We define a total order onQ(q) by f > g if and only iff−g∈

n∈Z

{qⁿ(c+qA)|c >0}

andf ≥g iff > gor f =g.

LetM andNbeU_q(g)-modules. A bilinear form (, ) :M⊗_Q(q)N →Q(q) is called an admissible pairing if it satisfies

(q^hu, v) = (u, q^hv), (e_iu, v) = (u, q⁻¹_i t⁻¹_i f_iv), (2.19)

(f_iu, v) = (u, q⁻¹_i t_ie_iv), for allu∈M andv∈N. (2.19) implies

(2.20) (e⁽ⁿ⁾_i u, v) = (u, q_i⁻ⁿ²t⁻ⁿ_i f_i⁽ⁿ⁾v), (f_i⁽ⁿ⁾u, v) = (u, q_i⁻ⁿ²tⁿ_ie⁽ⁿ⁾_i v).

A symmetric bilinear form ( , ) on M is called a preporlarization of M if it satisfies (2.19) foru, v∈M. A preporlarization is called a porlarization if it is positive definite with respective to the order onQ(q).

§2.3. Fusion construction

In what follows we assume that g is of affine type. Let P be the weight lattice, {Λ_i} the set of fundamental weights andδ the generator of null roots ofg. Then we haveP =

iZΛ_i⊕Zδ. We set P_cl=P/Zδ.

Similar to the quantum algebra U_q(g) which is associated withP, we can also consider U_q(g), which is associated with P_cl, namely, the subalgebra of U_q(g) generated bye_i, f_i, q^h (h∈(P_cl)^∗).

Let K be a commutative ring containing Q(q) and letxbe an invertible element ofK. We introduce aK⊗_Q(q)U_q(g)-moduleV_xby replacing the actions of e_i, f_i withx^δi0e_i, x^−δi0f_i. The action ofq^his not changed. Lety also be an invertible element of K. AK⊗_Q(q)U_q(g)-linear map

R(x, y) : V_x⊗V_y−→V_y⊗V_x

is called a R-matrix. Here we need to specify the coproduct ∆ ofU_q(g) we use in this paper. Our choice is the “lower” one (see [13]) given by

∆(q^h) =q^h⊗q^h forh∈(P_cl)^∗,

∆(e_i) =e_i⊗t⁻¹_i + 1⊗e_i,

∆(f_i) =f_i⊗1 +t_i⊗f_i.

For a finite-dimensionalU_q(g)-module V we assume the following.

V is irreducible.

(2.21)

There exists λ₀∈P_cl such thatwtV⊂λ₀+

i =0

Z_≤0α_i and dimV_λ0= 1.

(2.22)

Here {α_i} is the set of simple roots. Under these assumptions it is known that there exists a uniqueR-matrix up to a scalar multiple. Moreover,R(x, y) depends only on x/y. Take a non zero vector u₀ from V_λ0. We normalize R(x, y) in such a way thatR(x, y)(u₀⊗u₀) =u₀⊗u₀. It is known in [14] that ifV is a “good” module then the normalizedR-matrix does not have a pole at x/y=a∈A.

Next we review the fusion construction following section 3 of [12]. Let l be a positive integer and S_l the l-th symmetric group. Let s_i be the simple reflection which interchangesiandi+ 1, and let(w) be the length ofw∈S_l. LetV be a finite-dimensionalU_q(g)-module. LetR(x, y) denote theR-matrix forV⊗V. For anyw∈S_lwe construct a mapR_w(x₁, . . . , x_l) :V_x1⊗· · ·⊗V_xl → V_xw(1)⊗ · · · ⊗V_xw(l) by

R₁(x₁, . . . , x_l) = 1, R_si(x₁, . . . , x_l) =





j<i

id_Vxj



⊗R(x_i, x_i+1)⊗





j>i+1

id_Vxj



, R_ww(x₁, . . . , x_l) =R_w(x_w(1), . . . , x_w(l))◦R_w(x₁, . . . , x_l)

forw, w such that (ww) =(w) +(w).

Fixr∈Z>0. For eachl∈Z>0, we put R_l=R_w0(q^r(l−1), q^r(l−3), . . . , q^−r(l−1)) :

V_qr(l−1)⊗V_qr(l−3)⊗ · · · ⊗V_q−r(l−1) →V_q−r(l−1)⊗V_q−r(l−3)⊗ · · · ⊗V_qr(l−1), where w₀ is the longest element of S_l. Then R_l is aU_q(g)-linear homomor- phism. Define

V_l= ImR_l.

Let us denote byW the image of

R(q^r, q^−r) :V_qr⊗V_q−r −→V_q−r⊗V_qr

and byN its kernel. Then we have

V_l considered as a submodule ofV^⊗l=V_q−r(l−1)⊗ · · · ⊗V_qr(l−1)

(2.23)

is contained in

l−2

i=0

V^⊗i⊗W⊗V^{⊗(l−2−i)}.

Similarly, we have

V_l is a quotient ofV^⊗l/ l−2 i=0

V^⊗i⊗N⊗V^{⊗(l−2−i)}. (2.24)

§2.4. Preliminary propositions

In this subsection, following [12] we define a prepolarization on V_l and prepare a necessary proposition to show the main theorem. First we recall

Lemma 2.3. Let M_j andN_j beU_q(g)-modules and let(, )_j be an ad- missible pairing between M_j andN_j (j= 1,2). Then the pairing(, )between M₁⊗M₂ andN₁⊗N₂ defined by (u₁⊗u₂, v₁⊗v₂) = (u₁, v₁)₁(u₂, v₂)₂ for all u_j∈M_j andv_j∈N_j is admissible.

LetV be a finite-dimensionalU_q(g)-module satisfying (2.21),(2.22). Sup- pose V has a polarization. The polarization on V gives an admissible pair- ing between V_x and V_x−1. Hence it induces an admissible pairing between V_x1⊗ · · · ⊗V_xl andV_x−1

1 ⊗ · · · ⊗V_x−1 l .

Lemma 2.4. Ifx_j=x⁻¹_l+1−j forj = 1, . . . , l, then for anyu, u∈V_x1⊗

· · · ⊗V_xl, we have

(u, R_w0(x₁, . . . , x_l)u) = (u, R_w0(x₁, . . . , x_l)u).

By takingx₁=q^r(l−1), x₂=q^r(l−3), etc., we obtain the admissible pairing ( , ) between W = V_qr(l−1) ⊗V_qr(l−3) ⊗ · · · ⊗V_q−r(l−1) and W =V_q−r(l−1) ⊗ V_q−r(l−3)⊗ · · · ⊗V_qr(l−1) that satisfies

(2.25) (w, R_lw) = (w, R_lw) for anyw, w∈W.

This allows us to define a preporlarization (, )_l onV_l by (R_lu, R_lu)_l= (u, R_lu) foru, u∈V_qr(l−1)⊗V_qr(l−3)⊗ · · · ⊗V_q−r(l−1).

Next we introduce a Z-form of U_q(g). Recall that A is the subring of Q(q) consisting of rational functions without poles atq= 0. We introduce the subalgebrasA_Z andK_Z ofQ(q) by

A_Z={f(q)/g(q)|f(q), g(q)∈Z[q], g(0) = 1}, K_Z=A_Z[q⁻¹].

Then we have

K_Z∩A=A_Z, A_Z/qA_ZZ.

We then define U_q(g)_KZ as the K_Z-subalgebra ofU_q(g) generated by e_i, f_i, q^h (h∈(P_cl)^∗). SetV_KZ =U_q(g)_KZu₀and assume

(2.26) (V_KZ)_λ0 =K_Zu₀. Let us further set

(V_l)_KZ =R_l((V_KZ)^⊗l)∩(V_KZ)^⊗l. Then one can show

Proposition 2.5. (i) (, )_l is a nondegenerate prepolarization on V_l. (ii) (R_l(u^⊗l₀ ), R_l(u^⊗l₀ ))_l= 1.

(iii) ((V_l)_KZ,(V_l)_KZ)_l⊂K_Z.

LetIbe the index set of the vertices of the Dynkin diagram of gwith the vertex 0 as in [10]. Let g₀ be the finite-dimensional simple Lie algebra whose Dynkin diagram is obtained by removing the 0-vertex from that of g. Let P₊ be the set of dominant integral weights of g₀ and V(λ) be the irreducible highest weight U_q(g₀)-module of highest weightλfor λ∈P₊. The following proposition, which is essentially stated in Proposition 2.6.1 and 2.6.2 of [12], is a key to prove the main theorem.

Proposition 2.6. LetM be a finite-dimensional integrableU_q(g)-module.

Let (, )be a prepolarization onM, and M_KZ aU_q(g)_KZ-submodule of M such that (M_KZ, M_KZ)⊂ K_Z. Let λ₁, . . . , λ_m ∈ P₊, and we assume the following conditions.

(2.27) dimM_λk≤^m

j=1

dimV(λ_j)_λk fork= 1, . . . , m.

There existu_j∈(M_KZ)_λj (j= 1, . . . , m) such that(u_j, u_k)∈δ_jk+ (2.28)

qA, and(e_iu_j, e_iu_j)∈qq^−2(1+h_i ^i,λj)Afor any i∈I\ {0}.

Set L={u∈M |(u, u)∈A} and set B={b∈M_KZ∩L/M_KZ∩qL|(b, b)₀= 1}. Here (, )₀ is the Q-valued symmetric bilinear form onL/qL induced by (, ). Then we have the following.

(i) (, )is a polarization onM. (ii) M

jV(λ_j)asU_q(g₀)-modules.

(iii) (L, B)is a crystal pseudobase ofM.

§3. KR Module of Type D

§3.1. KR module W₁^(k)

First we review the Dynkin datum of typeD⁽¹⁾_n . LetI ={0,1, . . . , n} be the index set of the Dynkin diagram, {α_i}i∈I the set of simple roots, {Λ_i}i∈I

the set of fundamental weights. The standard null rootδ is given by (3.1) δ=α₀+α₁+ 2α₂+· · ·+ 2α_n−2+α_n−1+α_n. We denote the weight lattice byP, that is,P =

i∈IZΛ_i⊕Zδ. The sublattice P =

i∈I0ZΛ_i can be viewed as the weight lattice forD_n. HereI₀ =I\ {0} and Λ_i= Λ_i−a_iΛ₀ witha_ibeing the coefficient ofα_i in (3.1). It is sometimes useful to introduce an orthonormal basis {₁, ₂, . . . , _n} of Q⊗_ZP in such a way that we have

α_i=

_i−_i+1 (i= 1, . . . , n−1) _n−1+_n (i=n),

Λ_i=







₁+· · ·+_i (i= 1, . . . , n−2) (₁+· · ·+_n−1−_n)/2 (i=n−1) (₁+· · ·+_n−1+_n)/2 (i=n).

Then we have α₀ =δ−₁−₂. Since the lengths of the simple roots are all equal, we have q_i =qfor any i∈I. Hence we shall abbreviatei from [m]_i or [m]_i!.

LetW₁^(k)be thek-th fundamental representation ofU_q(D⁽¹⁾_n ). It is known that it has the following decomposition intoU_q(D_n)-modules.

(3.2) W₁^(k)

V(Λ_k)⊕V(Λ_k−2)⊕ · · · ⊕V(Λ₁ or 0) if 1≤k≤n−2,

V(Λ_k) ifk=n−1, n.

OnW₁^(k)the following results are known.

Proposition 3.1. (1) W₁^(k) is “good” in the sense of Kashiwara. In particular, it has a crystal base.

(2)W₁^(k)has a polarization.

The first claim is due to Kashiwara [14] and the second to Koga [18], who got the result by exploiting the fusion construction among the spin representa- tions.

§3.2. Crystal of W₁^(k)

We denote the crystal of W₁^(k) by B^k,1. We review in this subsection Schilling’s variation of Koga’s result on the crystal structure ofB^k,1. First we treat the case ofk= 1. The crystal graph ofB^1,1 is depicted as follows.

1 2 3 n−1 n

1 2 3 n−1 n

1 2 3 n−2 n−1

0 n

1 2 3 n−2 n−1

Here for b, b∈B^1,1 b−→ⁱ means ˜f_ib=b (⇔b= ˜e_ib).

Next in view of (3.2) we recall the U_q(D_n)-crystal structure of B(Λ_l), the crystal of U_q(D_n)-module V(Λ_l), by [15]. Consider the alphabet A = {1,2, . . . , n, n, n−1, . . . ,1} consisting of the crystal elements of B^1,1. It is given the following (partial) order.

1≺2≺ · · · ≺n−1≺n

n ≺n−1≺ · · · ≺1.

Then, for 1≤l≤n−2B(Λ_l) is identified with the set of columns m₁

m₂ ... m_l of height lsatisfying

m_jm_j+1 forj= 1, . . . , l−1, (3.3)

ifm_a=pandm_b =p, then dist(p, p)≤p.

(3.4)

Here dist(p, p) =a+l+ 1−b ifm_a =pandm_b =p. The column tableau as above is also written as m₁m₂· · ·m_l. Note that we allow (m_j, m_j+1) = (n, n) and (n, n). The actions of ˜e_i,f˜_i (i∈I₀) is given by considering the embedding

B(Λ_l) → (B^1,1)^⊗l m₁m₂· · ·m_l→m₁⊗m₂⊗ · · · ⊗m_l

and apply the tensor product rule of crystals on the r.h.s. B(0) is realized as {φ}with the trivial actions of ˜e_i,f˜_i (i∈I₀), that is, ˜e_iφ= ˜f_iφ= 0.

For 1≤k≤n−2, we are to representB^k,1as the set of column tableaux of heightksatisfying (3.3). By (3.2)B^k,1is the union of the sets corresponding toB(Λ_l) with 0≤l≤kandl≡k(mod 2). In [22] maps fromB(Λ_l) to column tableaux of heightk were defined. Ifb∈B(Λ_l), then fill the column of height l ofbsuccesively by a pair (i_j, i_j) for 1≤j≤(k−l)/2 in the following way to obtain a column of height k. Seti₀= 0. Leti_j−1< i_j be minimal such that

(1) neitheri_j or i_j is in the column;

(2) addingi_j andi_j to the column we have dist(i_j, i_j)≥i_j+j;

(3) addingi_jandi_jto the column, all other pairs (a, a) in the new column witha > i_j satisfy dist(a, a)≤a+j.

The filling map and ˜f_i fori∈I₀commute. Denote the filling map to heightk byF_k or simplyF. LetD_k orD, the dropping map, be the inverse ofF_k. Ex- plicitly, given a one-column tableau ofbof heightk, leti₀= 0 and successively findi_j> i_j−1minimal such that the pair (i_j, i_j) is inband dist(i_j, i_j)≥i_j+j.

Drop all such pairs (i_j, i_j) fromb. Thus we have

B^k,1

0≤l≤k, l≡k(2)

F_k(B(Λ_l)) asU_q(D_n)-crystals.

It is the set of all column tableaux of heightksatisfying (3.3) only.

We are left to give the rule of the actions of ˜e₀and ˜f₀. For this purpose we need slight variants ofF_k and D_k, denoted by ˜F_k and ˜D_k, respectively, which act on columns that do not contain 1,2,2,1. On these columns ˜F_k and ˜D_k are defined by replacing i→ i−2 and i→ i−2, then applying F_k and D_k, and finally replacing i→i+ 2 and i→i+ 2. The following proposition is given in [22].

Proposition 3.2. Forb∈B^k,1,

˜ e₀b=





























F_k( ˜D_k−2(x)) if b= 12x F˜_k−1(x)2 if b= 12x2 F˜_k−1(x)1 if b= 12x1 F˜_k−2(x)21 if b= 12x21 F_k( ˜D_k−1(x)2) if b= 1x F_k( ˜D_k−1(x)1) if b= 2x

x21 if b= 1x1 andD˜_k−2(x) =x

0 otherwise

f˜₀b=





























F_k( ˜D_k−2(x)) if b=x21 2 ˜F_k−1(x) if b= 2x21 1 ˜F_k−1(x) if b= 1x21 12 ˜F_k−2(x) if b= 12x21 F_k(2 ˜D_k−1(x)) if b=x1 F_k(1 ˜D_k−1(x)) if b=x2

12x if b= 1x1 andD˜_k−2(x) =x

0 otherwise

where xdoes not contain1,2,2,1.

§3.3. Existence of crystal pseudobase forW_l^(k)

In this subsection we prove our main theorem by using Proposition 2.6.

We prepare several lemmas and propositions.

Lemma 3.3. LetR(x/y)be theR-matrix fromW_1,x^(k)⊗W_1,y^(k)toW_1,y^(k)⊗ W_1,x^(k). Then it has the following form.

R(z) =P_2k+ z−q²

1−q²zP_k+1+k−1+· · · .

Here z =x/y, _j =₁+₂+· · ·+_j ∈P₊ for0≤j ≤n−1, and P_λ stands for the projector onto the irreducible U_q(D_n)-moduleV(λ)in(W₁^(k))^⊗2.

Proof. Letu₀be aU_q(D_n)-highest weight vector ofW₁^(k)of highest weight Λ_k(=_k). SinceV(_k+1+_k−1) is multiplicity free, a unique highest weight vector up to a scalar is given by

v=u₀⊗f_ku₀−qf_ku₀⊗u₀.

Sincef_iu₀= 0 fori∈I\ {k}, we have

F⁽¹⁾v=u₀⊗F⁽¹⁾u₀−qF⁽¹⁾u₀⊗u₀

where F⁽¹⁾=f_k+1· · ·f_n−2f_nf_n−1· · ·f_k+1f₁· · ·f_k−1. Hence we have F⁽²⁾v=qu₀⊗F⁽²⁾u₀−qF⁽²⁾u₀⊗u₀+ (unwanted terms)

where F⁽²⁾ = f₂· · ·f_kF⁽¹⁾ and we know f₀(unwanted terms) = 0 by weight consideration. Hence we have

F⁽³⁾v=q⁻¹y⁻¹u₀⊗F⁽³⁾u₀−qx⁻¹F⁽³⁾u₀⊗u₀ onV_x⊗V_y

whereF⁽³⁾=f₀F⁽²⁾ andV =W₁^(k). Note thatF⁽³⁾u₀=αu₀with someα= 0, since the corresponding crystal element is not killed from Proposition 3.2. Thus we have

F⁽³⁾v=α(q⁻¹y⁻¹−qx⁻¹)u₀⊗u₀. Now let

R(z)∝ϕ(z)P_2k+ϕ(z)P_k+1+k−1+· · ·. Then we have

R(z)F⁽³⁾v=α(q⁻¹y⁻¹−qx⁻¹)ϕ(z)(u₀⊗u₀)

=F⁽³⁾R(z)v=ϕ(z)F⁽³⁾v=α(q⁻¹x⁻¹−qy⁻¹)ϕ(z)(u₀⊗u₀).

Here byv we mean that it is considered to be inV_y⊗V_x. Thus we have ϕ(z)/ϕ(z) = z−q²

1−q²z.

Set W = ImR(q²), N = Ker R(q²). They areU_q(D_n⁽¹⁾)-modules. Using the main result of [20] one can show the following.

Lemma 3.4. We have

W W₂^(k), N

j∼k

W₁^(j)

asU_q(D⁽¹⁾_n )-modules. Herej∼kmeans that the corresponding vertices are tied by an edge in the Dynkin diagram. Moreover, both W andN are irreducible.

Proof. In [20] it is shown that there exists an exact sequence ofU_q(D_n⁽¹⁾)- modules

0−→

j∼k

W₁^(j)−→W_1,q^(k)⊗W_1,q^(k)₋₁ −→W₂^(k)−→0.

(An acute reader should have noticed that the exact sequence is different from [20]. It is because the definition of the KR modules and the choice of the coproduct are different.) Moreover, it is also known that

j∼kW₁^(j)andW₂^(k) are irreducible. Set W =

j∼kW₁^(j) and consider N ∩W. Since W is irreducible, we haveN∩W={0}orW. Recall thatW_1,q^(k)⊗W_1,q^(k)₋₁contains a unique irreducibleU_q(D_n)-moduleV(_k+1+_k−1). From the previous lemma and (3.2) we know it is contained both inNand inW. Hence we haveN ⊃W. Now suppose NW. Then we have a surjectiveU_q(D_n⁽¹⁾)-linear map

W_1,q^(k)⊗W_1,q^(k)₋₁/W−→W_1,q^(k)⊗W_1,q^(k)₋₁/N.

Since the l.h.s. is irreducible,N=W orW_1,q^(k)⊗W_1,q^(k)₋₁. SinceN cannot be the second choice by the previous lemma. One obtainsN=WandW W₂^(k).

SinceW is known to be a KR module by the previous lemma, we have Lemma 3.5. As aU_q(D_n)-moduleW has the following decomposition.

W

0≤m1≤m2≤[k/2]

V(Λ_k−2m1+ Λ_k−2m2)

We set B = B^k,1. We fix a basis {v_I}I∈B of W₁^(k) in such a way that v_I modqL=I as an element of B.

Proposition 3.6. N contains a vector of the form v_I1⊗v_I2−

J1⊗J2∈B1

a_J1J2v_J1⊗v_J2 (a_J1J2∈A)

for any I₁, I₂ such that I₁⊗I₂∈B^⊗2\B₁. See (4.2),(4.4) for the definition of B₁.

We now apply the fusion construction in Section 2.3 to V =W₁^(k) with r = 1. The assumptions (2.21),(2.22) are valid withλ₀= Λ_k. (2.26) can also be checked. Other necessary properties are guaranteed by Proposition 3.1. For l ∈ Z>0 we define W_l^(k)= ImR_l. Let k = [k/2]. Let c= (c₁, c₂, . . . , c_k) be