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Level One Perfect Crystals for B(1)n, C(1)n, and D(1)n
Journal of Algebra 217, 312᎐334 Ž1999. Article ID jabr.1998.7806, available online at http:rrwww.idealibrary.com on
Level One Perfect Crystals for BnŽ1., CnŽ1., and DnŽ1. Yoshiyuki Koga Department of Mathematics, Graduate School of Science, Osaka Uni¨ ersity, Toyonaka, Osaka 560-0043, Japan Communicated by Georgia Benkart Received June 18, 1998
0. INTRODUCTION Crystal base was first introduced by M. Kashiwara in w9x as a theory of the integrable representations for the quantum enveloping algebra Uq Ž ᒄ . at q s 0. In w1x S.-J. Kang et al. studied crystal bases of finite-dimensional modules over quantum affine algebras and introduced the notion of perfect crystals to identify the one-dimensional configuration sums of vertex models with the string functions of affine Lie algebras. In w2x they Ž1. Ž1. also constructed examples of perfect crystals in the cases of AŽ1. n , Bn , C n , Ž1. Ž2. Ž2. Ž2. Ž1. Ž1. Dn , A 2 n , A 2 ny1 , and Dnq1. Other examples for Cn and G 2 are treated in w3, 4x. Let ᒄ be an affine Lie algebra and ᒄ its classical part. We denote by B Ž ⌳ . the crystal base of the finite-dimensional Uq Ž ᒄ .-module with highest weight ⌳. In the case that ᒄ is of type AŽ1. n , there exists a perfect crystal B k, l for UqX Ž ᒄ . such that B k, l , B Ž l⌳ k . as a crystal for Uq Ž ᒄ .. For BnŽ1., CnŽ1., and DnŽ1., the known examples are isomorphic to either B Ž l⌳ 1 ., B Ž l⌳ ny 1 ., or B Ž l⌳ n . as a crystal for Uq Ž ᒄ .. The purpose of this paper is to give new examples of perfect crystals for UqX Ž BnŽ1. ., UqX Ž CnŽ1. ., and UqX Ž DnŽ1. . that are isomorphic to B Ž tk ⌳ k . [
ž[
⌳-t k ⌳ k
BŽ ⌳. ,
/
where t k s 1 if the kth simple root ␣ k of ᒄ is a long root, t k s 2 if ␣ k is a short root. The symbol - stands for the dominance order Žsee Sect. 3.. All of these crystals are expected to give the list of the perfect crystals of level one in the following sense: Any perfect crystal of level one is isomorphic to one of these crystals or the tensor product of them. 312 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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LEVEL ONE PERFECT CRYSTALS
This paper is organized as follows: In Section 1 we recall the theory of crystal base and summarize known results on the perfect crystals. In Sections 2 and 3 we give the main statements, and we prove them in the following sections. At the end of this paper we comment on the generic level perfect crystals.
1. REVIEW OF PERFECT CRYSTALS Let ᒄ be a symmetrizable Kac᎐Moody algebra over Q and ᒅ its Cartan subalgebra w14x. Let ␣ i N i g I 4 , h i N i g I 4 , and ⌳ i N i g I 4 be the set of the simple roots, the simple coroots, and the fundamental weights that are indexed by the finite set I. We denote the pairing between ᒅ and ᒅ* by ² , : and put P s Ý i Z⌳ i , P* s HomŽ P, Z.. Because ᒄ is symmetrizable, there is a symmetric bilinear form Ž , . on ᒅ* such that ² h i , : s 2Ž ␣ i , .rŽ ␣ i , ␣ i . for any i g I and g ᒅ*. The quantized enveloping algebra Uq Ž ᒄ . is an associative algebra over QŽ q . with generators e i , f i Ž i g I . and q h Ž h g P*. which satisfy the relations q 0 s 1,
q h ⭈ q h⬘ s q hq h⬘
q h e i qyh s q ² h , ␣ i : e i ,
w ei , fi x s ␦i , j
q h f i qyh s qy² h , ␣ i : f i ,
t i y ty1 i qi y qy1 i
,
where qi s q Ž ␣ i , ␣ i . and t i s q Ž ␣ i , ␣ i . h i , b
b
ms0
ms0
m m Ý Ž y1. eiŽ m. e j eiŽ bym. s Ý Ž y1. f iŽ m. f j f iŽ bym. s 0,
for i / j, where b s 1 y ² h i , ␣ j :. Here we set
w nx i s
qin y qyn i
e iŽ n. s
qi y qy1 i e in
w nx i!
n
,
,
w n x i !s Ł w k x i , ks1
f iŽ n. s
f in
w nx i!
.
The algebra Uq Ž ᒄ . has the following Hopf algebra structure, ⌬ Ž e i . s e i m ty1 q 1 m ei , i
⌬ Ž fi . s fi m 1 q ti m fi ,
⌬Ž qh . s qh m qh, a Ž e i . s ye i t i ,
a Ž f i . s yty1 i fi ,
a Ž q h . s qyh .
314
YOSHIYUKI KOGA
We denote by Uq Ž ᒄ . i the subalgebra of Uq Ž ᒄ . generated by e i , f i , and q h Ž h g P*.. For a Uq Ž ᒄ .-module M and g P, we set M s u g M N q h u s q ² h, : u for any h g P*4 . We say that M is integrable if it satisfies Ži. M s [ g P M, and Žii. M is a union of finite-dimensional Uq Ž ᒄ . i-modules for any i g I. Let u be any weight vector of an integrable Uq Ž ᒄ .-module M. It is easy to see that u can be uniquely expressed as u s Ý k f iŽ k . u k , where u k g Mqk ␣ i l ker e i and i g I. We define operators ˜ e i , f˜i by
˜e i u s Ý f iŽ ky1. u k ,
f˜i u s
Ý f iŽ kq1. u k .
For any vector space V and a subset B of V, we call B a pseudo-base if there exists a base B⬘ of V such that B s B⬘ [ ŽyB⬘.. Let us define a subring A of QŽ q . by A s f g QŽ q . N f : regular at q s 04 . A pair Ž L, B . is called a crystal Žpseudo-. base of an integrable module M if it satisfies 1. L is a free A-submodule of M such that QŽ q . mA L , M, 2. L s [g P L where L s L l M , 3. ˜ e i L ; L and f˜i L ; L. 4. B is a Žpseudo-.base of Q-vector space LrqL, 5. B s " g P B where B s B l Ž LrqL ., 6. ˜ e i B ; B " 04 and f˜i B ; B " 04 , 7. for b, b⬘ g B and i g I, b⬘ s f˜i b if and only if b s ˜ e i b⬘. We denote wtŽ b . s for b g M and wtŽ B . s wtŽ b . N b g B4 . Let us recall the tensor product of crystal bases. For b g B and i g I, we set
i Ž b . s max n G 0 N ˜ e n b g B4 , i Ž b . s max n G 0 N f˜in b g B 4 . It is easy to see that ² h i , wtŽ b .: s i Ž b . y i Ž b . for each i g I. Let B1 and B2 be crystal bases. The actions of ˜ e i and f˜i on the tensor product B1 m B2 can be written as f˜i b1 m b 2
if i Ž b1 . ) i Ž b 2 . ,
b1 m f˜i b 2
if i Ž b1 . F i Ž b 2 . ,
˜e i b1 m b 2 ˜e i Ž b1 m b 2 . s b1 m ˜ ei b2
if i Ž b1 . G i Ž b 2 . ,
f˜i Ž b1 m b 2 . s
½ ½
if i Ž b1 . - i Ž b 2 . .
Ž 1.
LEVEL ONE PERFECT CRYSTALS
315
Next we recall the definition of perfect crystals given in w1x. Let ᒄ be an affine Lie algebra and I s 0, 1, . . . , n4 the index set of simple roots of ᒄ. Let c and ␦ be the canonical central element and the generator of the imaginary roots. We take Ý nis0 Z⌳ i [ Z ␦ as the weight lattice P and then P* s Ý nis0 Zh i [ Z d. Set ᒅUcl s ᒅ*rQ ␦ and denote the canonical map ᒅ* ª ᒅUcl by cl. Because ² ␦ , h i : s 0 for any i g I, we can identify n Ž[is0 Qh i .* with ᒅUcl . Here we consider the fundamental weights ⌳ i as an element of ᒅUcl and define the classical weight lattice Pcl by Ý nis0 Q⌳ i . For a crystal B, we say that B is classical if wtŽ B . ; Pcl . Let us denote by UqX Ž ᒄ . the subalgebra of the Uq Ž ᒄ . generated by e i , f i , and t i for all i g I. From now on, we only consider UqX Ž ᒄ .-modules which have classical crystal bases. To give the definition of the perfect crystals, we need the following notations: Pclqs g Pcl N ² h i , : G 0 for any i g I 4 and Ž Pclq. l s g Pclq N ² c, : s l 4 . For an element b of the classical crystal B, set Ž b . s Ý i i Ž b . ⌳ i and Ž b . s Ý i i Ž b . ⌳ i . DEFINITION 1.1. For l g Z ) 0 , we call a classical crystal B a perfect crystal of level l if it satisfies the conditions: 1. B m B is connected. 2. There exists 0 g Pcl such that wtŽ B . ; 0 q Ý i/ 0 Z F 0 ␣ i and 噛Ž B0 . s 1. 3. There exist a finite-dimensional integrable UqX Ž ᒄ .-module with a crystal pseudo-base Ž L, B⬘. such that B , B⬘r "14 . 4. For any b g B, we have ² c, Ž b .: G l. . 5. The maps and from Bl s b N ² c, Ž b .: s l 4 to Ž pq cl l are bijective. An element of Bl is called minimal. Let B be a perfect crystal of level l. For g Ž Pclq. l , let bŽ . g B be the element that satisfies ŽbŽ .. s . We define the automorphism of Ž Pclq. l by s Ž bŽ ... We denote by BŽ . the crystal base of the integrable highest weight UqX Ž ᒄ .-module with highest weight . Then we have THEOREM 1.1 w1x. Let P Ž , B . be the set of sequences pŽ n.4 in B such that pŽ n. s bŽ ny 1 Ž .. for n 4 0. Then BŽ . is isomorphic to P Ž , B .. We recall the examples of perfect crystals given in w2, 3x in the case that Ž1. Ž1. Ž1. ᒄ is of type AŽ1. n , Bn , C n , and Dn . Let J be any subset of the index set I X and Uq Ž ᒄ J . the subalgebra of Uq Ž ᒄ . generated by e i , f i , and t i Ž i g J .. We sometimes use the notation J to denote the map defined by J Ž j . s j for any j g J. For a crystal B for UqX Ž ᒄ ., we denote by UJ Ž B . the crystal for
316
YOSHIYUKI KOGA
Uq Ž ᒄ J . such that UJ Ž B . s B as a set, and for j g J, f˜j Ž b . s b⬘ if and only if f˜j Ž b . s b⬘ in B. Here we list known results. Set J s 1, 2, . . . , n4 . THEOREM 1.2 w2x. Let ᒄ be of type AŽ1. n . For each integer k, l such that 1 F k F n, l G 1, there exists a perfect crystal B k, l of le¨ el l such that UJ Ž B k, l . , B Ž l⌳ k . as a crystal for Uq Ž ᒄ .. THEOREM 1.3 w2x. Let ᒄ be of type BnŽ1., CnŽ1., or DnŽ1.. For each integer l such that l G 1, there exist perfect crystals B 1, l for Uq Ž BnŽ1. ., B n, l for Uq Ž CnŽ1. . and B 1, l , B ny1, l , B n, l for Uq Ž DnŽ1. ., of le¨ el l such that UJ Ž B k, l . , B Ž l⌳ k . as a crystal for Uq Ž ᒄ .. THEOREM 1.4 w3x. Let ᒄ be of type CnŽ1.. For each integer l such that l G 1, there exist perfect a crystal B 1, l of le¨ el l such that UJ Ž B k, l . , B Ž2 l⌳ 1 . [ B Ž2Ž l y 1. ⌳ 1 . [ ⭈⭈⭈ [ B Ž2⌳ 1 . [ B Ž0. as a crystal for Uq Ž ᒄ .. 2. PERFECT CRYSTALS OF LEVEL ONE Through this paper, let J s 1, 2, . . . , n4 be a subset of the index set I. For ␣ ,  g Pcl , we define ␣ -  by  y ␣ g Ý nis1 Z G 0 ␣ i and ␣ /  . PROPOSITION 2.1. Let ᒄ be of type BnŽ1., CnŽ1., or DnŽ1.. For each integer k such that 1 F k F n, there exists a unique crystal B k, 1 for Uq Ž ᒄ . such that,
UJ Ž B k , 1 . , B Ž t k ⌳ k . [
ž[
⌳-t k ⌳ k
BŽ ⌳. ,
/
as crystals for Uq Ž ᒄ .. Here t k s 1 if ␣ k is a long root, t k s 2 if ␣ k is a short root. An explicit form of the crystal UJ Ž B k, 1 . as: For BnŽ1., 1 F k F n y 1,
UJ Ž B k , 1 . , B Ž ⌳ k . [ B Ž ⌳ ky2 . [ B Ž ⌳ ky4 . [ ⭈⭈⭈ [ Ž B Ž ⌳ 1 . or B Ž 0 . . . For BnŽ1., k s n,
UJ Ž B n , 1 . , B Ž 2⌳ n . [ B Ž ⌳ ny2 . [ B Ž ⌳ ny4 . [ ⭈⭈⭈ [ Ž B Ž ⌳ 1 . or B Ž 0 . . . For CnŽ1., 1 F k F n y 1,
UJ Ž B k , 1 . , B Ž 2⌳ k . [ B Ž 2⌳ ky1 . [ B Ž 2⌳ ky2 . [ ⭈⭈⭈ [ B Ž 2⌳ 1 . [ B Ž 0 . . For CnŽ1., k s n,
UJ Ž B n , 1 . , B Ž ⌳ n . .
317
LEVEL ONE PERFECT CRYSTALS
For DnŽ1., 1 F k F n y 2,
UJ Ž B k , 1 . , B Ž ⌳ k . [ B Ž ⌳ ky2 . [ B Ž ⌳ ky4 . [ ⭈⭈⭈ [ Ž B Ž ⌳ 1 . or B Ž 0 . . . For DnŽ1., k s n y 1, n.
UJ Ž B k , 1 . , B Ž ⌳ k . . We remark that in the case k s 1 Žfor BnŽ1. ., k s n Žfor CnŽ1. ., and k s 1, n y 1, n Žfor DnŽ1. ., these crystals coincide with those reviewed in the previous section. THEOREM 2.1. The crystal B k, 1 is a perfect crystal of le¨ el one. We prove Proposition 2.1 and Theorem 2.1 in Sections 4 and 5. Remark 2.1. By Theorem 1.2, the above proposition and theorem are valid for AŽ1. n . Remark 2.2. We conjecture that the above list of crystals exhausts the perfect crystals of level one; i.e., any level one perfect crystal is isomorphic to one of these crystals or the tensor product of them ŽSee Sect. 6..
3. CRYSTAL GRAPHS The crystal graph of the perfect crystal B k, 1 is given in this section. The minimal elements of B k, 1 and the automorphism of Ž Pclq. 1 are also described. Ž CnŽ1., B k, 1 ., Ž1 F k F n y 1.. Let ⑀ 1 , . . . , ⑀ n4 be the orthonormal base n of ᒅ s [is1 Q ␣ i such that ␣ i s ⑀ i y ⑀ iq1 Ž1 F i F n y 1. and ␣ n s 2 ⑀ n . We summarize crystal graphs for Uq Ž Cn . in w11x. Let B Ž ⌳ 1 . be the crystal base of the vector representation of Uq Ž Cn . that is labeled by Ž i ., Ž i . N 1 F i F n4 . Its crystal graph is given by f˜i Ž j . s ␦ i , j Ž j q 1 . ,
f˜i Ž j . s ␦ iq1, j Ž j y 1 .
f˜n Ž j . s ␦n , j Ž n . ,
f˜n Ž j . s 0.
for 1 F i - n,
Note that wtŽ i . s ⑀ i and wtŽ i . s y⑀ i . The crystal graph of B Ž ⌳ N . Ž1 F N F n. can be identified with the connected component of B Ž ⌳ 1 .m N containing the element Ž1. m Ž2. m ⭈⭈⭈ m Ž N .. In fact B Ž ⌳ N . s Ž m1 , . . . , m N . m i g 1, 2, . . . , n, n, . . . , 1 4 such that m1 - m 2 - ⭈⭈⭈ - m N , and i q Ž N y j q 1 . F m i if m i s m j Ž i - j . 4 ,
318
YOSHIYUKI KOGA
where Ž m1 , . . . , m N . s Ž m1 . m ⭈⭈⭈ m Ž m N . and the ordering - on 1, 2, ˜4 is defined by . . . ,1 1 - 2 - ⭈⭈⭈ - n - n - ⭈⭈⭈ - 1. Furthermore we recall the crystal graph of B Ž ⌳ M q ⌳ N . Ž1 F M F N F n.. Let u s Ž m1 , . . . , m M . and ¨ s Ž mX1 , . . . , mXN . be elements of B Ž ⌳ M . and B Ž ⌳ N .. For 1 F a F b F n, we say that u m ¨ g B Ž ⌳ M . m B Ž ⌳ N . is in the Ž a, b .-configuration if u m ¨ satisfies the following w12x; there exist 1 F p F q - r F s F M such that mXp s a, m q s b, m r s b, m s s a, or mXp s a, mXq s b, mXr s b, m s s a. Remark that the definition of the Ž a, b .-configuration includes the case of a s b, p s q, and r s s. Set pŽ a, b; u m ¨ . s Ž q y p . q Ž s y r ., then we have B Ž ⌳ M q ⌳ N . s u m ¨ g B Ž ⌳ M . m B Ž ⌳ N . mXi F m i for 1 F i F M and if u m ¨ is in the Ž a, b . -configuration, then p Ž a, b; u m ¨ . - b y a4 . The element u m ¨ is called a semi-standard C-tableau of shape ⌳ M q ⌳ N if u m ¨ g B Ž ⌳ M q ⌳ N .. From w2x we see that if the action of f˜0 on the crystal B Ž ⌳ n . is given by f˜0 b s
½
Ž 1, m1 , . . . , m ny1 .
if b s Ž m1 , . . . , m ny1 , 1 . ,
0
otherwise,
Ž 2.
then B Ž ⌳ n . is a perfect crystal of level one. We construct the UqX Ž CnŽ1. .-crystal B k, 1 by using the crystal B Ž ⌳ n .. For each s such that 0 F s F n and u s Ž m1 , . . . , m n ., ¨ s Ž mX1 , . . . , mXn . g B Ž ⌳ n ., we define sequences Ž g 1 , . . . , g nqs ., Ž g X1 , . . . , g Xnqs . as:
¡i if 1 F i F s, if s q 1 F i F s q n, m s j, ¢i q j if s q 1 F i F s q n, m s j, if 1 F i F n, m s j, ¡i q j ~ if 1 F i F n, m s j, g s iqj ¢n q s q 1 y i if n q 1 F i F n q s. g s~i q j
iys
i
iys
iys
X i
iys
Here we denote the elements Ž g 1 , . . . , g nqs . and Ž g X1 , . . . , g Xnqs . by u ˜ and ¨˜. Let us define the subset B˜s of B Ž ⌳ n .m 2 by B˜s s u m ¨ g B Ž ⌳ n .
½
m2
u ˜ m ¨˜ is a semi-standard C-tableau for Uq Ž Cnqs . of shape 2⌳ nqs .
5
LEVEL ONE PERFECT CRYSTALS
319
In the same way as w11x, we can show that B˜s is closed under the action of f˜i for i s 1, . . . , n and B˜s , B Ž 2⌳ n . [ B Ž 2⌳ ny1 . [ ⭈⭈⭈ [ B Ž 2⌳ nys . , as a crystal for Uq Ž Cn .. Set Bs s B˜s _ B˜sy1 , where B˜y1 s ⭋. From now on n we identify B k, 1 with [ssnyk Bs . We describe the crystal graph of the UqX Ž CnŽ1. .-crystal B k, 1 under the above identification. This is proven in the next section. For each j g J, the jth arrow in the crystal B k, 1 can be described by Ž1., because B k, 1 is identified with a subset of B Ž ⌳ n .m 2 . We give the 0th arrow of the crystal B k, 1. Set J⬘ s 2, 3, . . . , n4 . Let B Ž ⌳ n .q, B Ž ⌳ n .y, and B Ž ⌳ n . 0 be the subsets of B Ž ⌳ n . given by B Ž ⌳ n . q s Ž 1, m 2 , . . . , m n . g B Ž ⌳ n . 4 , B Ž ⌳ n . y s Ž m, . . . , m ny1 , 1 . g B Ž ⌳ n . 4 ,
Ž 3.
B Ž ⌳ n . 0 s B Ž ⌳ n . _ Ž B Ž ⌳ n . qj B Ž ⌳ n . y . . For 1 , 2 s " and 0, we define the subset B 1 , 2 of B Ž ⌳ n .m 2 by B 1 , 2 s b m b⬘ N b g B Ž ⌳ n . 1 , b⬘ g B Ž ⌳ n . 2 4 .
Ž 4.
The 0th arrow in B k, 1 can be described as: n
for b g
[
ssnykq1
Bs ,
0
b « f˜0 b, 0
for b g Bny k _ Ž Bqyj B0y . ,
b « f˜0 b,
for b g Bny k l Bqy ,
b « 0,
for b g Bny k l B0y ,
b « Ž b. ,
0 0
where f˜0 b is given by Ž2. Žand Ž1.., and the map is the Uq Ž ᒄ J ⬘ .-crystal isomorphism:
: Bny k l B0yª Bnyk l Bq0 . By the fact Ž10. in Section 4, this isomorphism is uniquely determined. PROPOSITION 3.1. The set of the minimal elements of B k, 1 is gi¨ en by 1 N i s 0, . . . , k 4 j bi2 m bi3 N i s k q 1, . . . , n4 , Ž B k , 1 . 1 s bi1 m bnyi
320
YOSHIYUKI KOGA
where bi1 s Ž i q 1, . . . , n, n, . . . , n y i q 1 . , bi2 s Ž 1, . . . , i y k, i q 1, . . . , n, n, . . . , n y k q 1 . , bi3 s Ž n y k q 1, . . . , n, n, . . . , i q 1, i y k, . . . , 1 . . We prove this proposition in Section 5. We give the description of the automorphism ⌳ 0 , ⌳ 1 , . . . , ⌳ n4 determined by the perfect crystal B k, 1.
of Ž Pclq. 1 s
Ž ⌳ i . s ⌳ i for any i such that 0 F i F n.
PROPOSITION 3.2.
The proof is similar to Proposition 1.2.5 in w2x. Ž BnŽ1., B k, 1 ., Ž1 F k F n.. Let ⑀ 1 , . . . , ⑀ n4 be the orthonormal base of n ᒅ s [is1 Q ␣ i such that ␣ i s ⑀ i y ⑀ iq1 Ž1 F i F n y 1. and ␣ n s ⑀ n . We recall the Uq Ž Bn .-crystal B Ž ⌳ n . in w11x. The elements of B Ž ⌳ n . can be labeled as: B Ž ⌳ n . s Ž m1 , . . . , m n . m i g 1, 2, . . . , n, n, . . . , 1 4 such that m1 - m 2 - ⭈⭈⭈ - m n , and m and m do not appear simultaneously ,
5
where the ordering - on 1, 2, . . . , 14 is defined by 1 - 2 - ⭈⭈⭈ - n - n - ⭈⭈⭈ - 1. Note that if we define wtŽ i . s ⑀ ir2 and wtŽ i . s y ⑀ ir2 then wtŽŽ m1 , . . . , m n .. s Ý nis1 wtŽ m i .. The action of f˜i on the crystal B Ž ⌳ n . is given by for 1 F i F n y 1,
¡ m ,..., iq1,..., sth
f˜i b s
ž ~
1
¢0
t th
i , . . . , mn
/
sth
t th
if b s m1 , . . . , i , . . . , ı q 1 , . . . , m n ,
ž
/
otherwise,
for i s n,
¡ ¢0
f˜ b s~ ž m , . . . , n
1
sth
n , . . . , mn
/
sth
if b s m1 , . . . , n , . . . , m n ,
ž
otherwise.
/
Ž 5.
321
LEVEL ONE PERFECT CRYSTALS
If we define the action of f˜0 on the crystal B Ž ⌳ n . by f˜0 b s
½
Ž 1, 2, m1 , . . . , m ny2 .
if b s Ž m1 , . . . , m ny2 , 2, 1 . ,
0
otherwise,
Ž 6.
then B Ž ⌳ n . is the crystal base of the UqX Ž BnŽ1. .-module V Ž ⌳ n y ⌳ 0 .. We construct the UqX Ž BnŽ1. .-crystal B k, 1. For each s such that 0 F s F n, we consider the following subset B˜s of B Ž ⌳ n .m 2 : B˜s s Ž m1 , . . . , m n . m Ž mX1 , . . . , mXn . g B Ž ⌳ n .
½
m2
mXi F m iqs
for 1 F i F n y s 4 . In a similar way to
CnŽ1.-case,
we see that
B˜s , B Ž 2⌳ n . [ B Ž ⌳ ny1 . [ ⭈⭈⭈ [ B Ž ⌳ nys . , as a crystal for Uq Ž Bn .. Set Bs s B˜s _ B˜sy1 , where B˜y1 s ⭋. From now on, we identify B k, 1 with [nyk F s F n, s ' nykŽ2. Bs . We describe the crystal graph of B k, 1. For each i such that 1 F i F n, the ith arrow in B k, 1 is determined by Ž5.. Then we give the 0th arrow in B k, 1. Set J⬘ s 3, . . . , n4 . First we consider the crystal B k, 1 for 1 F k F n y 1. Let B Ž ⌳ n .s Ž s s 1, 2, 3, 4. be the subsets of B Ž ⌳ n . given by B Ž ⌳ n . 1 s Ž 1, 2, m 3 , . . . , m n . g B Ž ⌳ n . 4 , B Ž ⌳ n . 2 s Ž 1, m 2 , . . . , m ny1 , 2 . g B Ž ⌳ n . 4 , B Ž ⌳ n . 3 s Ž 2, m 2 , . . . , m ny1 , 1 . g B Ž ⌳ n . 4 , BŽ ⌳n.4 s
½Žm , . . . , m 1
ny2 , 2, 1
. g BŽ ⌳n. 5 .
For s1 , s2 s 1, 2, 3, or 4, we take the subset Bs1 , s 2 of B Ž ⌳ n .m 2 defined by Bs1 , s 2 s b m b⬘ N b g B Ž ⌳ n . s1 , b⬘ g B Ž ⌳ n . s 2 4 . The 0th arrow in B k, 1 can be described as: for b g
[
nykq2FsFn , s'nyk Ž2 .
Bs ,
0
b « f˜0 b, 0
for b g Bny k _ Ž B14 j B24 j B34 . ,
b « f 0 b,
for b g Bny k l B14 ,
b « 0,
for b g Bny k l B24 ,
b « Ž b. ,
for b g Bny k l B34 ,
b « ⬘Ž b . ,
0 0 0
322
YOSHIYUKI KOGA
where f˜0 b is given by Ž6. Žand Ž1.., and the maps and ⬘ are the Uq Ž ᒄ J ⬘ .-crystal isomorphisms,
: Bny k l B24 ª Bnyk l B12 , ⬘: Bny k l B34 ª Bnyk l B13 . Next we consider the crystal B n, 1. Let B42 and B43 be the subsets of B42 and B43 given by B42 s B42 _
½Žm , . . . , m 1
ny2 , 2, 1
. m Ž 1, mX2 , . . . , mXny1 , 2.
mXiq1 F m i
for 1 F i F n y 2 B43 s B43 _
½Žm , . . . , m 1
ny2 , 2, 1
. m Ž 2, mX2 , . . . , mXny1 , 1.
5,
mXiq1 F m i
for 1 F i F n y 2
5.
The 0th arrow in B k, 1 can be described as: for b g
[
2FsFn , s'0 Ž2 .
Bs ,
0
b « f˜0 b, 0
for b g B0 _ Ž B42 j B43 . ,
b « f˜0 b,
for b g B0 l B42 ,
b « Ž b. ,
for b g B0 l B43 ,
b « ⬘Ž b . ,
0 0
where f˜0 b is given by Ž6. Žand Ž1.., and the maps and ⬘ are the Uq Ž ᒄ J ⬘ .-crystal isomorphisms,
: Bny k l B42 ª Bnyk l B21 , ⬘: Bny k l B43 ª Bnyk l B31 . Here B21 and B31 are the following subsets of B21 and B31 , B21 s B21 _ Ž 1, 2, m 3 , . . . , m n . m Ž 1, mX2 , . . . , mXny1 , 2 . mXiy1 F m i for 3 F i F n 4 , B31 s B31 _ Ž 1, 2, m 3 , . . . , m n . m Ž 2, mX2 , . . . , mXny1 , 1 . mXiy1 F m i for 3 F i F n 4 .
323
LEVEL ONE PERFECT CRYSTALS
PROPOSITION 3.3. The set of the minimal elements of B k, 1 is gi¨ en by
Ž Bk, 1.1 s
½
b1 m b 2 , b 3 m b 4 , b 5 m b 6 4 b1 m b 4 , b 3 m b 2 , b 5 m b 6 4
if k is e¨ en, if k is odd,
where b1 s Ž 1, . . . , n . ,
b 2 s Ž 1, . . . , n . ,
b 3 s Ž 2, . . . , n, 1 . ,
b 4 s Ž 1, n, . . . , 2 . ,
b 5 s Ž 1, . . . , n y k, n, . . . , n y k q 1 . , b 6 s Ž n y k q 1, . . . , n, n y k, . . . , 1 . . We give the description of the automorphism of Ž Pclq. 1 s ⌳ 0 , ⌳ 1 , ⌳ n4 determined by the perfect crystal B k, 1. PROPOSITION 3.4. If k is an e¨ en integer then Ž ⌳ i . s ⌳ i for any i s 0, 1, n. If k is an odd integer then Ž ⌳ i . s ⌳ 1yi for i s 0, 1 and Ž ⌳ n . s ⌳ n. The proof is similar to Proposition 1.2.5 in w2x. Ž DnŽ1., B k, 1 ., Ž1 F k F n y 2.. Let ⑀ 1 , . . . , ⑀ n4 be the orthonormal base n of ᒅ s [is1 Q ␣ i such that ␣ i s ⑀ i y ⑀ iq1 Ž1 F i F n y 1. and ␣ n s ⑀ ny1 q ⑀ n . We recall the Uq Ž Dn .-crystal B Ž ⌳ ny1 . and B Ž ⌳ n . in w11x. The elements of B Ž ⌳ n . Žresp., B Ž ⌳ ny1 .. can be labeled as: B Ž ⌳ n . Ž resp., B Ž ⌳ ny1 . . s Ž m1 , . . . , m n . m i g 1, 2, . . . , n, n, . . . , 1 4 such that m1 - m 2 - ⭈⭈⭈ - m n , m and m do not appear simultaneously, if m k s n then n y k is even Ž resp., odd. and if m k s n then n y k is odd Ž resp., even . 4 , where the ordering - on 1, 2, . . . , 14 is defined by 1 - 2 - ⭈⭈⭈ - n - n - ⭈⭈⭈ - 1. Note that if we define wtŽ i . s ⑀ ir2 and wtŽ i . s y⑀ ir2 then n
wt Ž Ž m1 , . . . , m n . . s
Ý wt Ž m i . . is1
324
YOSHIYUKI KOGA
The action of f˜i on the crystal B Ž ⌳ ny1 . and B Ž ⌳ n . is given by for 1 F i F n y 1,
¡ m ,..., iq1,..., sth
f˜i b s
ž ~
1
~
/
sth
t th
ž
/
otherwise,
¡ m ,...,
f˜n b s
i , . . . , mn
if b s m1 , . . . , i , . . . , i q 1 , . . . , m n ,
¢0
for i s n,
t th
ž
1
Ž sy1 .th
n
sth
, n y 1 , . . . , mn Ž sy1 .th
¢0
/
sth
if b s m1 , . . . , n y 1 , n , . . . , m n ,
ž
/
Ž 7.
otherwise.
From w2x we see that if the action of f˜0 on the crystals B Ž ⌳ ny1 . and B Ž ⌳ n . is given by f˜0 b s
½
Ž 1, 2, m1 , . . . , m ny2 .
if b s Ž m1 , . . . , m ny2 , 2, 1 . ,
0
otherwise,
Ž 8.
then B Ž ⌳ ny 1 . and B Ž ⌳ n . are perfect crystals of level one. For each s such that 0 F s F n, we consider the subset B˜s of B Ž ⌳ n .m 2 or B Ž ⌳ n . m B Ž ⌳ ny1 . defined by: if s is even: B˜s s Ž m1 , . . . , m n . m Ž mX1 , . . . , mXn . g B Ž ⌳ n . m B Ž ⌳ n . mXi F m iqs for 1 F i F n y s 4 , if s is odd: B˜s s Ž m1 , . . . , m n . m Ž mX1 , . . . , mXn . g B Ž ⌳ n . m B Ž ⌳ ny1 . mXi F m iqs for 1 F i F n y s 4 . In a similar way to CnŽ1.-case, we see that B˜s ,
½
B Ž 2⌳ n . [ B Ž ⌳ ny2 . [ ⭈⭈⭈ [ B Ž ⌳ nys .
if s is even,
B Ž ⌳ n q ⌳ ny1 . [ B Ž ⌳ ny3 . [ ⭈⭈⭈ [ B Ž ⌳ nys .
if s is odd,
325
LEVEL ONE PERFECT CRYSTALS
as a crystal for Uq Ž Dn .. Set Bs s B˜s _ B˜sy2 , where B˜y1 s B˜y2 s ⭋. From now on, we identify B k, 1 with [nyk F s F n, s ' nykŽ2. Bs . We describe the crystal graph of B k, 1 for 1 F k F n y 2. The ith arrow in B k, 1 is determined by Ž7. for 1 F i F n. Then we give the 0th arrow in B k, 1. Set J⬘ s 3, . . . , n4 . We set n⬘ by n⬘ s n if n ' k Žmod 2. and n⬘ s n y 1 if n k k Žmod 2.. For r s n, n y 1 and s s 1, 2, 3, 4, let B Ž ⌳ r .s be the subset of B Ž ⌳ r . given by B Ž ⌳ r . 1 s Ž 1, 2, m 3 , . . . , m n . g B Ž ⌳ r . 4 , B Ž ⌳ r . 2 s Ž 1, m 2 , . . . , m ny1 , 2 . g B Ž ⌳ r . 4 , B Ž ⌳ r . 3 s Ž 2, m 2 , . . . , m ny1 , 1 . g B Ž ⌳ r . 4 , BŽ ⌳r .4 s
½Žm , . . . , m 1
ny2 , 2, 1
. g BŽ ⌳r . 5 .
For s1 , s2 s 1, 2, 3, or 4, we take the subset Bs1 , s 2 of B Ž ⌳ n . m B Ž ⌳ n⬘ . defined by Bs1 , s 2 s b m b⬘ N b g B Ž ⌳ n . s1 , b⬘ g B Ž ⌳ n⬘ . s 2 4 . The 0th arrow in B k, 1 can be described as: for b g
[
nykq2FsFn , s'nyk Ž2 .
Bs ,
0
b « f˜0 b, 0
for b g Bny k _ Ž B14 j B24 j B34 . ,
b « f˜0 b,
for b g Bny k j B14 ,
b « 0,
for b g Bny k j B24 ,
b « Ž b. ,
for b g Bny k j B34 ,
b « ⬘Ž b . ,
0 0 0
where f˜0 b is given by Ž8. Žand Ž1.., and the maps and ⬘ are the Uq Ž ᒄ J ⬘ .-crystal isomorphisms,
: Bny k l B24 ª Bnyk l B12 , ⬘: Bny k l B34 ª Bnyk l B13 .
326
YOSHIYUKI KOGA
Proposition 3.5. The set of the minimal elements of B k, 1 is gi¨ en by
Ž Bk, 1.1 s
½
b 1 m b 2 , b 3 m b 4 , b 5 m b 6 , b7 m b 8 4 b1 m b 4 , b 3 m b 2 , b 9 m b 8 , b10 m b 6 4
if k is e¨ en, if k is odd,
where b1 s Ž 1, . . . , n . ,
b 2 s Ž 1, . . . , n . ,
b 3 s Ž 2, . . . , n, 1 . ,
b 4 s Ž 1, n, . . . , 2 . ,
b 5 s Ž 1, . . . , n y k y 1, n, n y 1, . . . , n y k . , b 6 s Ž n y k, . . . , n y 1, n, n y k y 1, . . . , 1 . , b7 s Ž 1, . . . , n y k, n, . . . , n y k q 1 . , b 8 s Ž n, . . . , n y k q 1, n y k, . . . , 1 . , b 9 s Ž 1, . . . , n y k, n, n y 1, . . . , n y k q 1 . , b10 s Ž 1, . . . , n y k y 1, n, . . . , n y k . . We gi¨ e the description of the automorphism of Ž Pclq. 1 s ⌳ 0 , ⌳ 1 , ⌳ ny 1 , ⌳ n4 determined by the perfect crystal B k, 1. PROPOSITION 3.6. If k is an e¨ en integer then Ž ⌳ i . s ⌳ i for any i s 0, 1, n y 1, n. If k is an odd integer then Ž ⌳ i . s ⌳ 1yi for i s 0, 1 and Ž ⌳ i . s ⌳ 2 nyiy1 for i s n y 1, n. 4. PROOF OF PROPOSITION 2.1 We prove Proposition 2.1 especially in case of CnŽ1., because the other cases can be proved similarly. We show the existence and uniqueness of the crystal B k, 1 for each k. Proof of the Existence. Let V be the nth fundamental representation of Uq Ž Cn . and Vz s QŽ q .w z, zy1 x m V the affinization of V. As a Uq Ž Cn .-module, V m V decomposes as: V m V , V Ž 2⌳ n . [ V Ž 2⌳ ny1 . [ ⭈⭈⭈ [ V Ž 2⌳ 1 . [ V Ž 0 . . The R-matrix Vx m Vy ª Vy m Vx is of the form n
R Ž xry . s
Ý ␥s Ž xry . P2 ⌳ ss0
ny s
,
327
LEVEL ONE PERFECT CRYSTALS
where P2 ⌳ ny s denotes the projection from V m V to V Ž2⌳ nys . and ␥s Ž z . is given by s
␥s Ž z . s
n
Ł Ž z y q 2 iq2 . Ł Ž1 y q 2 iq2 z . .
is1
Ž 9.
issq1
For each integer k such that 1 F k F n y 1, we consider the UqX Ž CnŽ1. .module Vk defined by Vk s R Ž qy2 nq2 ky2 . Ž Vqyn q ky 1 m Vq ny kq 1 . . From Ž9. we see that Vk , V Ž 2 Ž ⌳ k y ⌳ 0 . . [ V Ž 2 Ž ⌳ ky1 y ⌳ 0 . . [ ⭈⭈⭈ [ V Ž2 Ž ⌳ 1 y ⌳ 0 . . [ V Ž 0. , as a Uq Ž Cn .-module. Because the UqX Ž CnŽ1. .-module V has a polarization Ž , ., we can define a prepolarization Ž , . of Vk in the same way as w2x. The existence of a pseudo-crystal base of the UqX Ž CnŽ1. .-module Vk can be proved by an argument similar to the proof of Proposition 3.4.5 in w2x. Proof of the Uniqueness. We set B s [ssnyk Bs . Let B⬘ be any such that UJ Ž B⬘. , B as a crystal for Uq Ž Cn .. We denote the isomorphism UJ Ž B⬘. ª B by . Let us consider the subset X B ⬘, of B = B defined by n
UqX Ž CnŽ1. .-crystal
X B ⬘, s Ž b1 , b 2 . N b1 , b 2 g B, such that f˜0y1 Ž b1 . s y1 Ž b 2 . 4 . To prove the uniqueness, it suffices to show the proposition: PROPOSITION 4.1. The set X B ⬘, does not depend on the choice of the crystal B⬘ and the isomorphism . Proof. Put J⬘ s 2, 3, . . . , n4 . We observe that B Ž ⌳ n . " and B Ž ⌳ n . 0 in Section 3 are closed under the action of f˜j Ž j g J⬘.. We have B Ž ⌳ n . ", B Ž ⌳ ny 1 . and B Ž ⌳ n . 0 , B Ž ⌳ ny2 . as a crystal for Uq Ž ᒄ J ⬘ . , Uq Ž Cny1 .. Moreover, the subset B1 , 2 in Ž4. decomposes ny1
B1 , 2 ,
[ B Ž 2⌳ .
for 1 , 2 s " ,
i
is0
ny2
B , 0 , B0, ,
[ BŽ ⌳ q ⌳ is0
i
iq1
ny2
B0, 0 ,
[ is0
.
for s " ,
Ž 10 .
ny3
B Ž 2⌳ i . [
[ BŽ ⌳ q ⌳ is0
i
iq2
.,
328
YOSHIYUKI KOGA
as a crystal for Uq Ž Cny1 .. Here we consider ⌳ 0 s 0. For Ž1 , 2 . / Ž0, 0., let b1 , 2 Ž i . be the highest weight element of Uq Ž Cny1 .-crystal B 1 , 2 with highest weight 2⌳ i or ⌳ i q ⌳ iq1. We also denote by b 0, 0 Ž i . Žresp., ˜ b 0, 0 Ž i .. Ž the highest weight element of B0, 0 with highest weight 2⌳ i resp., ⌳ i q ⌳ iq2 .. By the definition of Bs in Section 3, we have
¡B
ny i
b1 , 2Ž i . g
~B
ny iy1
¢B
ny iy2
if Ž 1 , 2 . s Ž q, y . , if Ž 1 , 2 . s Ž q, q . , Ž y, y . , Ž 0, y . ,
Ž q, 0 . , Ž 0, 0 . , if Ž 1 , 2 . s Ž y, q . , Ž y, 0 . , Ž 0, q . ,
Ž 11 .
˜b 0, 0 Ž i . g Bnyiy2 . LEMMA 4.1. For any B⬘ and such that : B , B⬘ as a crystal for Uq Ž Cn ., we ha¨ e f˜0y1 Ž bqy Ž i . . s
½
for i s k,
0 y1
Ž bqq Ž i . .
for i - k,
f˜0y1 Ž byy Ž i y 1 . . s y1 Ž bqy Ž i y 1 . . , f˜0y1 Ž b 0y Ž i y 1 . . s
½
y1 Ž bq0 Ž i y 1 . . y1
Ž b0q Ž i y 1. .
for i s k , for i - k ,
f˜0y1 Ž by0 Ž i y 2 . . s y1 Ž bq0 Ž i y 2 . . , f˜0y1 Ž b 1 , 2Ž i . . s f˜0y1 Ž ˜ b 0, 0 Ž i . . s 0
for the others.
Proof. We can show this lemma by downward induction on i. Q.E.D. From this lemma, we see that Ž f˜0y1 Ž b 1 , 2Ž i ... does not depend on the choice of B⬘ and for any highest weight element b1 , 2Ž i . g B s n [ssnyk Bs . Because the action of f˜0 on B⬘ commutes with that of f˜j for any j g J⬘, we obtain Proposition 4.1. Remark 4.1. By the above lemma we have the crystal graph of B k, 1 in Section 3.
5. PROOF OF THEOREM 2.1 In this section we prove Theorem 2.1. By Proposition 2.1 the crystal B k, 1 satisfies the second condition in Definition 1.1. For the level one perfect crystals, the fourth condition is trivial. Therefore, we have to show the following two propositions.
LEVEL ONE PERFECT CRYSTALS
PROPOSITION 5.1.
329
B k, 1 m B k, 1 is connected.
PROPOSITION 5.2. The maps and gi¨ e bijections from Ž B k, 1 .1 to ⌳ 0 , ⌳ 1 , . . . , ⌳ n4. PROOF
OF
PROPOSITION 5.1. For s s 1, 2, . . . , n, we set
u s s Ž 1, 2, . . . , n . m Ž 1, 2, . . . , s, n, n y 1, . . . , s q 1 . , and u 0 s Ž 1, 2, . . . , n . m Ž n, n y 1, . . . , 1 . . We notice that u s g Bnys and it is the highest weight element of the Uq Ž ᒄ J .-crystal Bnys . We prove that any element b m b⬘ g Ž B k, 1 .m 2 is connected with u 0 m u 0 . If b g Bnys then there exists an element b1 g B k, 1 such that b m b⬘ is connected with u s m b1. Moreover, from Section 3, we see
˜e0 Ž u s . s Ž 1, 2, . . . , n . m Ž 2, . . . , s, n, n y 1, . . . , s q 1, 1 . g Bnysq1 , and 0 Ž u s . s 0 for each s G 1. By Ž1. we have ˜ e0tq1 Ž u s m b1 . s ˜ e0 u s m ˜ e0t b1 t Ž . Ž . for t s 0 b1 . Because ˜ e0 u s belongs to Bnysq1 , ˜ e0 u s m ˜ e0 b1 is connected with u sy 1 m b 2 for some b 2 g B k, 1 and then u s m b1 is connected with u sy 1 m b 2 . By induction we can show that u s m b1 is connected with u 0 m b 3 for some b 3 g B k, 1. Note that Bn is the crystal of the trivial representation for Uq Ž ᒄ J .. Therefore Bn m Bm , Bm as a Uq Ž ᒄ J .-crystal. Because the Uq Ž ᒄ J .-crystal Bm is connected, we see that u 0 m b 3 is connected with u 0 m u s . By the crystal graph of B k, 1 in Section 3, we have 0 Ž u 0 . s 1 and 0 Ž u s . s 2 for s G 2, and then ˜ e0 Ž u 0 m u s . s u 0 m ˜ e0 u s g Bn m Bnysq1. Hence u 0 m ˜ e0 u s is connected with u 0 m u sy1. By induction we can show that u 0 m u s is connected with u 0 m u 0 . Now we complete the proof. Q.E.D. Proof of Proposition 5.2. We denote the set of the minimal elements of the Uq Ž CnŽ1. .-crystal B Ž ⌳ n .m 2 by M. The minimal elements of B Ž ⌳ n . are given by
Ž B Ž ⌳ n . . 1 s Ž i q 1, . . . , n, n, . . . ,
n y i q 1 . N i s 0, 1, . . . , n4 .
Therefore, by Lemma 4.6.2 in w2x, we have M s b m b⬘ g B Ž ⌳ n .
m2
N Ž b . s Ž b⬘ . , ² c, Ž b . : s 1 4
s Ž i q 1, . . . , n y i q 1 . m Ž n y i q 1, . . . , i q 1 . N i s 0, 1, . . . , n4 .
330
YOSHIYUKI KOGA
From the crystal graph of B k, 1, we see that if b g [ssnykq1 Bs then the values of 0 Ž b . and 0 Ž b . in the crystal B k, 1 are equal to those of 0 Ž b . and 0 Ž b . in the crystal B Ž ⌳ n .m 2 . Hence we have n
n
ŽB
k, 1
.1 l
ž
[
n
ssnykq1
Bs s M l
/
ž
[
ssnykq1
Bs
/
s bi N i s 0, 1, . . . , k 4 , where bi s Ž i q 1, . . . , n y i q 1. m Ž n y i q 1, . . . , i q 1.. Because n
ŽB
k, 1
ž
.1 s Ž B
k, 1
.1 l
ž
[
ssnykq1
Bs
//
j Ž Ž B k , 1 . 1 l Bnyk . ,
we have to determine the set Ž B k, 1 .1 l Bnyk . For each i such that k q 1 F i F n, we set bi s Ž 1, 2, . . . , i y k, i q 1, . . . , n y k q 1 . m Ž n y k q 1, . . . , i q 1, i y k, . . . , 1 . . To prove Proposition 5.2, we need the lemma. LEMMA 5.1. Thus,
Ž B k , 1 . 1 l Bnyk s bi N i s k q 1, . . . , n4 . Before giving the proof of this lemma, we complete the proof of the proposition. By direct calculation we have
i Ž bj . s i Ž bj . s
½
1 0
if i s j, if i / j,
for any i, j such that 1 F i, j F n. Now it is clear that the maps and are bijective. Q.E.D. Proof of Lemma 5.1. For each i such that 1 F i F n, we set Ji s i q 1, . . . , n4 and consider the subsets X i and X i of B Ž ⌳ n . defined by X i s Ž 1, 2, . . . , i , m iq1 , . . . , m n . 4 , Xi s
½žm ,..., m 1
nyi ,
i , . . . , 2, 1
/5.
We see that X i , X i , B Ž ⌳ nyi . as a crystal for Uq Ž ᒄ J i . , Uq Ž Cnyi .. We denote the Uq Ž Cnyi .-isomorphisms X i ª B Ž ⌳ nyi . and X i ª B Ž ⌳ nyi . by i
LEVEL ONE PERFECT CRYSTALS
331
and i . For s ", 0, we set
Ž X i . s y1 i Ž B Ž ⌳ nyi . . ,
Ž X i . s y1 i Ž B Ž ⌳ nyi . . ,
where we define B Ž ⌳ ny i . in the same way as Ž3.. We can explicitly write the maps and as:
i : Ž 1, 2, . . . , i , m iq1 , . . . , m n . ¬ Ž mXiq1 , . . . , mXn . , i : m1 , . . . , m nyi , i , . . . , 2, 1 ¬ Ž mX1 , . . . , mXnyi . ,
ž
Ž 12 .
/
where mXj s
½
myi
if m jqi s m,
myi
if m jqi s m.
For any subsets X and X ⬘ of B Ž ⌳ n ., we denote the set b m b⬘ g B Ž ⌳ n .m 2 N b g X, b⬘ g X ⬘4 by X m X ⬘. First we prove that, for any b m b⬘ g Bny k l ŽŽ X i m X i . _ Ž X iq1 m X iq1 .., the condition b m b⬘ g Ž B k, 1 .1 is equivalent to i Ž b . m i Ž b⬘. g Ž B Ž ⌳ nyi .m 2 .1. To show this, it suffices to prove the facts,
j Ž b m b⬘ . s 0
if 0 F j F i y 1,
i Ž b m b⬘ . s 0 Ž i Ž b . m i Ž b⬘ . . ,
Ž 13 . Ž 14 .
j Ž b m b⬘ . s jyi Ž i Ž b . m i Ž b⬘ . .
if i q 1 F j F n.
Ž 15 .
Fact Ž13. is clear from the crystal graph of B k, 1 and the definition of X i and X i . Fact Ž15. is also obvious, because i and i are isomorphisms of Uq Ž ᒄ J i .. We prove Fact Ž14.. First we notice that X iq1 m X iq1 s Ž X i .q mŽ X i .y and
Ž Xi m X˜i . _ Ž Xiq1 m Xiq1 . s
D
1 , 2s" , 0 Ž 1 , 2 ./ Žq , y .
Ž X i . 1 m Ž X i . 2 .
For each 1 , 2 s ", 0, and b m b⬘ g Ž X i . 1 m Ž X i . 2 , the value of i Ž b m b⬘. as:
2 _ 1 q 0 y q
2
1
0
0
1
0
0
y
0
0
0
.
332
YOSHIYUKI KOGA
For i Ž b . m i Ž b⬘. g B Ž ⌳ nyi . 1 m B Ž ⌳ nyi . 2 , the value of 0 Ž i Ž b . m i Ž b⬘.. as:
2 _ 1 q 0 y q
2
1
0
0
1
0
0
y
1
0
0
.
Therefore, if b m b⬘ g Ž X i . 1 m Ž X i . 2 for Ž1 , 2 . / Žq, y. then i Ž b m b⬘. s 0 Ž i Ž b . m i Ž b⬘... To complete the proof, it is enough to show that
Ž B k , 1 . 1 l Bnyk l Ž Ž X i m X i . _ Ž X iq1 m X iq1 . . s bkqi 4 ,
Ž 16 .
for each i such that 1 F i F n y k, and Bny k l Ž X nykq1 m X nykq1 . s ⭋.
Ž 17 .
By the fact that
Ž B Ž ⌳ ny i . m2 . 1 s Ž j q 1, . . . , n y i , n y i, . . . , Ž n y i . y j q 1. j s 0, 1, . . . , n y i
4,
and Ž12., we have
Ž B k , 1 . 1 l Bnyk l Ž Ž X i m X i . _ Ž X iq1 m X iq1 . . s b m b⬘ g Bny k l Ž Ž X i m X i . _ Ž X iq1 m X iq1 . .
½
i Ž b . m i Ž b⬘ . g Ž B Ž ⌳ nyi .
m2
.1 5
s bkq i 4 . The second statement Ž17. is clear from the definition of Bny k .
Q.E.D.
6. DISCUSSION At the end of this paper we state a conjecture on the perfect crystals of generic level that was given by A. Kuniba et al. in w5x. Through the research for Bethe ansatz systems, they obtain a conjecture on the exis-
333
LEVEL ONE PERFECT CRYSTALS
tence of the level l perfect crystal B k, l such that, as a Uq Ž ᒄ 1, 2, . . . , n4 .-crystal, B k , l , B Ž l␥ k ⌳ k . [ ⭈⭈⭈ , where ␥ k is some positive integer. To be precise let a k Žresp., a k . be the kth label Žresp., colabel. of the Dynkin diagram of ᒄ w14x. The integer ␥ k is given by ␥ k s maxŽ1, a kra ˇk .. In the cases of BnŽ1., CnŽ1., and DnŽ1., this integer is equal to the integer t k in the Introduction and Proposition 2.1. From w6, 7x Žand w8x for the simply laced case. the UqX Ž ᒄ .-crystal B k, l is expected to decompose into Uq Ž ᒄ 1, 2, . . . , n4 .-crystals as: For BnŽ1., Bk, l ,
[
m1 , . . . , m k
B Ž m1 ␥ 1 ⌳ 1 q ⭈⭈⭈ qm k ␥ k ⌳ k . ,
where the sum is taken over m1 q ⭈⭈⭈ qm k F l and m i s 0 for i k k Žmod 2.. For CnŽ1.,
Bk, l ,
¡ ¢B Ž l⌳ .
~m[ ,..., m 1
B Ž m1 ␥ 1 ⌳ 1 q ⭈⭈⭈ qm k ␥ k ⌳ k .
if 1 F k F n y 1,
k
if k s n,
n
where the sum is taken over m1 q ⭈⭈⭈ qm k F l. For DnŽ1.,
B
k, l
¡ ¢B Ž l⌳ .
[ ,~ m , . . . , m 1
B Ž m1 ␥ 1 ⌳ 1 q ⭈⭈⭈ qm k ␥ k ⌳ k .
if 1 F k F n y 2,
k
if k s n y 1, n,
k
where the sum is taken over m1 q ⭈⭈⭈ qm k F l and m i s 0 for i k k Žmod 2.. In case of l s 1, these crystals coincide with those of Proposition 2.1.
ACKNOWLEDGMENTS The author would like to express his gratitude to Masato Okado for his suggestion of this problem and advice. He is grateful to Atsuo Kuniba for his comments on perfect crystals of generic level and also to Etsuro Date, Kenji Iohara, Mikami Hirasawa, Toshiki Nakashima, and Shigenori Yamane for their valuable advice. Furthermore he thanks Mitsuru Ikawa and Kiyokazu Nagatomo for giving him continuous encouragement.
334
YOSHIYUKI KOGA
REFERENCES 1. S.-J. Kang, M. Kashiwara, K. C. Misra, T. Miwa, T. Nakashima, and A. Nakayashiki, Affine crystals and vertex models, Internat. J. Modern Phys. A 7, Suppl. 1A Ž1992., 449᎐484. 2. 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 Ž1992., 499᎐607. 3. S.-J. Kang, M. Kashiwara, and K. C. Misra, Crystal bases of Verma modules for quantum affine Lie algebras, Compositio Math. 92 Ž1994., 299᎐325. . Ž . 4. S. Yamane, Perfect crystals of Uq Ž G Ž1. 2 , J. Algebra 210 1998 , 440᎐486. 5. A. Kuniba, K. C. Misra, M. Okado, T. Takagi, and J. Uchiyama, Crystals for Demazure modules of classical affine Lie algebras, q-algr9707014. 6. A. N. Kirillov and N. Yu. Reshetikhin, Representations of Yangians and multiplicities of occurrence of the irreducible components of the tensor product of representations of simple Lie algebras, J. So¨ iet Math. 52 Ž1990., 3156᎐3164. 7. A. Kuniba, Thermodynamics of the Uq Ž X rŽ1. . Bethe ansatz system with q a root of unity, Nucl. Phys. B 389 Ž1993., 209᎐244. 8. M. Kleber, Combinatorial structure of infinite-dimensional representations of Yangians: simply-laced case, Internat. Math. Res. Notices 4 Ž1997., 187᎐201. 9. M. Kashiwara, Crystallizing the q-analogue of universal enveloping algebras, Comm. Math. Phys. 133 Ž1990., 246᎐260. 10. M. Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras, Duke Math. J. 63 Ž1991., 465᎐516. 11. M. Kashiwara and T. Nakashima, Crystal graphs for representations of the q-analogue of classical Lie algebras, J. Algebra 165 Ž1994., 295᎐345. 12. T. Nakashima, Crystal base and a generalization of the Littlewood᎐Richardson rule for the classical Lie algebras Comm. Math. Phys. 154 Ž1993., 215᎐243. 13. M. Okado, Quantum R matrices related to the spin representations of Bn and Dn , Comm. Math. Phys. 134 Ž1990., 467᎐486. 14. V. G. Kac, Infinite dimensional Lie algebras, 3rd ed., Cambridge Univ. Press, Cambridge, U.K., 1990.
Level One Perfect Crystals for BnŽ1., CnŽ1., and DnŽ1. Yoshiyuki Koga Department of Mathematics, Graduate School of Science, Osaka Uni¨ ersity, Toyonaka, Osaka 560-0043, Japan Communicated by Georgia Benkart Received June 18, 1998
0. INTRODUCTION Crystal base was first introduced by M. Kashiwara in w9x as a theory of the integrable representations for the quantum enveloping algebra Uq Ž ᒄ . at q s 0. In w1x S.-J. Kang et al. studied crystal bases of finite-dimensional modules over quantum affine algebras and introduced the notion of perfect crystals to identify the one-dimensional configuration sums of vertex models with the string functions of affine Lie algebras. In w2x they Ž1. Ž1. also constructed examples of perfect crystals in the cases of AŽ1. n , Bn , C n , Ž1. Ž2. Ž2. Ž2. Ž1. Ž1. Dn , A 2 n , A 2 ny1 , and Dnq1. Other examples for Cn and G 2 are treated in w3, 4x. Let ᒄ be an affine Lie algebra and ᒄ its classical part. We denote by B Ž ⌳ . the crystal base of the finite-dimensional Uq Ž ᒄ .-module with highest weight ⌳. In the case that ᒄ is of type AŽ1. n , there exists a perfect crystal B k, l for UqX Ž ᒄ . such that B k, l , B Ž l⌳ k . as a crystal for Uq Ž ᒄ .. For BnŽ1., CnŽ1., and DnŽ1., the known examples are isomorphic to either B Ž l⌳ 1 ., B Ž l⌳ ny 1 ., or B Ž l⌳ n . as a crystal for Uq Ž ᒄ .. The purpose of this paper is to give new examples of perfect crystals for UqX Ž BnŽ1. ., UqX Ž CnŽ1. ., and UqX Ž DnŽ1. . that are isomorphic to B Ž tk ⌳ k . [
ž[
⌳-t k ⌳ k
BŽ ⌳. ,
/
where t k s 1 if the kth simple root ␣ k of ᒄ is a long root, t k s 2 if ␣ k is a short root. The symbol - stands for the dominance order Žsee Sect. 3.. All of these crystals are expected to give the list of the perfect crystals of level one in the following sense: Any perfect crystal of level one is isomorphic to one of these crystals or the tensor product of them. 312 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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LEVEL ONE PERFECT CRYSTALS
This paper is organized as follows: In Section 1 we recall the theory of crystal base and summarize known results on the perfect crystals. In Sections 2 and 3 we give the main statements, and we prove them in the following sections. At the end of this paper we comment on the generic level perfect crystals.
1. REVIEW OF PERFECT CRYSTALS Let ᒄ be a symmetrizable Kac᎐Moody algebra over Q and ᒅ its Cartan subalgebra w14x. Let ␣ i N i g I 4 , h i N i g I 4 , and ⌳ i N i g I 4 be the set of the simple roots, the simple coroots, and the fundamental weights that are indexed by the finite set I. We denote the pairing between ᒅ and ᒅ* by ² , : and put P s Ý i Z⌳ i , P* s HomŽ P, Z.. Because ᒄ is symmetrizable, there is a symmetric bilinear form Ž , . on ᒅ* such that ² h i , : s 2Ž ␣ i , .rŽ ␣ i , ␣ i . for any i g I and g ᒅ*. The quantized enveloping algebra Uq Ž ᒄ . is an associative algebra over QŽ q . with generators e i , f i Ž i g I . and q h Ž h g P*. which satisfy the relations q 0 s 1,
q h ⭈ q h⬘ s q hq h⬘
q h e i qyh s q ² h , ␣ i : e i ,
w ei , fi x s ␦i , j
q h f i qyh s qy² h , ␣ i : f i ,
t i y ty1 i qi y qy1 i
,
where qi s q Ž ␣ i , ␣ i . and t i s q Ž ␣ i , ␣ i . h i , b
b
ms0
ms0
m m Ý Ž y1. eiŽ m. e j eiŽ bym. s Ý Ž y1. f iŽ m. f j f iŽ bym. s 0,
for i / j, where b s 1 y ² h i , ␣ j :. Here we set
w nx i s
qin y qyn i
e iŽ n. s
qi y qy1 i e in
w nx i!
n
,
,
w n x i !s Ł w k x i , ks1
f iŽ n. s
f in
w nx i!
.
The algebra Uq Ž ᒄ . has the following Hopf algebra structure, ⌬ Ž e i . s e i m ty1 q 1 m ei , i
⌬ Ž fi . s fi m 1 q ti m fi ,
⌬Ž qh . s qh m qh, a Ž e i . s ye i t i ,
a Ž f i . s yty1 i fi ,
a Ž q h . s qyh .
314
YOSHIYUKI KOGA
We denote by Uq Ž ᒄ . i the subalgebra of Uq Ž ᒄ . generated by e i , f i , and q h Ž h g P*.. For a Uq Ž ᒄ .-module M and g P, we set M s u g M N q h u s q ² h, : u for any h g P*4 . We say that M is integrable if it satisfies Ži. M s [ g P M, and Žii. M is a union of finite-dimensional Uq Ž ᒄ . i-modules for any i g I. Let u be any weight vector of an integrable Uq Ž ᒄ .-module M. It is easy to see that u can be uniquely expressed as u s Ý k f iŽ k . u k , where u k g Mqk ␣ i l ker e i and i g I. We define operators ˜ e i , f˜i by
˜e i u s Ý f iŽ ky1. u k ,
f˜i u s
Ý f iŽ kq1. u k .
For any vector space V and a subset B of V, we call B a pseudo-base if there exists a base B⬘ of V such that B s B⬘ [ ŽyB⬘.. Let us define a subring A of QŽ q . by A s f g QŽ q . N f : regular at q s 04 . A pair Ž L, B . is called a crystal Žpseudo-. base of an integrable module M if it satisfies 1. L is a free A-submodule of M such that QŽ q . mA L , M, 2. L s [g P L where L s L l M , 3. ˜ e i L ; L and f˜i L ; L. 4. B is a Žpseudo-.base of Q-vector space LrqL, 5. B s " g P B where B s B l Ž LrqL ., 6. ˜ e i B ; B " 04 and f˜i B ; B " 04 , 7. for b, b⬘ g B and i g I, b⬘ s f˜i b if and only if b s ˜ e i b⬘. We denote wtŽ b . s for b g M and wtŽ B . s wtŽ b . N b g B4 . Let us recall the tensor product of crystal bases. For b g B and i g I, we set
i Ž b . s max n G 0 N ˜ e n b g B4 , i Ž b . s max n G 0 N f˜in b g B 4 . It is easy to see that ² h i , wtŽ b .: s i Ž b . y i Ž b . for each i g I. Let B1 and B2 be crystal bases. The actions of ˜ e i and f˜i on the tensor product B1 m B2 can be written as f˜i b1 m b 2
if i Ž b1 . ) i Ž b 2 . ,
b1 m f˜i b 2
if i Ž b1 . F i Ž b 2 . ,
˜e i b1 m b 2 ˜e i Ž b1 m b 2 . s b1 m ˜ ei b2
if i Ž b1 . G i Ž b 2 . ,
f˜i Ž b1 m b 2 . s
½ ½
if i Ž b1 . - i Ž b 2 . .
Ž 1.
LEVEL ONE PERFECT CRYSTALS
315
Next we recall the definition of perfect crystals given in w1x. Let ᒄ be an affine Lie algebra and I s 0, 1, . . . , n4 the index set of simple roots of ᒄ. Let c and ␦ be the canonical central element and the generator of the imaginary roots. We take Ý nis0 Z⌳ i [ Z ␦ as the weight lattice P and then P* s Ý nis0 Zh i [ Z d. Set ᒅUcl s ᒅ*rQ ␦ and denote the canonical map ᒅ* ª ᒅUcl by cl. Because ² ␦ , h i : s 0 for any i g I, we can identify n Ž[is0 Qh i .* with ᒅUcl . Here we consider the fundamental weights ⌳ i as an element of ᒅUcl and define the classical weight lattice Pcl by Ý nis0 Q⌳ i . For a crystal B, we say that B is classical if wtŽ B . ; Pcl . Let us denote by UqX Ž ᒄ . the subalgebra of the Uq Ž ᒄ . generated by e i , f i , and t i for all i g I. From now on, we only consider UqX Ž ᒄ .-modules which have classical crystal bases. To give the definition of the perfect crystals, we need the following notations: Pclqs g Pcl N ² h i , : G 0 for any i g I 4 and Ž Pclq. l s g Pclq N ² c, : s l 4 . For an element b of the classical crystal B, set Ž b . s Ý i i Ž b . ⌳ i and Ž b . s Ý i i Ž b . ⌳ i . DEFINITION 1.1. For l g Z ) 0 , we call a classical crystal B a perfect crystal of level l if it satisfies the conditions: 1. B m B is connected. 2. There exists 0 g Pcl such that wtŽ B . ; 0 q Ý i/ 0 Z F 0 ␣ i and 噛Ž B0 . s 1. 3. There exist a finite-dimensional integrable UqX Ž ᒄ .-module with a crystal pseudo-base Ž L, B⬘. such that B , B⬘r "14 . 4. For any b g B, we have ² c, Ž b .: G l. . 5. The maps and from Bl s b N ² c, Ž b .: s l 4 to Ž pq cl l are bijective. An element of Bl is called minimal. Let B be a perfect crystal of level l. For g Ž Pclq. l , let bŽ . g B be the element that satisfies ŽbŽ .. s . We define the automorphism of Ž Pclq. l by s Ž bŽ ... We denote by BŽ . the crystal base of the integrable highest weight UqX Ž ᒄ .-module with highest weight . Then we have THEOREM 1.1 w1x. Let P Ž , B . be the set of sequences pŽ n.4 in B such that pŽ n. s bŽ ny 1 Ž .. for n 4 0. Then BŽ . is isomorphic to P Ž , B .. We recall the examples of perfect crystals given in w2, 3x in the case that Ž1. Ž1. Ž1. ᒄ is of type AŽ1. n , Bn , C n , and Dn . Let J be any subset of the index set I X and Uq Ž ᒄ J . the subalgebra of Uq Ž ᒄ . generated by e i , f i , and t i Ž i g J .. We sometimes use the notation J to denote the map defined by J Ž j . s j for any j g J. For a crystal B for UqX Ž ᒄ ., we denote by UJ Ž B . the crystal for
316
YOSHIYUKI KOGA
Uq Ž ᒄ J . such that UJ Ž B . s B as a set, and for j g J, f˜j Ž b . s b⬘ if and only if f˜j Ž b . s b⬘ in B. Here we list known results. Set J s 1, 2, . . . , n4 . THEOREM 1.2 w2x. Let ᒄ be of type AŽ1. n . For each integer k, l such that 1 F k F n, l G 1, there exists a perfect crystal B k, l of le¨ el l such that UJ Ž B k, l . , B Ž l⌳ k . as a crystal for Uq Ž ᒄ .. THEOREM 1.3 w2x. Let ᒄ be of type BnŽ1., CnŽ1., or DnŽ1.. For each integer l such that l G 1, there exist perfect crystals B 1, l for Uq Ž BnŽ1. ., B n, l for Uq Ž CnŽ1. . and B 1, l , B ny1, l , B n, l for Uq Ž DnŽ1. ., of le¨ el l such that UJ Ž B k, l . , B Ž l⌳ k . as a crystal for Uq Ž ᒄ .. THEOREM 1.4 w3x. Let ᒄ be of type CnŽ1.. For each integer l such that l G 1, there exist perfect a crystal B 1, l of le¨ el l such that UJ Ž B k, l . , B Ž2 l⌳ 1 . [ B Ž2Ž l y 1. ⌳ 1 . [ ⭈⭈⭈ [ B Ž2⌳ 1 . [ B Ž0. as a crystal for Uq Ž ᒄ .. 2. PERFECT CRYSTALS OF LEVEL ONE Through this paper, let J s 1, 2, . . . , n4 be a subset of the index set I. For ␣ ,  g Pcl , we define ␣ -  by  y ␣ g Ý nis1 Z G 0 ␣ i and ␣ /  . PROPOSITION 2.1. Let ᒄ be of type BnŽ1., CnŽ1., or DnŽ1.. For each integer k such that 1 F k F n, there exists a unique crystal B k, 1 for Uq Ž ᒄ . such that,
UJ Ž B k , 1 . , B Ž t k ⌳ k . [
ž[
⌳-t k ⌳ k
BŽ ⌳. ,
/
as crystals for Uq Ž ᒄ .. Here t k s 1 if ␣ k is a long root, t k s 2 if ␣ k is a short root. An explicit form of the crystal UJ Ž B k, 1 . as: For BnŽ1., 1 F k F n y 1,
UJ Ž B k , 1 . , B Ž ⌳ k . [ B Ž ⌳ ky2 . [ B Ž ⌳ ky4 . [ ⭈⭈⭈ [ Ž B Ž ⌳ 1 . or B Ž 0 . . . For BnŽ1., k s n,
UJ Ž B n , 1 . , B Ž 2⌳ n . [ B Ž ⌳ ny2 . [ B Ž ⌳ ny4 . [ ⭈⭈⭈ [ Ž B Ž ⌳ 1 . or B Ž 0 . . . For CnŽ1., 1 F k F n y 1,
UJ Ž B k , 1 . , B Ž 2⌳ k . [ B Ž 2⌳ ky1 . [ B Ž 2⌳ ky2 . [ ⭈⭈⭈ [ B Ž 2⌳ 1 . [ B Ž 0 . . For CnŽ1., k s n,
UJ Ž B n , 1 . , B Ž ⌳ n . .
317
LEVEL ONE PERFECT CRYSTALS
For DnŽ1., 1 F k F n y 2,
UJ Ž B k , 1 . , B Ž ⌳ k . [ B Ž ⌳ ky2 . [ B Ž ⌳ ky4 . [ ⭈⭈⭈ [ Ž B Ž ⌳ 1 . or B Ž 0 . . . For DnŽ1., k s n y 1, n.
UJ Ž B k , 1 . , B Ž ⌳ k . . We remark that in the case k s 1 Žfor BnŽ1. ., k s n Žfor CnŽ1. ., and k s 1, n y 1, n Žfor DnŽ1. ., these crystals coincide with those reviewed in the previous section. THEOREM 2.1. The crystal B k, 1 is a perfect crystal of le¨ el one. We prove Proposition 2.1 and Theorem 2.1 in Sections 4 and 5. Remark 2.1. By Theorem 1.2, the above proposition and theorem are valid for AŽ1. n . Remark 2.2. We conjecture that the above list of crystals exhausts the perfect crystals of level one; i.e., any level one perfect crystal is isomorphic to one of these crystals or the tensor product of them ŽSee Sect. 6..
3. CRYSTAL GRAPHS The crystal graph of the perfect crystal B k, 1 is given in this section. The minimal elements of B k, 1 and the automorphism of Ž Pclq. 1 are also described. Ž CnŽ1., B k, 1 ., Ž1 F k F n y 1.. Let ⑀ 1 , . . . , ⑀ n4 be the orthonormal base n of ᒅ s [is1 Q ␣ i such that ␣ i s ⑀ i y ⑀ iq1 Ž1 F i F n y 1. and ␣ n s 2 ⑀ n . We summarize crystal graphs for Uq Ž Cn . in w11x. Let B Ž ⌳ 1 . be the crystal base of the vector representation of Uq Ž Cn . that is labeled by Ž i ., Ž i . N 1 F i F n4 . Its crystal graph is given by f˜i Ž j . s ␦ i , j Ž j q 1 . ,
f˜i Ž j . s ␦ iq1, j Ž j y 1 .
f˜n Ž j . s ␦n , j Ž n . ,
f˜n Ž j . s 0.
for 1 F i - n,
Note that wtŽ i . s ⑀ i and wtŽ i . s y⑀ i . The crystal graph of B Ž ⌳ N . Ž1 F N F n. can be identified with the connected component of B Ž ⌳ 1 .m N containing the element Ž1. m Ž2. m ⭈⭈⭈ m Ž N .. In fact B Ž ⌳ N . s Ž m1 , . . . , m N . m i g 1, 2, . . . , n, n, . . . , 1 4 such that m1 - m 2 - ⭈⭈⭈ - m N , and i q Ž N y j q 1 . F m i if m i s m j Ž i - j . 4 ,
318
YOSHIYUKI KOGA
where Ž m1 , . . . , m N . s Ž m1 . m ⭈⭈⭈ m Ž m N . and the ordering - on 1, 2, ˜4 is defined by . . . ,1 1 - 2 - ⭈⭈⭈ - n - n - ⭈⭈⭈ - 1. Furthermore we recall the crystal graph of B Ž ⌳ M q ⌳ N . Ž1 F M F N F n.. Let u s Ž m1 , . . . , m M . and ¨ s Ž mX1 , . . . , mXN . be elements of B Ž ⌳ M . and B Ž ⌳ N .. For 1 F a F b F n, we say that u m ¨ g B Ž ⌳ M . m B Ž ⌳ N . is in the Ž a, b .-configuration if u m ¨ satisfies the following w12x; there exist 1 F p F q - r F s F M such that mXp s a, m q s b, m r s b, m s s a, or mXp s a, mXq s b, mXr s b, m s s a. Remark that the definition of the Ž a, b .-configuration includes the case of a s b, p s q, and r s s. Set pŽ a, b; u m ¨ . s Ž q y p . q Ž s y r ., then we have B Ž ⌳ M q ⌳ N . s u m ¨ g B Ž ⌳ M . m B Ž ⌳ N . mXi F m i for 1 F i F M and if u m ¨ is in the Ž a, b . -configuration, then p Ž a, b; u m ¨ . - b y a4 . The element u m ¨ is called a semi-standard C-tableau of shape ⌳ M q ⌳ N if u m ¨ g B Ž ⌳ M q ⌳ N .. From w2x we see that if the action of f˜0 on the crystal B Ž ⌳ n . is given by f˜0 b s
½
Ž 1, m1 , . . . , m ny1 .
if b s Ž m1 , . . . , m ny1 , 1 . ,
0
otherwise,
Ž 2.
then B Ž ⌳ n . is a perfect crystal of level one. We construct the UqX Ž CnŽ1. .-crystal B k, 1 by using the crystal B Ž ⌳ n .. For each s such that 0 F s F n and u s Ž m1 , . . . , m n ., ¨ s Ž mX1 , . . . , mXn . g B Ž ⌳ n ., we define sequences Ž g 1 , . . . , g nqs ., Ž g X1 , . . . , g Xnqs . as:
¡i if 1 F i F s, if s q 1 F i F s q n, m s j, ¢i q j if s q 1 F i F s q n, m s j, if 1 F i F n, m s j, ¡i q j ~ if 1 F i F n, m s j, g s iqj ¢n q s q 1 y i if n q 1 F i F n q s. g s~i q j
iys
i
iys
iys
X i
iys
Here we denote the elements Ž g 1 , . . . , g nqs . and Ž g X1 , . . . , g Xnqs . by u ˜ and ¨˜. Let us define the subset B˜s of B Ž ⌳ n .m 2 by B˜s s u m ¨ g B Ž ⌳ n .
½
m2
u ˜ m ¨˜ is a semi-standard C-tableau for Uq Ž Cnqs . of shape 2⌳ nqs .
5
LEVEL ONE PERFECT CRYSTALS
319
In the same way as w11x, we can show that B˜s is closed under the action of f˜i for i s 1, . . . , n and B˜s , B Ž 2⌳ n . [ B Ž 2⌳ ny1 . [ ⭈⭈⭈ [ B Ž 2⌳ nys . , as a crystal for Uq Ž Cn .. Set Bs s B˜s _ B˜sy1 , where B˜y1 s ⭋. From now on n we identify B k, 1 with [ssnyk Bs . We describe the crystal graph of the UqX Ž CnŽ1. .-crystal B k, 1 under the above identification. This is proven in the next section. For each j g J, the jth arrow in the crystal B k, 1 can be described by Ž1., because B k, 1 is identified with a subset of B Ž ⌳ n .m 2 . We give the 0th arrow of the crystal B k, 1. Set J⬘ s 2, 3, . . . , n4 . Let B Ž ⌳ n .q, B Ž ⌳ n .y, and B Ž ⌳ n . 0 be the subsets of B Ž ⌳ n . given by B Ž ⌳ n . q s Ž 1, m 2 , . . . , m n . g B Ž ⌳ n . 4 , B Ž ⌳ n . y s Ž m, . . . , m ny1 , 1 . g B Ž ⌳ n . 4 ,
Ž 3.
B Ž ⌳ n . 0 s B Ž ⌳ n . _ Ž B Ž ⌳ n . qj B Ž ⌳ n . y . . For 1 , 2 s " and 0, we define the subset B 1 , 2 of B Ž ⌳ n .m 2 by B 1 , 2 s b m b⬘ N b g B Ž ⌳ n . 1 , b⬘ g B Ž ⌳ n . 2 4 .
Ž 4.
The 0th arrow in B k, 1 can be described as: n
for b g
[
ssnykq1
Bs ,
0
b « f˜0 b, 0
for b g Bny k _ Ž Bqyj B0y . ,
b « f˜0 b,
for b g Bny k l Bqy ,
b « 0,
for b g Bny k l B0y ,
b « Ž b. ,
0 0
where f˜0 b is given by Ž2. Žand Ž1.., and the map is the Uq Ž ᒄ J ⬘ .-crystal isomorphism:
: Bny k l B0yª Bnyk l Bq0 . By the fact Ž10. in Section 4, this isomorphism is uniquely determined. PROPOSITION 3.1. The set of the minimal elements of B k, 1 is gi¨ en by 1 N i s 0, . . . , k 4 j bi2 m bi3 N i s k q 1, . . . , n4 , Ž B k , 1 . 1 s bi1 m bnyi
320
YOSHIYUKI KOGA
where bi1 s Ž i q 1, . . . , n, n, . . . , n y i q 1 . , bi2 s Ž 1, . . . , i y k, i q 1, . . . , n, n, . . . , n y k q 1 . , bi3 s Ž n y k q 1, . . . , n, n, . . . , i q 1, i y k, . . . , 1 . . We prove this proposition in Section 5. We give the description of the automorphism ⌳ 0 , ⌳ 1 , . . . , ⌳ n4 determined by the perfect crystal B k, 1.
of Ž Pclq. 1 s
Ž ⌳ i . s ⌳ i for any i such that 0 F i F n.
PROPOSITION 3.2.
The proof is similar to Proposition 1.2.5 in w2x. Ž BnŽ1., B k, 1 ., Ž1 F k F n.. Let ⑀ 1 , . . . , ⑀ n4 be the orthonormal base of n ᒅ s [is1 Q ␣ i such that ␣ i s ⑀ i y ⑀ iq1 Ž1 F i F n y 1. and ␣ n s ⑀ n . We recall the Uq Ž Bn .-crystal B Ž ⌳ n . in w11x. The elements of B Ž ⌳ n . can be labeled as: B Ž ⌳ n . s Ž m1 , . . . , m n . m i g 1, 2, . . . , n, n, . . . , 1 4 such that m1 - m 2 - ⭈⭈⭈ - m n , and m and m do not appear simultaneously ,
5
where the ordering - on 1, 2, . . . , 14 is defined by 1 - 2 - ⭈⭈⭈ - n - n - ⭈⭈⭈ - 1. Note that if we define wtŽ i . s ⑀ ir2 and wtŽ i . s y ⑀ ir2 then wtŽŽ m1 , . . . , m n .. s Ý nis1 wtŽ m i .. The action of f˜i on the crystal B Ž ⌳ n . is given by for 1 F i F n y 1,
¡ m ,..., iq1,..., sth
f˜i b s
ž ~
1
¢0
t th
i , . . . , mn
/
sth
t th
if b s m1 , . . . , i , . . . , ı q 1 , . . . , m n ,
ž
/
otherwise,
for i s n,
¡ ¢0
f˜ b s~ ž m , . . . , n
1
sth
n , . . . , mn
/
sth
if b s m1 , . . . , n , . . . , m n ,
ž
otherwise.
/
Ž 5.
321
LEVEL ONE PERFECT CRYSTALS
If we define the action of f˜0 on the crystal B Ž ⌳ n . by f˜0 b s
½
Ž 1, 2, m1 , . . . , m ny2 .
if b s Ž m1 , . . . , m ny2 , 2, 1 . ,
0
otherwise,
Ž 6.
then B Ž ⌳ n . is the crystal base of the UqX Ž BnŽ1. .-module V Ž ⌳ n y ⌳ 0 .. We construct the UqX Ž BnŽ1. .-crystal B k, 1. For each s such that 0 F s F n, we consider the following subset B˜s of B Ž ⌳ n .m 2 : B˜s s Ž m1 , . . . , m n . m Ž mX1 , . . . , mXn . g B Ž ⌳ n .
½
m2
mXi F m iqs
for 1 F i F n y s 4 . In a similar way to
CnŽ1.-case,
we see that
B˜s , B Ž 2⌳ n . [ B Ž ⌳ ny1 . [ ⭈⭈⭈ [ B Ž ⌳ nys . , as a crystal for Uq Ž Bn .. Set Bs s B˜s _ B˜sy1 , where B˜y1 s ⭋. From now on, we identify B k, 1 with [nyk F s F n, s ' nykŽ2. Bs . We describe the crystal graph of B k, 1. For each i such that 1 F i F n, the ith arrow in B k, 1 is determined by Ž5.. Then we give the 0th arrow in B k, 1. Set J⬘ s 3, . . . , n4 . First we consider the crystal B k, 1 for 1 F k F n y 1. Let B Ž ⌳ n .s Ž s s 1, 2, 3, 4. be the subsets of B Ž ⌳ n . given by B Ž ⌳ n . 1 s Ž 1, 2, m 3 , . . . , m n . g B Ž ⌳ n . 4 , B Ž ⌳ n . 2 s Ž 1, m 2 , . . . , m ny1 , 2 . g B Ž ⌳ n . 4 , B Ž ⌳ n . 3 s Ž 2, m 2 , . . . , m ny1 , 1 . g B Ž ⌳ n . 4 , BŽ ⌳n.4 s
½Žm , . . . , m 1
ny2 , 2, 1
. g BŽ ⌳n. 5 .
For s1 , s2 s 1, 2, 3, or 4, we take the subset Bs1 , s 2 of B Ž ⌳ n .m 2 defined by Bs1 , s 2 s b m b⬘ N b g B Ž ⌳ n . s1 , b⬘ g B Ž ⌳ n . s 2 4 . The 0th arrow in B k, 1 can be described as: for b g
[
nykq2FsFn , s'nyk Ž2 .
Bs ,
0
b « f˜0 b, 0
for b g Bny k _ Ž B14 j B24 j B34 . ,
b « f 0 b,
for b g Bny k l B14 ,
b « 0,
for b g Bny k l B24 ,
b « Ž b. ,
for b g Bny k l B34 ,
b « ⬘Ž b . ,
0 0 0
322
YOSHIYUKI KOGA
where f˜0 b is given by Ž6. Žand Ž1.., and the maps and ⬘ are the Uq Ž ᒄ J ⬘ .-crystal isomorphisms,
: Bny k l B24 ª Bnyk l B12 , ⬘: Bny k l B34 ª Bnyk l B13 . Next we consider the crystal B n, 1. Let B42 and B43 be the subsets of B42 and B43 given by B42 s B42 _
½Žm , . . . , m 1
ny2 , 2, 1
. m Ž 1, mX2 , . . . , mXny1 , 2.
mXiq1 F m i
for 1 F i F n y 2 B43 s B43 _
½Žm , . . . , m 1
ny2 , 2, 1
. m Ž 2, mX2 , . . . , mXny1 , 1.
5,
mXiq1 F m i
for 1 F i F n y 2
5.
The 0th arrow in B k, 1 can be described as: for b g
[
2FsFn , s'0 Ž2 .
Bs ,
0
b « f˜0 b, 0
for b g B0 _ Ž B42 j B43 . ,
b « f˜0 b,
for b g B0 l B42 ,
b « Ž b. ,
for b g B0 l B43 ,
b « ⬘Ž b . ,
0 0
where f˜0 b is given by Ž6. Žand Ž1.., and the maps and ⬘ are the Uq Ž ᒄ J ⬘ .-crystal isomorphisms,
: Bny k l B42 ª Bnyk l B21 , ⬘: Bny k l B43 ª Bnyk l B31 . Here B21 and B31 are the following subsets of B21 and B31 , B21 s B21 _ Ž 1, 2, m 3 , . . . , m n . m Ž 1, mX2 , . . . , mXny1 , 2 . mXiy1 F m i for 3 F i F n 4 , B31 s B31 _ Ž 1, 2, m 3 , . . . , m n . m Ž 2, mX2 , . . . , mXny1 , 1 . mXiy1 F m i for 3 F i F n 4 .
323
LEVEL ONE PERFECT CRYSTALS
PROPOSITION 3.3. The set of the minimal elements of B k, 1 is gi¨ en by
Ž Bk, 1.1 s
½
b1 m b 2 , b 3 m b 4 , b 5 m b 6 4 b1 m b 4 , b 3 m b 2 , b 5 m b 6 4
if k is e¨ en, if k is odd,
where b1 s Ž 1, . . . , n . ,
b 2 s Ž 1, . . . , n . ,
b 3 s Ž 2, . . . , n, 1 . ,
b 4 s Ž 1, n, . . . , 2 . ,
b 5 s Ž 1, . . . , n y k, n, . . . , n y k q 1 . , b 6 s Ž n y k q 1, . . . , n, n y k, . . . , 1 . . We give the description of the automorphism of Ž Pclq. 1 s ⌳ 0 , ⌳ 1 , ⌳ n4 determined by the perfect crystal B k, 1. PROPOSITION 3.4. If k is an e¨ en integer then Ž ⌳ i . s ⌳ i for any i s 0, 1, n. If k is an odd integer then Ž ⌳ i . s ⌳ 1yi for i s 0, 1 and Ž ⌳ n . s ⌳ n. The proof is similar to Proposition 1.2.5 in w2x. Ž DnŽ1., B k, 1 ., Ž1 F k F n y 2.. Let ⑀ 1 , . . . , ⑀ n4 be the orthonormal base n of ᒅ s [is1 Q ␣ i such that ␣ i s ⑀ i y ⑀ iq1 Ž1 F i F n y 1. and ␣ n s ⑀ ny1 q ⑀ n . We recall the Uq Ž Dn .-crystal B Ž ⌳ ny1 . and B Ž ⌳ n . in w11x. The elements of B Ž ⌳ n . Žresp., B Ž ⌳ ny1 .. can be labeled as: B Ž ⌳ n . Ž resp., B Ž ⌳ ny1 . . s Ž m1 , . . . , m n . m i g 1, 2, . . . , n, n, . . . , 1 4 such that m1 - m 2 - ⭈⭈⭈ - m n , m and m do not appear simultaneously, if m k s n then n y k is even Ž resp., odd. and if m k s n then n y k is odd Ž resp., even . 4 , where the ordering - on 1, 2, . . . , 14 is defined by 1 - 2 - ⭈⭈⭈ - n - n - ⭈⭈⭈ - 1. Note that if we define wtŽ i . s ⑀ ir2 and wtŽ i . s y⑀ ir2 then n
wt Ž Ž m1 , . . . , m n . . s
Ý wt Ž m i . . is1
324
YOSHIYUKI KOGA
The action of f˜i on the crystal B Ž ⌳ ny1 . and B Ž ⌳ n . is given by for 1 F i F n y 1,
¡ m ,..., iq1,..., sth
f˜i b s
ž ~
1
~
/
sth
t th
ž
/
otherwise,
¡ m ,...,
f˜n b s
i , . . . , mn
if b s m1 , . . . , i , . . . , i q 1 , . . . , m n ,
¢0
for i s n,
t th
ž
1
Ž sy1 .th
n
sth
, n y 1 , . . . , mn Ž sy1 .th
¢0
/
sth
if b s m1 , . . . , n y 1 , n , . . . , m n ,
ž
/
Ž 7.
otherwise.
From w2x we see that if the action of f˜0 on the crystals B Ž ⌳ ny1 . and B Ž ⌳ n . is given by f˜0 b s
½
Ž 1, 2, m1 , . . . , m ny2 .
if b s Ž m1 , . . . , m ny2 , 2, 1 . ,
0
otherwise,
Ž 8.
then B Ž ⌳ ny 1 . and B Ž ⌳ n . are perfect crystals of level one. For each s such that 0 F s F n, we consider the subset B˜s of B Ž ⌳ n .m 2 or B Ž ⌳ n . m B Ž ⌳ ny1 . defined by: if s is even: B˜s s Ž m1 , . . . , m n . m Ž mX1 , . . . , mXn . g B Ž ⌳ n . m B Ž ⌳ n . mXi F m iqs for 1 F i F n y s 4 , if s is odd: B˜s s Ž m1 , . . . , m n . m Ž mX1 , . . . , mXn . g B Ž ⌳ n . m B Ž ⌳ ny1 . mXi F m iqs for 1 F i F n y s 4 . In a similar way to CnŽ1.-case, we see that B˜s ,
½
B Ž 2⌳ n . [ B Ž ⌳ ny2 . [ ⭈⭈⭈ [ B Ž ⌳ nys .
if s is even,
B Ž ⌳ n q ⌳ ny1 . [ B Ž ⌳ ny3 . [ ⭈⭈⭈ [ B Ž ⌳ nys .
if s is odd,
325
LEVEL ONE PERFECT CRYSTALS
as a crystal for Uq Ž Dn .. Set Bs s B˜s _ B˜sy2 , where B˜y1 s B˜y2 s ⭋. From now on, we identify B k, 1 with [nyk F s F n, s ' nykŽ2. Bs . We describe the crystal graph of B k, 1 for 1 F k F n y 2. The ith arrow in B k, 1 is determined by Ž7. for 1 F i F n. Then we give the 0th arrow in B k, 1. Set J⬘ s 3, . . . , n4 . We set n⬘ by n⬘ s n if n ' k Žmod 2. and n⬘ s n y 1 if n k k Žmod 2.. For r s n, n y 1 and s s 1, 2, 3, 4, let B Ž ⌳ r .s be the subset of B Ž ⌳ r . given by B Ž ⌳ r . 1 s Ž 1, 2, m 3 , . . . , m n . g B Ž ⌳ r . 4 , B Ž ⌳ r . 2 s Ž 1, m 2 , . . . , m ny1 , 2 . g B Ž ⌳ r . 4 , B Ž ⌳ r . 3 s Ž 2, m 2 , . . . , m ny1 , 1 . g B Ž ⌳ r . 4 , BŽ ⌳r .4 s
½Žm , . . . , m 1
ny2 , 2, 1
. g BŽ ⌳r . 5 .
For s1 , s2 s 1, 2, 3, or 4, we take the subset Bs1 , s 2 of B Ž ⌳ n . m B Ž ⌳ n⬘ . defined by Bs1 , s 2 s b m b⬘ N b g B Ž ⌳ n . s1 , b⬘ g B Ž ⌳ n⬘ . s 2 4 . The 0th arrow in B k, 1 can be described as: for b g
[
nykq2FsFn , s'nyk Ž2 .
Bs ,
0
b « f˜0 b, 0
for b g Bny k _ Ž B14 j B24 j B34 . ,
b « f˜0 b,
for b g Bny k j B14 ,
b « 0,
for b g Bny k j B24 ,
b « Ž b. ,
for b g Bny k j B34 ,
b « ⬘Ž b . ,
0 0 0
where f˜0 b is given by Ž8. Žand Ž1.., and the maps and ⬘ are the Uq Ž ᒄ J ⬘ .-crystal isomorphisms,
: Bny k l B24 ª Bnyk l B12 , ⬘: Bny k l B34 ª Bnyk l B13 .
326
YOSHIYUKI KOGA
Proposition 3.5. The set of the minimal elements of B k, 1 is gi¨ en by
Ž Bk, 1.1 s
½
b 1 m b 2 , b 3 m b 4 , b 5 m b 6 , b7 m b 8 4 b1 m b 4 , b 3 m b 2 , b 9 m b 8 , b10 m b 6 4
if k is e¨ en, if k is odd,
where b1 s Ž 1, . . . , n . ,
b 2 s Ž 1, . . . , n . ,
b 3 s Ž 2, . . . , n, 1 . ,
b 4 s Ž 1, n, . . . , 2 . ,
b 5 s Ž 1, . . . , n y k y 1, n, n y 1, . . . , n y k . , b 6 s Ž n y k, . . . , n y 1, n, n y k y 1, . . . , 1 . , b7 s Ž 1, . . . , n y k, n, . . . , n y k q 1 . , b 8 s Ž n, . . . , n y k q 1, n y k, . . . , 1 . , b 9 s Ž 1, . . . , n y k, n, n y 1, . . . , n y k q 1 . , b10 s Ž 1, . . . , n y k y 1, n, . . . , n y k . . We gi¨ e the description of the automorphism of Ž Pclq. 1 s ⌳ 0 , ⌳ 1 , ⌳ ny 1 , ⌳ n4 determined by the perfect crystal B k, 1. PROPOSITION 3.6. If k is an e¨ en integer then Ž ⌳ i . s ⌳ i for any i s 0, 1, n y 1, n. If k is an odd integer then Ž ⌳ i . s ⌳ 1yi for i s 0, 1 and Ž ⌳ i . s ⌳ 2 nyiy1 for i s n y 1, n. 4. PROOF OF PROPOSITION 2.1 We prove Proposition 2.1 especially in case of CnŽ1., because the other cases can be proved similarly. We show the existence and uniqueness of the crystal B k, 1 for each k. Proof of the Existence. Let V be the nth fundamental representation of Uq Ž Cn . and Vz s QŽ q .w z, zy1 x m V the affinization of V. As a Uq Ž Cn .-module, V m V decomposes as: V m V , V Ž 2⌳ n . [ V Ž 2⌳ ny1 . [ ⭈⭈⭈ [ V Ž 2⌳ 1 . [ V Ž 0 . . The R-matrix Vx m Vy ª Vy m Vx is of the form n
R Ž xry . s
Ý ␥s Ž xry . P2 ⌳ ss0
ny s
,
327
LEVEL ONE PERFECT CRYSTALS
where P2 ⌳ ny s denotes the projection from V m V to V Ž2⌳ nys . and ␥s Ž z . is given by s
␥s Ž z . s
n
Ł Ž z y q 2 iq2 . Ł Ž1 y q 2 iq2 z . .
is1
Ž 9.
issq1
For each integer k such that 1 F k F n y 1, we consider the UqX Ž CnŽ1. .module Vk defined by Vk s R Ž qy2 nq2 ky2 . Ž Vqyn q ky 1 m Vq ny kq 1 . . From Ž9. we see that Vk , V Ž 2 Ž ⌳ k y ⌳ 0 . . [ V Ž 2 Ž ⌳ ky1 y ⌳ 0 . . [ ⭈⭈⭈ [ V Ž2 Ž ⌳ 1 y ⌳ 0 . . [ V Ž 0. , as a Uq Ž Cn .-module. Because the UqX Ž CnŽ1. .-module V has a polarization Ž , ., we can define a prepolarization Ž , . of Vk in the same way as w2x. The existence of a pseudo-crystal base of the UqX Ž CnŽ1. .-module Vk can be proved by an argument similar to the proof of Proposition 3.4.5 in w2x. Proof of the Uniqueness. We set B s [ssnyk Bs . Let B⬘ be any such that UJ Ž B⬘. , B as a crystal for Uq Ž Cn .. We denote the isomorphism UJ Ž B⬘. ª B by . Let us consider the subset X B ⬘, of B = B defined by n
UqX Ž CnŽ1. .-crystal
X B ⬘, s Ž b1 , b 2 . N b1 , b 2 g B, such that f˜0y1 Ž b1 . s y1 Ž b 2 . 4 . To prove the uniqueness, it suffices to show the proposition: PROPOSITION 4.1. The set X B ⬘, does not depend on the choice of the crystal B⬘ and the isomorphism . Proof. Put J⬘ s 2, 3, . . . , n4 . We observe that B Ž ⌳ n . " and B Ž ⌳ n . 0 in Section 3 are closed under the action of f˜j Ž j g J⬘.. We have B Ž ⌳ n . ", B Ž ⌳ ny 1 . and B Ž ⌳ n . 0 , B Ž ⌳ ny2 . as a crystal for Uq Ž ᒄ J ⬘ . , Uq Ž Cny1 .. Moreover, the subset B1 , 2 in Ž4. decomposes ny1
B1 , 2 ,
[ B Ž 2⌳ .
for 1 , 2 s " ,
i
is0
ny2
B , 0 , B0, ,
[ BŽ ⌳ q ⌳ is0
i
iq1
ny2
B0, 0 ,
[ is0
.
for s " ,
Ž 10 .
ny3
B Ž 2⌳ i . [
[ BŽ ⌳ q ⌳ is0
i
iq2
.,
328
YOSHIYUKI KOGA
as a crystal for Uq Ž Cny1 .. Here we consider ⌳ 0 s 0. For Ž1 , 2 . / Ž0, 0., let b1 , 2 Ž i . be the highest weight element of Uq Ž Cny1 .-crystal B 1 , 2 with highest weight 2⌳ i or ⌳ i q ⌳ iq1. We also denote by b 0, 0 Ž i . Žresp., ˜ b 0, 0 Ž i .. Ž the highest weight element of B0, 0 with highest weight 2⌳ i resp., ⌳ i q ⌳ iq2 .. By the definition of Bs in Section 3, we have
¡B
ny i
b1 , 2Ž i . g
~B
ny iy1
¢B
ny iy2
if Ž 1 , 2 . s Ž q, y . , if Ž 1 , 2 . s Ž q, q . , Ž y, y . , Ž 0, y . ,
Ž q, 0 . , Ž 0, 0 . , if Ž 1 , 2 . s Ž y, q . , Ž y, 0 . , Ž 0, q . ,
Ž 11 .
˜b 0, 0 Ž i . g Bnyiy2 . LEMMA 4.1. For any B⬘ and such that : B , B⬘ as a crystal for Uq Ž Cn ., we ha¨ e f˜0y1 Ž bqy Ž i . . s
½
for i s k,
0 y1
Ž bqq Ž i . .
for i - k,
f˜0y1 Ž byy Ž i y 1 . . s y1 Ž bqy Ž i y 1 . . , f˜0y1 Ž b 0y Ž i y 1 . . s
½
y1 Ž bq0 Ž i y 1 . . y1
Ž b0q Ž i y 1. .
for i s k , for i - k ,
f˜0y1 Ž by0 Ž i y 2 . . s y1 Ž bq0 Ž i y 2 . . , f˜0y1 Ž b 1 , 2Ž i . . s f˜0y1 Ž ˜ b 0, 0 Ž i . . s 0
for the others.
Proof. We can show this lemma by downward induction on i. Q.E.D. From this lemma, we see that Ž f˜0y1 Ž b 1 , 2Ž i ... does not depend on the choice of B⬘ and for any highest weight element b1 , 2Ž i . g B s n [ssnyk Bs . Because the action of f˜0 on B⬘ commutes with that of f˜j for any j g J⬘, we obtain Proposition 4.1. Remark 4.1. By the above lemma we have the crystal graph of B k, 1 in Section 3.
5. PROOF OF THEOREM 2.1 In this section we prove Theorem 2.1. By Proposition 2.1 the crystal B k, 1 satisfies the second condition in Definition 1.1. For the level one perfect crystals, the fourth condition is trivial. Therefore, we have to show the following two propositions.
LEVEL ONE PERFECT CRYSTALS
PROPOSITION 5.1.
329
B k, 1 m B k, 1 is connected.
PROPOSITION 5.2. The maps and gi¨ e bijections from Ž B k, 1 .1 to ⌳ 0 , ⌳ 1 , . . . , ⌳ n4. PROOF
OF
PROPOSITION 5.1. For s s 1, 2, . . . , n, we set
u s s Ž 1, 2, . . . , n . m Ž 1, 2, . . . , s, n, n y 1, . . . , s q 1 . , and u 0 s Ž 1, 2, . . . , n . m Ž n, n y 1, . . . , 1 . . We notice that u s g Bnys and it is the highest weight element of the Uq Ž ᒄ J .-crystal Bnys . We prove that any element b m b⬘ g Ž B k, 1 .m 2 is connected with u 0 m u 0 . If b g Bnys then there exists an element b1 g B k, 1 such that b m b⬘ is connected with u s m b1. Moreover, from Section 3, we see
˜e0 Ž u s . s Ž 1, 2, . . . , n . m Ž 2, . . . , s, n, n y 1, . . . , s q 1, 1 . g Bnysq1 , and 0 Ž u s . s 0 for each s G 1. By Ž1. we have ˜ e0tq1 Ž u s m b1 . s ˜ e0 u s m ˜ e0t b1 t Ž . Ž . for t s 0 b1 . Because ˜ e0 u s belongs to Bnysq1 , ˜ e0 u s m ˜ e0 b1 is connected with u sy 1 m b 2 for some b 2 g B k, 1 and then u s m b1 is connected with u sy 1 m b 2 . By induction we can show that u s m b1 is connected with u 0 m b 3 for some b 3 g B k, 1. Note that Bn is the crystal of the trivial representation for Uq Ž ᒄ J .. Therefore Bn m Bm , Bm as a Uq Ž ᒄ J .-crystal. Because the Uq Ž ᒄ J .-crystal Bm is connected, we see that u 0 m b 3 is connected with u 0 m u s . By the crystal graph of B k, 1 in Section 3, we have 0 Ž u 0 . s 1 and 0 Ž u s . s 2 for s G 2, and then ˜ e0 Ž u 0 m u s . s u 0 m ˜ e0 u s g Bn m Bnysq1. Hence u 0 m ˜ e0 u s is connected with u 0 m u sy1. By induction we can show that u 0 m u s is connected with u 0 m u 0 . Now we complete the proof. Q.E.D. Proof of Proposition 5.2. We denote the set of the minimal elements of the Uq Ž CnŽ1. .-crystal B Ž ⌳ n .m 2 by M. The minimal elements of B Ž ⌳ n . are given by
Ž B Ž ⌳ n . . 1 s Ž i q 1, . . . , n, n, . . . ,
n y i q 1 . N i s 0, 1, . . . , n4 .
Therefore, by Lemma 4.6.2 in w2x, we have M s b m b⬘ g B Ž ⌳ n .
m2
N Ž b . s Ž b⬘ . , ² c, Ž b . : s 1 4
s Ž i q 1, . . . , n y i q 1 . m Ž n y i q 1, . . . , i q 1 . N i s 0, 1, . . . , n4 .
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YOSHIYUKI KOGA
From the crystal graph of B k, 1, we see that if b g [ssnykq1 Bs then the values of 0 Ž b . and 0 Ž b . in the crystal B k, 1 are equal to those of 0 Ž b . and 0 Ž b . in the crystal B Ž ⌳ n .m 2 . Hence we have n
n
ŽB
k, 1
.1 l
ž
[
n
ssnykq1
Bs s M l
/
ž
[
ssnykq1
Bs
/
s bi N i s 0, 1, . . . , k 4 , where bi s Ž i q 1, . . . , n y i q 1. m Ž n y i q 1, . . . , i q 1.. Because n
ŽB
k, 1
ž
.1 s Ž B
k, 1
.1 l
ž
[
ssnykq1
Bs
//
j Ž Ž B k , 1 . 1 l Bnyk . ,
we have to determine the set Ž B k, 1 .1 l Bnyk . For each i such that k q 1 F i F n, we set bi s Ž 1, 2, . . . , i y k, i q 1, . . . , n y k q 1 . m Ž n y k q 1, . . . , i q 1, i y k, . . . , 1 . . To prove Proposition 5.2, we need the lemma. LEMMA 5.1. Thus,
Ž B k , 1 . 1 l Bnyk s bi N i s k q 1, . . . , n4 . Before giving the proof of this lemma, we complete the proof of the proposition. By direct calculation we have
i Ž bj . s i Ž bj . s
½
1 0
if i s j, if i / j,
for any i, j such that 1 F i, j F n. Now it is clear that the maps and are bijective. Q.E.D. Proof of Lemma 5.1. For each i such that 1 F i F n, we set Ji s i q 1, . . . , n4 and consider the subsets X i and X i of B Ž ⌳ n . defined by X i s Ž 1, 2, . . . , i , m iq1 , . . . , m n . 4 , Xi s
½žm ,..., m 1
nyi ,
i , . . . , 2, 1
/5.
We see that X i , X i , B Ž ⌳ nyi . as a crystal for Uq Ž ᒄ J i . , Uq Ž Cnyi .. We denote the Uq Ž Cnyi .-isomorphisms X i ª B Ž ⌳ nyi . and X i ª B Ž ⌳ nyi . by i
LEVEL ONE PERFECT CRYSTALS
331
and i . For s ", 0, we set
Ž X i . s y1 i Ž B Ž ⌳ nyi . . ,
Ž X i . s y1 i Ž B Ž ⌳ nyi . . ,
where we define B Ž ⌳ ny i . in the same way as Ž3.. We can explicitly write the maps and as:
i : Ž 1, 2, . . . , i , m iq1 , . . . , m n . ¬ Ž mXiq1 , . . . , mXn . , i : m1 , . . . , m nyi , i , . . . , 2, 1 ¬ Ž mX1 , . . . , mXnyi . ,
ž
Ž 12 .
/
where mXj s
½
myi
if m jqi s m,
myi
if m jqi s m.
For any subsets X and X ⬘ of B Ž ⌳ n ., we denote the set b m b⬘ g B Ž ⌳ n .m 2 N b g X, b⬘ g X ⬘4 by X m X ⬘. First we prove that, for any b m b⬘ g Bny k l ŽŽ X i m X i . _ Ž X iq1 m X iq1 .., the condition b m b⬘ g Ž B k, 1 .1 is equivalent to i Ž b . m i Ž b⬘. g Ž B Ž ⌳ nyi .m 2 .1. To show this, it suffices to prove the facts,
j Ž b m b⬘ . s 0
if 0 F j F i y 1,
i Ž b m b⬘ . s 0 Ž i Ž b . m i Ž b⬘ . . ,
Ž 13 . Ž 14 .
j Ž b m b⬘ . s jyi Ž i Ž b . m i Ž b⬘ . .
if i q 1 F j F n.
Ž 15 .
Fact Ž13. is clear from the crystal graph of B k, 1 and the definition of X i and X i . Fact Ž15. is also obvious, because i and i are isomorphisms of Uq Ž ᒄ J i .. We prove Fact Ž14.. First we notice that X iq1 m X iq1 s Ž X i .q mŽ X i .y and
Ž Xi m X˜i . _ Ž Xiq1 m Xiq1 . s
D
1 , 2s" , 0 Ž 1 , 2 ./ Žq , y .
Ž X i . 1 m Ž X i . 2 .
For each 1 , 2 s ", 0, and b m b⬘ g Ž X i . 1 m Ž X i . 2 , the value of i Ž b m b⬘. as:
2 _ 1 q 0 y q
2
1
0
0
1
0
0
y
0
0
0
.
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YOSHIYUKI KOGA
For i Ž b . m i Ž b⬘. g B Ž ⌳ nyi . 1 m B Ž ⌳ nyi . 2 , the value of 0 Ž i Ž b . m i Ž b⬘.. as:
2 _ 1 q 0 y q
2
1
0
0
1
0
0
y
1
0
0
.
Therefore, if b m b⬘ g Ž X i . 1 m Ž X i . 2 for Ž1 , 2 . / Žq, y. then i Ž b m b⬘. s 0 Ž i Ž b . m i Ž b⬘... To complete the proof, it is enough to show that
Ž B k , 1 . 1 l Bnyk l Ž Ž X i m X i . _ Ž X iq1 m X iq1 . . s bkqi 4 ,
Ž 16 .
for each i such that 1 F i F n y k, and Bny k l Ž X nykq1 m X nykq1 . s ⭋.
Ž 17 .
By the fact that
Ž B Ž ⌳ ny i . m2 . 1 s Ž j q 1, . . . , n y i , n y i, . . . , Ž n y i . y j q 1. j s 0, 1, . . . , n y i
4,
and Ž12., we have
Ž B k , 1 . 1 l Bnyk l Ž Ž X i m X i . _ Ž X iq1 m X iq1 . . s b m b⬘ g Bny k l Ž Ž X i m X i . _ Ž X iq1 m X iq1 . .
½
i Ž b . m i Ž b⬘ . g Ž B Ž ⌳ nyi .
m2
.1 5
s bkq i 4 . The second statement Ž17. is clear from the definition of Bny k .
Q.E.D.
6. DISCUSSION At the end of this paper we state a conjecture on the perfect crystals of generic level that was given by A. Kuniba et al. in w5x. Through the research for Bethe ansatz systems, they obtain a conjecture on the exis-
333
LEVEL ONE PERFECT CRYSTALS
tence of the level l perfect crystal B k, l such that, as a Uq Ž ᒄ 1, 2, . . . , n4 .-crystal, B k , l , B Ž l␥ k ⌳ k . [ ⭈⭈⭈ , where ␥ k is some positive integer. To be precise let a k Žresp., a k . be the kth label Žresp., colabel. of the Dynkin diagram of ᒄ w14x. The integer ␥ k is given by ␥ k s maxŽ1, a kra ˇk .. In the cases of BnŽ1., CnŽ1., and DnŽ1., this integer is equal to the integer t k in the Introduction and Proposition 2.1. From w6, 7x Žand w8x for the simply laced case. the UqX Ž ᒄ .-crystal B k, l is expected to decompose into Uq Ž ᒄ 1, 2, . . . , n4 .-crystals as: For BnŽ1., Bk, l ,
[
m1 , . . . , m k
B Ž m1 ␥ 1 ⌳ 1 q ⭈⭈⭈ qm k ␥ k ⌳ k . ,
where the sum is taken over m1 q ⭈⭈⭈ qm k F l and m i s 0 for i k k Žmod 2.. For CnŽ1.,
Bk, l ,
¡ ¢B Ž l⌳ .
~m[ ,..., m 1
B Ž m1 ␥ 1 ⌳ 1 q ⭈⭈⭈ qm k ␥ k ⌳ k .
if 1 F k F n y 1,
k
if k s n,
n
where the sum is taken over m1 q ⭈⭈⭈ qm k F l. For DnŽ1.,
B
k, l
¡ ¢B Ž l⌳ .
[ ,~ m , . . . , m 1
B Ž m1 ␥ 1 ⌳ 1 q ⭈⭈⭈ qm k ␥ k ⌳ k .
if 1 F k F n y 2,
k
if k s n y 1, n,
k
where the sum is taken over m1 q ⭈⭈⭈ qm k F l and m i s 0 for i k k Žmod 2.. In case of l s 1, these crystals coincide with those of Proposition 2.1.
ACKNOWLEDGMENTS The author would like to express his gratitude to Masato Okado for his suggestion of this problem and advice. He is grateful to Atsuo Kuniba for his comments on perfect crystals of generic level and also to Etsuro Date, Kenji Iohara, Mikami Hirasawa, Toshiki Nakashima, and Shigenori Yamane for their valuable advice. Furthermore he thanks Mitsuru Ikawa and Kiyokazu Nagatomo for giving him continuous encouragement.
334
YOSHIYUKI KOGA
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