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Schur–Weyl Reciprocity for Ariki–Koike Algebras
Journal of Algebra 221, 293᎐314 Ž1999. Article ID jabr.1999.7973, available online at http:rrwww.idealibrary.com on
Schur᎐Weyl Reciprocity for Ariki᎐Koike AlgebrasU Masahiro Sakamoto and Toshiaki Shoji Department of Mathematics, Science Uni¨ ersity of Tokyo, Noda, Chiba 278-8510, Japan Communicated by Michel Broue´ Received May 1, 1998 Let ᒄ s gl m 1 [ ⭈⭈⭈ [ gl m r be a Levi subalgebra of gl m , with m s Ý ris1 m i , and V the natural representation of the quantum group Uq Ž ᒄ .. We construct a representation of the Ariki᎐Koike algebra Hn, r on the n-fold tensor space of V, commuting with the action of Uq Ž ᒄ ., and prove the Schur᎐Weyl reciprocity for the actions of Uq Ž ᒄ . and Hn, r on it. 䊚 1999 Academic Press
1. INTRODUCTION The natural representation of UŽ gl m . on V s ⺓ m induces the action of V , which commutes with the permutation action of the symmetric group ᑭ n . The classical Schur᎐Weyl reciprocity asserts that the images of UŽ gl m . and of ᑭ n in End V mn are mutually fully centralizers of each other. Let ᒄ s gl m 1 [ ⭈⭈⭈ [ gl m r be a Levi subalgebra of gl m , with m s Ý ris1 m i , and Wn, r the complex reflection group ᑭ n h Ž⺪rr⺪. n. The action of ᑭ n on V mn can be extended to an action of Wn, r , commuting with the action of UŽ ᒄ . obtained as the restriction of UŽ gl m .. The Schur᎐Weyl reciprocity also holds between the action of UŽ ᒄ . and Wn, r on V mn. The q-analogue of the Schur᎐Weyl reciprocity, i.e., the reciprocity between the quantum group Uq Ž gl m . and the Hecke algebra Hn of type A ny 1 , was obtained by Jimbo wJx. In this paper, we establish the q-analogue of the latter case; i.e., we consider the quantum group Uq Ž ᒄ . and the Ariki᎐Koike algebra Hn, r which is the Hecke algebra of Wn, r . More precisely, we consider the natural representation of Uq Ž gl m . of an m-dimensional vector space V. It induces an action of the subalgebra Uq Ž ᒄ . of Uq Ž gl m . on V mn. We construct a representation of Hn, r on V mn commuting with the action of Uq Ž ᒄ ., and prove the Schur᎐Weyl reciprocity between mn
U
This paper is a contribution to the Joint Research Project ‘‘Representation Theory of Finite and Algebraic Groups’’ 1997᎐1999 under the Japanese᎐German Cooperative Science Promotion Program supported by JSPS and DFG. 293 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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the actions of Uq Ž ᒄ . and Hn, r on V mn. In wATYx, Ariki, Terasoma, and Yamada treated a similar problem in the special case where m1 s ⭈⭈⭈ s m r s 1. Our approach was inspired by their arguments. For the proof of the Schur᎐Weyl reciprocity, we need to use the classification of irreducible representations of Hn, r together with some properties of representations obtained by Ariki and Koike wAKx. However, for the construction of the representation of Hn, r on V mn, we only need the generators and relations for Hn, r . This representation provides us an alternative approach to some results given in wAKx. In fact, it is verified that in the case where m i G n for any i G 1, the representation of Wn, r on V mn is faithful. By using a specialization argument, one can show in this case that the representation of Hn, r on V mn is also faithful. It follows from this by a simple argument that we have dim Hn, r s < Wn, r < and Hn, r is semisimple. Note that in wAKx, these results are obtained by constructing all the irreducible representations of Hn, r and by counting their dimensions. Using the result of our paper, Ariki wAx defined certain type of cyclotomic q-Schur algebras as quotients of quantum groups, and studied the relationship with the cyclotomic q-Schur algebras defined by Dipper, James, and Mathas wDJMx. 2. A REVIEW OF KNOWN RESULTS 2.1. Let Uq Ž gl m . be the quantized universal enveloping algebra of gl m over K s ⺡Ž q, u1 , . . . , u r ., a field of rational functions in variables q, u1 , . . . , u r , which is an associative algebra defined by generators e i , f i , Ž1 F i - m., q " i Ž1 F i F m. and by the relations qi qj s qj qi ,
q i qy i s qy i q i s 1,
¡q
y1
i
y i
q ej q
s ~ qe
¢e ¡qf
ej
j
j
q i f j qy i s qy1 f j
¢f
if j s i , otherwise,
j
~
if j s i y 1,
j
ei f j y f j ei s ␦i j
if j s i y 1, if j s i , otherwise, K i y Ky1 i q y qy1
,
e i " 1 e i2 y Ž q q qy1 . e i e i " 1 e i q e i2 e i " 1 s 0, f i " 1 f i2 y Ž q q qy1 . f i f i " 1 f i q f i2 f i " 1 s 0, ei e j s e j ei , fi f j s f j fi if < i y j < G 2,
Ž 2.1.1.
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where in the above formula, K i stand for q iy iq 1 for i s 1, . . . , m y 1. It is known that Uq Ž gl m . has a Hopf algebra structure with comultiplication ⌬: Uq Ž gl m . ª Uq Ž gl m . m Uq Ž gl m . defined by ⌬Ž q" i . s q" i m q" i , ⌬ Ž ei . s ei m 1 q K i m ei , ⌬ Ž f i . s f i m Ky1 q 1 m fi . i Let V be an m-dimensional vector space over K with a basis E s ¨ 1 , . . . , ¨ m 4 . The natural representation of Uq Ž gl m . on V is defined by
Ž ei . ¨ j s Ž fi . ¨ j s Ž q" i .¨j s
½ ½
½
¨ jy1
if j s i q 1,
0
if j / i q 1,
¨ jq1
if j s i ,
0
if j / i ,
q"1¨j
if j s i ,
0
if j / i.
For a positive integer n, we consider the tensor product space V mn, on which Uq Ž gl m .mn acts naturally. We define inductively an algebra homomorphism ⌬Ž k . : Uq Ž gl m . ª Uq Ž gl m .mk by ⌬Ž k . s Ž ⌬Ž ky1. m id.( ⌬ for each k G 3, where ⌬Ž2. s ⌬. Combined with ⌬Ž n., Uq Ž gl m . acts naturally on V mn. We denote this action also by . It is expressed as follows: ny1
Ž ei . s
Ý
K im p m e i m 1mŽ ny1yp. ,
ps0 ny1
Ž fi . s
Ý
1m p m f i m Ž Ky1 i .
m Ž ny1yp .
,
Ž 2.1.2.
ps0
Ž q " i . s q " i m ⭈⭈⭈ m q " i . 2.2. We fix positive integers m1 , m 2 , . . . , m r such that Ý i m i s m. Let Vi be an m i-dimensional vector space with a basis Ei s ¨ jŽ i. N j s 1, . . . , m i 4 for i s 1, . . . , r. We define a total order on the set @ Ei by the lexicographic order on the set of pairs Ž i, j . such that 1 F i F r, 1 F j F m i , and identify the set @ Ei with E by this total order. Hence we have V s r [is1 Vi . We now consider the Lie algebra ᒄ s gl m 1 [ ⭈⭈⭈ [ gl m r. The Lie algebra gl m i acts naturally on Vi , and by the above identification, ᒄ is regarded as a subalgebra of gl m . Hence Uq Ž ᒄ . is a subalgebra of Uq Ž gl m ., and acts on V mn by the restriction of the action of Uq Ž gl m ..
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For a Young diagram , we denote by < < the number of boxes in , and by l Ž . the number of rows of . Irreducible representations of Uq Ž gl m . occurring in V mn are parametrized by Young diagrams with l Ž . F m. Let V be an irreducible Uq Ž gl m .-module with highest weight vector ¨ corresponding to s Ž 1 G 2 G ⭈⭈⭈ G m G 0.. Then we have q i ¨ s q i ¨ for i s 1, . . . , m. Since Uq Ž ᒄ . s Uq Ž gl m 1 . m ⭈⭈⭈ m Uq Ž gl m r ., irreducible representations of Uq Ž ᒄ . are parametrized by r-tuples s Ž Ž1., . . . , Ž r . . of Young diagrams Ž i. with l Ž Ž i. . F m i . The corresponding irreducible Uq Ž ᒄ .-module V is given as V s VŽ1. m ⭈⭈⭈ m VŽ r . . For an r-tuple , we denote by < < s Ý ris1 < Ž i. < the size of . Let ⌳ m 1 , . . . , m r be the set of r-tuples of Young diagrams of size n such that l Ž Ž i. . F m i . Then it is known that irreducible representations of Uq Ž ᒄ . occurring in V mn are parametrized by ⌳ m 1 , . . . , m r. We define a function b: E ª ⺞ by bŽ ¨ jŽ i. . s i. Note that our definition of the total order on E implies that if bŽ x 1 . - bŽ x 2 . for x 1 , x 2 g E , then we have x 1 - x 2 . 2.3. We recall here the definition and fundamental properties of the Ariki᎐Koike algebra Hn, r . Let Wn, r be the complex reflection group ᑭ n h Ž⺪rr⺪. n. Ariki and Koike introduced in wAKx a Hecke algebra Hn, r associated to Wn, r , which is a q-analogue of the group algebra of Wn, r and is defined by generators and relations as follows: Hn, r is the K-algebra with generators a1 , . . . , a n and relations
Ž a1 y u1 . Ž a1 y u 2 . ⭈⭈⭈ Ž a1 y u r . s 0, Ž ai y q . Ž ai q qy1 . s 0 Ž 2 F i F n . , a1 a2 a1 a2 s a2 a1 a2 a1 , a i a iq1 a i s a iq1 a i a iq1 ai a j s a j ai
Ž 2.3.1.
Ž 2 F i - n. ,
Ž < i y j < G 2. .
It is known by wAKx that Hn, r is a semisimple algebra with dim K Hn, r s < Wn, r < s n!r n. We define a subalgebra Tn, r of Hn, r as follows: Let us define t 1 , t 2 , . . . , t n recursively by t 1 s a1 , t j s a j t jy1 a j for j G 2, and let Tn, r be the subalgebra of Hn, r generated by t 1 , . . . , t n . It is shown in wAKx that Tn, r is a commutative semisimple algebra. Irreducible representations of Hn, r and Tn, r are determined in wAKx. In order to describe their results, we prepare some notations. Let s Ž Ž1., . . . , Ž r . . be an r-tuple of Young diagrams with size n. An r-tuple ⺣ s Ž S Ž1., . . . , S Ž r . . of tableaus is called a standard tableau of shape if each jŽ1 F j F n. occurs exactly once as an entry in one tableau of ⺣, and each S Ž i. is a standard tableau of shape Ž i.. If the
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number i appears in the intersection of the jth row and the kth column of S Ž p., we write Ž⺣; i . s p and cŽ⺣; i . s k y j. For each standard tableau ⺣ of size n, one can construct a linear character ⺣ of Tn, r by
⺣ Ž t i . s u Ž⺣ ; i. q 2 cŽ⺣ ; i. PROPOSITION 2.4 ŽwAKx..
Ž 1 F i F n. .
Ž 2.3.2.
Let Hn, r and Tn, r be as abo¨ e. Then
Ži. The complete set of irreducible representations of Tn, r is gi¨ en by ⺣ 4 , where ⺣ runs o¨ er all the r-tuple standard tableaus of size n. Žii. Irreducible representations of Hn, r are parametrized by r-tuple Young diagrams of size n. We denote by Z the irreducible Hn, r-module corresponding to . The decomposition of Z into Tn, r-modules is described as follows. PROPOSITION 2.5 ŽwAK, Proposition 3.16x.. Let Z be an irreducible Hn, r-module. Then Z s [⺣ ⺣ , where ⺣ runs o¨ er all the standard tableaus of shape . As a corollary to the above propositions, we have COROLLARY 2.6. Ži. Let Z be an irreducible Hn, r-module. Then for each standard tableau ⺣ of shape , there exists a common eigen¨ ector ¨ ⺣ in Z for t 1 , . . . , t n with eigen¨ alues gi¨ en in the right hand side of Ž2.3.2.. Žii. Con¨ ersely let Z be an Hn, r-module. Assume that Z contains a common eigen¨ ector ¨ ⺣ for t 1 , . . . , t n with eigen¨ alues gi¨ en as in Ž2.3.2. for some ⺣ of shape . Then the Hn, r-submodule of Z generated by ¨ ⺣ is a sum of copies of Z . 2.7. Here we recall the classical Schur᎐Weyl reciprocity for Wn, r and UŽ ᒄ .. So, let UŽ ᒄ . be the universal enveloping algebra of ᒄ. As in the case of Uq Ž ᒄ ., we have the natural representation of UŽ ᒄ . on an m-dimensional vector space V over ⺓, with basis E s ¨ 1 , . . . , ¨ m 4 . Hence UŽ ᒄ . acts naturally on V mn. We denote this action by . On the other hand, Wn, r acts on V mn as follows: Wn, r is generated by s1 , s2 , . . . , sn , where s2 , . . . , sn are generators of ᑭ n corresponding to transpositions, Ž1, 2., . . . , Ž n y 1, n., and s1 satisfies the relation s1r s 1, s1 s2 s1 s2 s s2 s1 s2 s1 , s1 si s si s1 for i G 3. The group ᑭ n acts on V mn by permuting the components of the tensor product, while s1 acts on V mn by s1 Ž x 1 m ⭈⭈⭈ m x n . s bŽ x 1 . x 1 m ⭈⭈⭈ m x n , for x i g E , where is a fixed primitive r th root of unity. We denote this action by . The action of Wn, r on V mn commutes with that of UŽ ᒄ .. Irreducible representations of UŽ ᒄ . and Wn, r are parametrized by r-tuple
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Young diagrams as before. We denote by V Žresp. Z . the irreducible UŽ ᒄ .-module Žresp. the irreducible Wn, r-module. corresponding to . Let ⺓Wn, r by the group algebra of Wn, r over ⺓. It is well known that the following Schur᎐Weyl reciprocity holds for the ⺓Wn, r m UŽ ᒄ .-module V mn. PROPOSITION 2.8. ŽUŽ ᒄ .. and Ž⺓Wn, r . are mutually the full centralizer algebras of each other; i.e., we ha¨ e
Ž U Ž ᒄ . . s End W n , r V mn ,
Ž ⺓Wn , r . s End UŽ ᒄ . V mn .
More precisely, we ha¨ e the decomposition of the ⺓Wn, r m UŽ ᒄ .-module V mn as V mn s
[
g⌳ m1 , . . . , m r
Z m V .
Since we could not find a reference for this result, and since this proposition plays an essential role in our later discussion, we give here a brief outline of the proof. For the first statement, we may consider the action of GLm 1Ž⺓. = ⭈⭈⭈ = GLm rŽ⺓. on V mn instead of UŽ ᒄ .. Then Weyl’s original proof wWx can be applied in our case with minor modifications, which gives the first assertion. For the second assertion, we note the following. First consider the ⺓ ᑭ n m UŽ gl m . module V mn. Let s Ž 1 G ⭈⭈⭈ G k . be a partition of n, and T the standard tableau of shape obtained by inserting the numbers 1, 2, . . . , n in order along consecutive rows. Let wX s ¨ i1 m ⭈⭈⭈ m ¨ i n be a vector of V mn such that the first 1 components are equal to ¨ 1 , and the next 2 components are equal to ¨ 2 , and so on. Let c g ⺓ ᑭ n be the Young symmetrizer with respect to T Žfor the definition, see, e.g., wF, Section 7x., and set w s cwX . Then w is a highest weight vector with highest weight in V mn, and w generates the Specht module S of ᑭ n . Now we consider the general case. For a given s Ž Ž1., . . . , Ž r . . of size n, we choose wi g Vimn i with n i s < Ž i. <, a highest weight vector for UŽ gl m i . with highest weight Ž i. as above. Set w s wr m ⭈⭈⭈ m w 1 g V mn. Then w is a highest weight vector with highest weight in V mn, and the Wn, r-module generated by w coincides with Z . The second assertion follows from this. 3. AN ACTION OF Hn, r ON V mn 3.1. In this section we shall construct a representation of Hn, r on V mn commuting with the action of Uq Ž ᒄ .. First we recall a result of Jimbo, who constructed a representation of the Hecke algebra of type A ny 1 on V mn. Let Hn be the subalgebra of Hn, r generated by a2 , . . . , a n . Then by wAKx, Hn is isomorphic to the Hecke algebra associated to the symmetric group ᑭ n .
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299
It is known by Jimbo wJx that Hn acts on V mn by nyk . ky2. Ž a k . s idmŽ m Rˇ m idmŽ , V V
where Rˇ g End K Ž V m V . is defined by
¡q¨ m ¨ j
~
if i s j,
i
RˇŽ ¨ i m ¨ j . s ¨ j m ¨ i
¢¨ m ¨ q Ž q y q j
i
y1
if i ) j,
. ¨i m ¨ j
Ž 3.1.1.
if i - j.
This action of Hn commutes with the action of Uq Ž gl m . on V mn. We set Ti s Ž a i . for i s 2, . . . , n. Let i be the action of ᑭ n on V mn corresponding to the transposition Ž i y 1, i . as in 2.4. We define, for i s 2, . . . , n, an endomorphism Si g End K Ž V mn . as follows; Let x s x 1 m x 2 m ⭈⭈⭈ m x n be a basis element in V mn with x i g E . Then we define Si Ž x . s
½
Ti Ž x .
if b Ž x iy1 . s b Ž x i . ,
i Ž x .
if b Ž x iy1 . / b Ž x i . .
Ž 3.1.2.
Note that the two conditions on x in Ž3.1.2. remain unchanged by the Ž x . is given by Tiy1 Ž x . Žresp i Ž x .. action of Si . Hence Si is invertible; Sy1 i if x satisfies the former Žresp. the latter . condition in Ž3.1.2.. We define an endomorphism g End K Ž V mn . by Ž x . s u bŽ x 1 . x for a basis element x s x 1 m ⭈⭈⭈ m x n . We have the following theorem. THEOREM 3.2.
Let Ž a i . g End K Ž V mn . be the elements gi¨ en by
Ž a1 . s Ty1 ⭈⭈⭈ Tny1 Sn ⭈⭈⭈ S2 , 2 Ž a i . s Ti
for i s 2, . . . , n.
Then the action of a i gi¨ es rise to a representation of Hn, r on V mn commuting with the action of Uq Ž ᒄ .. Remark 3.3. In wATYx, Ariki, Terasoma, and Yamada constructed a representation of Hn, r on V mn satisfying a similar property, in the special case where m1 s ⭈⭈⭈ s m r s 1, i.e., in the case where ᒄ is a Cartan subalgebra ᒅ of gl m . In this case our definition of Ž a1 . coincides with theirs. In fact, in this case, the first case in Ž3.1.2. occurs only when x iy1 s x i , and so S i Ž x . s qx by Ž3.1.1.. Hence Sn ⭈⭈⭈ S2 Ž x . s u bŽ x 1 . q ␣ x 2 m ⭈⭈⭈ m x n m x 1 ,
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where ␣ is the number of x i such that x i s x 1 for 2 F i F n. This agrees with Ž x . in wATY, Prop. 2.1x. The remainder of this section is devoted to the proof of the theorem. Before proving the existence of the representation, we shall show that the thus obtained representation of Hn, r actually commutes with . LEMMA 3.4. The actions of Hn, r and Uq Ž ᒄ . on V mn commute with each other. Proof. Since Uq Ž ᒄ . commutes with Ti for i s 2, . . . , n Žcf. 3.1., it is enough to show that Uq Ž ᒄ . commutes with Ž a1 .. Clearly commutes with Uq Ž ᒄ .. So the lemma is reduced to showing the commutativity of Si with e j , f j , q " j g ᒄ. Since S i preserves the weights of the vectors in V mn, Si commutes with q " j. We show that S i commutes with e j . Let x s x 1 m ⭈⭈⭈ m x n be a basis element. First we note that Ž e j . preserves the condition in Ž3.1.2.. If bŽ x iy1 . s bŽ x i . we have S i Ž e j .Ž x . s Ž e j . S i Ž x . since S i s Ti on x and on Ž e j .Ž x .. Now assume that bŽ x iy1 . / bŽ x i .. We p show that i Ž e j .Ž x . s Ž e j . i Ž x .. Set X p s K m m e j m 1mŽ ny1yp. for j p s 0, . . . , n y 1. Then i commutes with X p unless p s i y 2, i y 1. Hence in view of Ž2.1.2., we have only to verify the following equation:
i Ž X iy2 q X iy1 . Ž x . s Ž X iy2 q X iy1 . Ž i Ž x . . .
Ž 3.4.1.
Since i affects only the i y 1 and ith components of the tensor product, and since the remaining parts are the same for X iy2 Ž x . and X iy1Ž x ., we may only consider the parts corresponding to i y 1 and ith components. The corresponding part of the left hand side is equal to e j x i m K j x iy1 q x i m e j x iy1 . The corresponding part of the right hand side is equal to K j x i m e j x iy1 q e j x i m x iy1 . We must show that these two expressions are equal. Now assume that e j g gl m k for some k. If bŽ x i . / k, bŽ x iy1 . / k, we have e j x iy1 s e j x i s 0 and so both expressions are equal to 0. If bŽ x i . s k, then bŽ x iy1 . / k, and so e j x iy1 s 0, K j x iy1 s x iy1. Again we can check that expressions are equal. The case where bŽ x iy1 . s k is done similarly. Hence we have verified the formula Ž3.4.1. and the commutativity for e j follows. The case for f j is dealt with similarly. 3.5. We put T1 s Ž a1 .. In order to prove the theorem, we have to show that T1 , . . . , Tn satisfy the same relations as in Ž2.3.1.. By a result of Jimbo wJx Žsee 3.1., we know already that T2 , . . . , Tn satisfy the relations in
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Ž2.3.1.. So, we have only to show the following three relations:
Ž T1 y u1 . Ž T1 y u 2 . ⭈⭈⭈ Ž T1 y u r . s 0, T1T2 T1T2 s T2 T1T2 T1 , T1Tj s Tj T1
Ž 3.5.1.
for j s 3, . . . , n.
We shall verify these relations by subsequent lemmas. Note that the outline of the proof is quite similar to that of wATYx, although our situation is more complicated than theirs. First we show the following lemma. LEMMA 3.6. Let Yj, p be the subspace of V mn generated by basis elements x s x 1 m ⭈⭈⭈ m x n such that bŽ x p . G j for j G 1. Then y1 Ži. If x g Yj, p , then Tpq1 ⭈⭈⭈ Tny1 Sn ⭈⭈⭈ S pq1Ž x . g x q Yjq1, p . Žii. ŽT1 y u j .Yj, 1 ; Yjq1, 1. Žiii. ŽT1 y u1 .ŽT1 y u 2 . ⭈⭈⭈ ŽT1 y u r . s 0.
In particular, the first relation in Ž3.5.1. holds. Proof. We shall prove the first assertion by backward induction on p. The assertion is clear for p s n. Assume that it holds for p, and take x g Yj, py1. It is easily checked that Tp Ž Yj, py1 . s Yj, p for any j G 1. Then we have S p Ž x . g Yj, p . In fact, if S p Ž x . s Tp Ž x ., the above relation is applied. If S p Ž x . s p Ž x ., this is also clear. Hence by applying the induction hypothesis to S p Ž x ., we have y1 Tpy1 Tpq1 ⭈⭈⭈ Tny1 Sn ⭈⭈⭈ S pq1 S p Ž x . g Tpy1 Ž S p Ž x . q Yjq1, p . ,
s Tpy1 Ž S p Ž x . . q Yjq1, py1 . Ž 3.6.1. Assume now that bŽ x p . / bŽ x py1 .. Then S p Ž x . s p Ž x . and Tpy1 Ž p Ž x . . s
½
x y Ž q y qy1 . p Ž x .
if b Ž x p . ) b Ž x py1 . ,
x
if b Ž x p . - b Ž x py1 . .
Note in the first case of the above formula, we have p Ž x . g Yjq1, py1 and so Tpy1 Ž p Ž x .. is contained in x q Yjq1, py1. It follows that the right hand side of Ž3.6.1. is contained in x q Yjq1, py1. Next assume that bŽ x p . s bŽ x py1 .. Then we have Tpy1 S p Ž x . s x, and again the right hand side of Ž3.6.1. is contained in x q Yjq1, py1. Hence Ži. follows. We show the second assertion of the lemma. By applying Ži. to the case p s 1, we have for any x g Yj, 1 , T1 Ž x . s Ty1 ⭈⭈⭈ Tny1 Sn ⭈⭈⭈ S2 Ž x . g u bŽ x 1 . x q Yjq1, 1 . 2
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It follows that
Ž T1 y u j . Ž x . g Ž u bŽ x . y u j . x q Yjq1, 1 . 1
Now since x g Yj, 1 , we have bŽ x 1 . G j. Then Ž u bŽ x 1 . y u j . x s 0 if bŽ x 1 . s j, and x g Yjq1, 1 if bŽ x 1 . ) j. Hence the right hand side of the above formula is contained in Yjq1, 1. This proves the second assertion. The third assertion is immediate from Žii. if we notice that Y1, 1 s V mn and Yrq1, 1 s 04 . Before verifying the second relation in Ž3.5.1., we prove some properties of Si . LEMMA 3.7. The following formulae hold for j G 3. Ži. S j S jy1Tj s Tjy1 S j S jy1. y1 Žii. S j S jy1 S j Sy1 jy1Tjy1 s Tj S j S jy1 S j S jy1 . Žiii. S j S jy1 S j S jy1Tjy1 s Tj S j S jy1 S j S jy1. Proof. It is enough to show the lemma in the case where n s 3 and j s 3. The formula is verified by evaluating both sides at basis elements x s x 1 m x 2 m x 3 according to the patterns of bŽ x i .. We define Rˇi j g End K Ž V mn . as an endomorphism which acts as Rˇ in Ž3.1.1. on the i and jth component of x and leaves other factors invariant. Then for g ᑭ n , we have Rˇi jy1 s Rˇ Ž i. Ž j. , where the left hand side stands for the action of on V mn as in 2.7. First we consider the formula Ži.. In the case where bŽ x i . are all the same, the formula follows from the relation T3 T2 T3 s T2 T3 T2 , while in the case where all the bŽ x i . are distinct, we have S3 S2 T3 Ž x . s 3 2 Rˇ23 Ž x . s Rˇ12 3 2 Ž x . s T2 S 3 S 2 Ž x . . A similar computation works also for the case where bŽ x 2 . s bŽ x 3 ., and bŽ x 1 . / bŽ x 2 .. Next consider the case where bŽ x 1 . s bŽ x 2 ., and bŽ x 3 . / bŽ x 1 .. We use the following notation. Set w q x s q y qy1 , and define
w qxij s qi j s
½ ½
w qx
if x i - x j ,
0
otherwise,
q
if x i s x j ,
1
otherwise,
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where x i - x j denotes the total order on E . Then we have S 3 S 2 T3 Ž x . s
½
Rˇ23 Ž x 3 m x 1 m x 2 .
if x 2 ) x 3 ,
Rˇ23 Ž x 3 m x 1 m x 2 . q w q x Rˇ13 Ž x 1 m x 3 m x 2 .
if x 2 - x 3 .
It follows that S3 S2 T3 Ž x . s q12 x 3 m x 2 m x 1 q w q x 12 x 3 m x 1 m x 2 q w q x 23 Ž q12 x 2 m x 3 m x 1 q w q x 12 x 1 m x 3 m x 2 . . On the other hand, since T2 S3 S2 Ž x . s Rˇ12 3 Rˇ12 Ž x . s Rˇ12 Rˇ13 Ž x 1 m x 3 m x 2 . , we have T2 S3 S2 Ž x . s q12 Ž x 3 m x 2 m x 1 q w q x 23 x 2 m x 3 m x 1 . q w q x 12 Ž x 3 m x 1 m x 2 q w q x 23 x 1 m x 3 m x 2 . . Hence we get the desired equality. Finally the case where bŽ x 1 . s bŽ x 3 . and bŽ x 2 . / Ž x 1 . is verified in a similar way. Thus the formula Ži. is proved. Next we consider the formula Žii.. In the case where all the bŽ x i . are the y1 same, this follows from the relation T3T2 T3Ty1 2 T2 s T3 T3 T2 T3 T2 . In the case where all the bŽ x i . are distinct, this follows from the relation 3 2 3 2 Rˇ12 s Rˇ23 3 2 3 2 . We assume that bŽ x 1 . s bŽ x 2 . and bŽ x 3 . / bŽ x 1 .. Then we have S3 S2 S3 Sy1 2 T2 Ž x . s S 3 S 2 S 3 Ž x . s T3 2 3 Ž x . . On the other hand, we have y1 T3 S3 S2 S3 Sy1 2 Ž x . s T3 T3 2 3 T2 Ž x . s T3 2 3 Ž x . .
Hence we get the desired equality. Next assume that bŽ x 1 . s bŽ x 3 . and bŽ x 2 . / bŽ x 1 .. Then we have T3 S3 S2 S3 Sy1 2 Ž x . s T3 3 2 T3 Ž x 2 m x 1 m x 3 . s T3 T2 Ž x 1 m x 3 m x 2 . . On the other hand, we have S3 S2 S3 Sy1 2 T2 Ž x . s S3 S2 S3 Ž x 1 m x 2 m x 3 q w q x 12 x 2 m x 1 m x 3 . s S3 S2 Ž x 1 m x 3 m x 2 q w q x 12 T3 Ž x 2 m x 1 m x 3 . . s 3T2 Ž x 1 m x 3 m x 2 . q w q x 12 3 2 T3 Ž x 2 m x 1 m x 3 . s 3T2 Ž x 1 m x 3 m x 2 . q w q x 12 T2 Ž x 1 m x 3 m x 2 . s T3 T2 Ž x 1 m x 3 m x 2 . .
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Hence we get the desired equality. Finally we assume that bŽ x 2 . s bŽ x 3 . and bŽ x 1 . / bŽ x 2 ..l Then in a similar way as above, we can show that both Ž x . and S3 S2 S3 Sy1 Ž . Ž . T3 S3 S2 S3 Sy1 2 2 T2 x coincides with T3 3 T2 x 2 m x 3 m x 1 . Hence the formula Žii. holds. Finally we consider the formula Žiii.. This case is dealt with almost similarly as in the case Žii.. In fact, the same argument is applied to the case where all the bŽ x i . are the same, or are distinct. Moreover, it is the same for the last two cases of the above computation. In the case where bŽ x 1 . s bŽ x 2 . and bŽ x 3 . / bŽ x 1 ., we see that both of S3 S2 S3 S2 T2 Ž x . and T3 S3 S2 S3 S2 Ž x . coincide with T33 2 3 Ž x .. Thus the formula Žiii. holds, and the lemma is proved. We can now prove the second relation of Ž3.5.1.. LEMMA 3.8. Ži. Žii. Žiii. Živ. Žv.
Set s Sn ⭈⭈⭈ S2 . Then the following formulae hold.
Tj s Tjy1 for j s 3, . . . , n. 2 T2 s Tn 2 . y1 T2 s T2 y1 . Ž . 2 T2 s TnŽ . 2 . T1T2 T1T2 s T2 T1T2 T1.
In particular the second relation of Ž3.5.1. holds. Proof. First we show Ži.. Note that Si Tj s Tj Si for < i y j < G 2. Hence by using the relation Ži. in Lemma 3.7, we have
Tj s Sn ⭈⭈⭈ S jq1 Ž S j S jy1Tj . S jy2 ⭈⭈⭈ S2 s S n ⭈⭈⭈ S jq1 Ž Tjy1 S j S jy1 . S jy2 ⭈⭈⭈ S2 s Tjy1 . This proves Ži.. We show Žii.. Again we note that Si S j s S j Si for < i y j < G 2. Then we have
2 T2 s Ž Sn ⭈⭈⭈ S2 . Ž Sn ⭈⭈⭈ S2 . T2 s Sn Ž Sny1 Sn . ⭈⭈⭈ Ž S3 S4 . Ž S2 S3 . S2 T2 y1 s Ž Sn Sny1 Sn Sy1 ny1 .Ž S ny1 S ny2 S ny1 S ny2 . ⭈⭈⭈
⭈⭈⭈ Ž S4 S3 S4 Sy1 3 . Ž S 3 S 2 S 3 S 2 . T2 . Now we have Ž S3 S2 S3 S2 .T2 s T3 Ž S3 S2 S3 S2 . by applying Lemma 3.7 Žiii. with j s 3. Then by applying Lemma 3.7 Žii., for j s 4, . . . , n successively,
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305
we see that the last formula is equal to y1 2 Tn Ž Sn Sny1 Sn Sy1 ny1 . ⭈⭈⭈ Ž S4 S3 S4 S3 . Ž S3 S2 S3 S2 . s Tn .
This proves Žii.. We show Žiii.. First we note that commutes with S3 , . . . , Sn by definition. Then we see easily that y1 T2 s Sy1 2 S2 T2 . Similarly we have T2 y1 s T2 Sy1 S . Hence in order to prove the state2 2 ment, it is enough to show that y1 Sy1 2 S 2 T2 s T2 S 2 S 2 .
Ž 3.8.1.
We show Ž3.8.1.. We may consider the case n s 2. Let x s x 1 m x 2 be a basis element in V m V. First assume that bŽ x 1 . s bŽ x 2 .. Then clearly we have 2 y1 Sy1 2 S 2 T2 Ž x . s T2 Ž x . s T2 S 2 S 2 Ž x . .
Next assume that bŽ x 1 . / bŽ x 2 .. We write bŽ x 1 . s ␣ and bŽ x 2 . s  . Then we have Sy1 2 S2 T2 Ž x . s 2 2 T2 Ž x . s u␣ u T2 Ž x . , T2 Sy1 2 S2 Ž x . s T2 2 2 Ž x . s T2 Ž u␣ u x . . Thus Ž3.8.1. holds and the assertion Žiii. is proved. We show Živ.. By Žiii. and Žii., we have 2 2 Ž . T2 s 2 Ž y1 . T2 s 2 T2 Ž y1 . s Tn Ž . .
We show Žv.. We note that commutes with T3 , . . . , Tn . By using Ži., we have T1T2 T1 s Ž Ty1 ⭈⭈⭈ Tny1 . Ž Ty1 ⭈⭈⭈ Tny1 . 2 3 y1 s Ž Ty1 ⭈⭈⭈ Tny1 .Ž Ty1 ⭈⭈⭈ Tny1 . Ž . . 2 2 2
Hene by Živ., we have y1 T1T2 T1T2 s Ž Ty1 ⭈⭈⭈ Tny1 .Ž Ty1 ⭈⭈⭈ Tny1 . Tn Ž . . 2 2 2
Ž 3.8.2.
y1 s Ž Ty1 ⭈⭈⭈ Tny1 .Ž Ty1 ⭈⭈⭈ Tny1 . Ž . . 3 2
Ž 3.8.3.
On the other hand, again by using Ži., we have T2 T1T2 T1 s Ž Ty1 ⭈⭈⭈ Tny1 . Ž Ty1 ⭈⭈⭈ Tny1 . 3 3 2
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In view of Ž3.8.2. and Ž3.8.3., the proof of Žv. is reduced to showing the following formula, which is easily obtained by using the standard argument on the generators of Hn Žsee wATY, Lemma 2.3x.:
Ž Ty1 2
y1 y1 ⭈⭈⭈ Tny1 .Ž Ty1 ⭈⭈⭈ Tny1 ⭈⭈⭈ Tny1 .Ž Ty1 ⭈⭈⭈ Tny1 . Tn s Ž Ty1 .. 2 3 2
This shows Žv., and the lemma is now proved. Finally we show LEMMA 3.9. For each j G 3, we ha¨ e T1Tj s Tj T1. Hence the third relation of Ž3.5.1. holds. Proof. By using Ži., we have T1Tj s Ty1 ⭈⭈⭈ Tny1 Tj 2 s Ty1 ⭈⭈⭈ Tny1 Tjy1 2 y1 y1 y1 s Ty1 ⭈⭈⭈ Ž Tjy1 Tj Tjy1 . Tjq1 ⭈⭈⭈ Tny1 2 y1 y1 y1 s Ty1 ⭈⭈⭈ Tjy2 Tj . ⭈⭈⭈ Tny1 Ž TjTjy1 2
s Tj T1 . The lemma is proved. Theorem 3.2 is now proved by Lemma 3.6, Lemma 3.8, and Lemma 3.9. Remark 3.10. Ži. For the construction of the representation of Hn, r on V mn, no properties established in wAKx are required. We do not need to assume that Hn, r is semisimple nor dim Hn, r is finite. In fact, the construction proceeds by just using the description of Hn, r in terms of generators and relations. In the beginning of 3.1, it was used that Hn is isomorphic to the Hecke algebra of type A ny 1. This was proved in wAKx. But for the construction of our representation, this is not essential. One can define Ti s Ž a i . for i G 2 as in 3.1. Then it satisfies the last two conditions in Ž2.3.1. by Jimbo wJx, which is enough for our later discussion. Žii. The construction of the representation of Hn, r works in a more general setting. Let R s ⺪w q, qy1 , u1 , . . . , u r x. We denote by VR the Rlattice of V spanned by the basis E s ¨ 1 , . . . , ¨ m 4 . The algebra Hn, r has an R-structure, Hn, r s K mR HR , where HR is the R-algebra generated by a1 , . . . , a n subject to the conditions in Ž2.3.1.. The quantum group Uq Ž ᒄ . also has the Kostant form Uq, R which acts on VRmn. Then the previous arguments can be applied to this setting, and one gets a representation of HR on VRmn commuting with the actions of Uq, R . We now consider an integral domain RX , with elements uX1 , . . . , uXr g RX and an invertible ele-
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307
ment qX g RX . By the specialization, u i ¬ uXi , q ¬ qX , we can consider the algebras HRX s RX mR HR , Uq, RX s RX mR Uq, R , and the representation of X Uq, RX on VRX s RX mR VR . Then clearly we have an action of HRX on VRmn commuting with the action of Uq, RX . This is the situation considered in wAx. 4. SCHUR᎐WEYL RECIPROCITY FOR Hn, r AND Uq Ž ᒄ . In this section, we prove the Schur᎐Weyl reciprocity for the Hn, r m Uq Ž ᒄ .-module V mn. THEOREM 4.1. Ž Hn, r . and ŽUq Ž ᒄ .. are mutually the full centralizer algebras of each other. Moreo¨ er, the Hn, r m Uq Ž ᒄ .-module V mn is decomposed as V mn s
[
g⌳ m1 , . . . , m r
Z m V .
Ž 4.1.1.
The theorem will be proved in 4.8 after some preliminaries. First we prepare some lemmas. LEMMA 4.2. Let g ⌳ m 1 , . . . , m r. Then V mn contains an irreducible Hn, rmodule isomorphic to Z , consisting of highest weight ¨ ectors of Uq Ž ᒄ . with highest weight . Proof. First we consider the case where r s 1. Note that in this case Hn,1 is identified with Hn since a1 s u1 ⭈ 1. Then the subalgebra Tn, 1 is nothing but the subalgebra of Hn generated by tX2 s a22 , tX3 s a3 a22 a3 , . . . , tXn s a n tXny1 a n . By Jimbo ŽwJx., it is known that the Hn m Uq Ž gl m .-module V mn is decomposed as V mn s
[Z
g⌳ m
m V ,
Ž 4.2.1.
where ⌳ m is the set of Young diagrams of size n with l Ž . F m, and V Žresp. Z . is the irreducible Uq Ž gl m .-module with highest weight Žresp. the irreducible Hn-module corresponding to ., respectively. Now by the decomposition Ž4.2.1., for each g ⌳ m one can find an irreducible Hn, 1-module Z consisting of highest weight vectors for V . Let S be a standard tableau of type . Then by applying Corollary 2.6 Ži., we can find a common eigenvector ¨ S for tX2 , . . . , tXn satisfying Ž2.3.2. which is a highest weight vector for V. We now return to the general setting. In view of Corollary 2.6 Žii., to prove the lemma it is enough to show the existence of a common
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eigenvector ¨ ⺣ for some ⺣ of shape , which is also a highest weight vector for V . We choose ⺣ s Ž S Ž1., . . . , S Ž r . . as follows: set n i s < Ž i. < and pi s n i q ⭈⭈⭈ n r for i s 1, . . . , r. Also we set prq1 s 0. We define S Ž i. by filling n i numbers piq1 q 1 - piq1 q 2 - ⭈⭈⭈ - pi in the boxes in Ž i., first to all the entries of the first row, and then to all the entries of the second row, and so on, in increasing order. We consider the action of Hn i m Uq Ž gl m i . on Vimn i . The standard tableau S Ž i. obviously determines a standard tableau S of type Ž i. with numbers 1, . . . , n i 4 . Hence by the previous argument, one can find a common eigenvector wi g Vimn i for tX2 , . . . , tXn i g Tn i , 1 satisfying Ž2.3.2. with respect to S , which is also a highest weight vector for VŽ i. . We now set mn ry 1 ¨ ⺣ s wr m wry1 m ⭈⭈⭈ m w 1 g Vrmn r m Vry1 m ⭈⭈⭈ m V1mn1 .
Since wi is a highest weight vector of VŽ i. , ¨ ⺣ is a highest weight vector of V s VŽ r . m ⭈⭈⭈ m VŽ1. . We shall show that ¨ ⺣ is a common eigenvector for t 1 , . . . , t n with eigenvalues as in the right hand side of Ž2.3.2., which is equivalent to the following statement. Ž4.2.2.. Assume that pkq 1 - j F pk . Then we have t j¨ ⺣ s u k q 2 c ¨ ⺣
with c s c Ž S Ž k . ; j y pkq1 . .
By Corollary 2.6 Žii. and by the commutativity of the actions of Hn, r and Uq Ž ᒄ ., the proof of the lemma is reduced to showing Ž4.2.2.. We show Ž4.2.2.. Assume that pkq 1 - j F pk , and set pkq1 s p, pk s q. Now t j ¨ ⺣ can be written as y1 ⭈⭈⭈ Tny1 S n ⭈⭈⭈ S2 T2 ⭈⭈⭈ Tj ¨ ⺣ . t j ¨ ⺣ s Tjq1
First we note that Tpq 1 ⭈⭈⭈ Tj ¨ ⺣ g Vrmn r m ⭈⭈⭈ m V1mn1 .
Ž 4.2.3.
Let x s x 1 m ⭈⭈⭈ m x n be a basis element of V mn occurring in the expression of Tpq 1 ⭈⭈⭈ Tj ¨ ⺣ . Then x p g Vk , and x i ) x p for all i - p. Moreover we have x i f Vk for i - p. This implies that T2 ⭈⭈⭈ Tp Ž x . s x p m x 1 m ⭈⭈⭈ m x py1 m x pq1 m ⭈⭈⭈ m x n . and we see that S p ⭈⭈⭈ S2 T2 ⭈⭈⭈ Tp Ž x . s u k x. It follows that y1 t j ¨ ⺣ s u k Tjq1 ⭈⭈⭈ Tny1 Sn ⭈⭈⭈ S pq1Tpq1 ⭈⭈⭈ Tj ¨ ⺣ .
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On the other hand we have S q ⭈⭈⭈ S pq1Tpq1 ⭈⭈⭈ Tj ¨ ⺣ g Vrmn r m ⭈⭈⭈ m V1mn1 . Let y s y 1 m ⭈⭈⭈ m yn be a basis element of V mn occurring in the expression of S q ⭈⭈⭈ S pq1Tpq1 ⭈⭈⭈ Tj ¨ ⺣ . Then we see that yq g Vk , yi f Vk for all i ) q. Moreover, yi - yq for all i ) q. This implies, by a similar argument y1 y1 Ž . as above, that we have Tqq 1 ⭈⭈⭈ Tn S n ⭈⭈⭈ S qq1 y s y. Hence y1 t j ¨ ⺣ s u k Tjq1 ⭈⭈⭈ Tqy1 S q ⭈⭈⭈ S pq1Tpq1 ⭈⭈⭈ Tj ¨ ⺣ .
Now by Ž4.2.3. we see that S pq 1 , . . . , S q acts successively on Tpq1 ⭈⭈⭈ Tj ¨ ⺣ by Tpq 1 ⭈⭈⭈ Tq . Hence we have t j ¨ ⺣ s u k Tj ⭈⭈⭈ Tpq1Tpq1 ⭈⭈⭈ Tj ¨ ⺣ . But Tj ⭈⭈⭈ Tpq1Tpq1 ⭈⭈⭈ Tj ¨ ⺣ coincides with wr m ⭈⭈⭈ m wXk m ⭈⭈⭈ m w 1 , where wXk s tXjyp w k . Now by the definition of w k , we have wXk s q 2 c w k with c s cŽ S Ž k . ; j y p .. It follows that t j ¨ ⺣ s u k q 2 c ¨ ⺣ . This implies Ž4.2.2., and so the lemma is proved. 4.3. Next we shall compute the multiplicity of V in V mn. We perform this by considering the restriction of Uq Ž ᒄ . to a smaller subalgebra. So, assume that m r ) 1, and let ᒄX s gl m 1 [ ⭈⭈⭈ [ Ž gl m ry1 [ gl 1 .. We regard ᒄX as a subalgebra of ᒄ by letting gl m ry1 be a subalgebra of gl m r corresponding to the basis ¨ 1Ž r ., . . . , ¨ mŽ r r.y1 , and gl 1 to be that corresponding to ¨ mŽ r r.. For g ⌳ m 1 , . . . , m r and g ⌳ m 1 , . . . , m ry1, 1 , we define a relation $ by the following condition: let s Ž Ž1., . . . , Ž r . . and s Ž Ž1., . . . , Ž r ., Ž rq1. .. Then Ž i. s Ž i. for i s 1, . . . , r y 1 and
Ž1r . G Ž1r . G ⭈⭈⭈ G Žmr .ry1 G Žmr .ry1 G Žmr .r ,
Ž 4.3.1.
where Ž r . s Ž Ž1r . G Ž2r . G ⭈⭈⭈ G Žmr .r G 0. Žresp. Ž r . s Ž Ž1r . G Ž2r . G ⭈⭈⭈ G Žmr .ry1 G 0.. is a partition with l Ž Ž r . . F m r Žresp. a partition with l Ž Ž r . . F m r y 1.. Note that we have < < s < < s n, and l Ž Ž rq1. . F 1. Moreover the condition Ž4.3.1. implies that < Ž r . < F < Ž r . <. Hence Ž rq1. is determined uniquely once Ž1., . . . , Ž r . is determined as above. Now Uq Ž ᒄX . is regarded as a subalgebra of Uq Ž ᒄ .. The following lemma describes the restriction of V to Uq Ž ᒄX .-submodules. LEMMA 4.4.
Let g ⌳ m 1 , . . . , m r. Then we ha¨ e V < Uq Ž ᒄ X . s
[V . $
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Proof. This lemma seems to be well known. But since we could not find an appropriate reference for it, we give a proof below. The lemma is easily reduced to the case where r s 1. So we consider the subalgebra gl my 1 [ gl 1 of gl m . Then Uq Ž gl my1 . is a subalgebra of Uq Ž gl m .. For a partition s Ž 1 G ⭈⭈⭈ G m G 0. with l Ž . F m and for X s Ž X1 G ⭈⭈⭈ G Xmy1 G 0. with l Ž X . F m y 1, we define a relation X $ by a similar formula as Ž4.3.1.. Let V be an irreducible Uq Ž gl m .-module. Then the restriction of V to Uq Ž gl my 1 . is described by the branching rule due to Ueno᎐Takebayashi᎐Shibukawa wUTSx as follows: V < Uq Ž g l my 1 . s
[V . X
X$
Hence the restriction to Uq Ž gl my1 [ gl 1 . s Uq Ž gl my1 . m Uq Ž gl 1 . is given as V < Uq Ž g l my 1[ g l 1 . s
[V X$
X
m VY ,
where VY , K is an irreducible Uq Ž gl 1 .-module with weight Y which is determined uniquely from X . Now assume that g ⌳ m , i.e., < < s n and l Ž . F m. Then V occurs in V mn. It follows that the highest weight s Ž X , Y . of V s VX m VY should satisfy < X < q < Y < s n. Since l Ž Y . F 1, this condition determines uniquely Y , and actually the condition X $ is equivalent to the condition $ , where s Ž ., 1-tuple partition. Thus the lemma is proved. For g ⌳ m 1 , . . . , m r Žresp. g ⌳ m 1 , . . . , m ry1, 1 ., we denote by mult V Žresp. mult V . the multiplicity of V Žresp. V . in V mn. As a corollary to Lemma 4.4, we have the following. COROLLARY 4.5.
For g ⌳ m 1 , . . . , m ry1, 1 , we ha¨ e mult V s
Ý
%
mult V .
Recall that Z is an irreducible Wn, r-module corresponding to . We now prove PROPOSITION 4.6.
For each g ⌳ m 1 , . . . , m r, we ha¨ e mult V s dim ⺓ Z .
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Proof. First we note the following formula. For g ⌳ m 1 , . . . , m ry1, 1 , let Z be an irreducible Wn, rq1-module. Then we have dim Z s
Ý
%
dim Z .
Ž 4.6.1.
In fact, under the setting in 2.7, we can consider the restriction of UŽ ᒄ .-module V to UŽ ᒄX .. Then the discussion in Lemma 4.4 can be applied also for this case, and we have a similar formula as Corollary 4.5 with respect to mult V and mult V , where mult V stands for the multiplicity of V in V mn. But by Proposition 2.8, we know that mult V s dim Z , and mult V s dim Z . This implies Ž4.6.1.. We prove the proposition by induction on the semisimple rank of ᒄ, i.e., on M s Ý ris1Ž m i y 1.. First assume that M s 0. Then m1 s ⭈⭈⭈ s m r s 1 and s Ž Ž1., . . . , Ž r . ., with l Ž Ž i. . F 1. Hence each Ž i. can be identified with a non-negative integer. Now dim V s 1 and mult V coincides with the dimension of the weight space in V mn for the weight . It follows that r mult V s n!rŁ is1 Ž i.!, which is equal to dim Z . Next we consider the case where ᒄ is not a Cartan subalgebra of gl m . Then there exists some k such that m k ) 1. Clearly we may assume that k s r. Let ᒄX be as in 4.3. Now for a given g ⌳ m 1 , . . . , m r, choose g ⌳ m 1 , . . . , m ry1, 1 such that $ by Ži r . s Ži r . for i s 1, . . . , m r y 1 and Ž rq1. s Ž Žmr .r ., one part partition. Then for X g ⌳ m 1 , . . . , m r, the condition $ X implies that Ž r . F Ž X .Ž r . with respect to the usual partial order of weights, and Ž i. s Ž X .Ž i. for i s 1, . . . , r y 1, where X s ŽŽ X .Ž1., . . . , Ž X .Ž r . .. Now by induction hypothesis, we may assume that mult V s dim Z for g ⌳ m 1 , . . . , m ry1, 1. By Corollary 4.5 and Ž4.6.1. together with the above remark, we have
Ý
X
$
mult VX s
Ý
X
$
dim Z X ,
Ž 4.6.2.
Here we may assume, by backward induction on the partial order of weights Ž X .Ž r ., that mult VX s dim Z X for any X / occurring in the expression of Ž4.6.2.. This implies that mult V s dim Z as asserted. The proposition is now proved. Remark 4.7. In the above proof, the formula Ž4.6.1. was deduced from Proposition 2.8. However, Ž4.6.1. can be formulated in terms of the combinatorics of Young diagrams. Then it is possible to prove Ž4.6.1. directly in a framework of colored Young diagrams. 4.8. We shall prove Theorem 4.1. Let A s End Uq Ž ᒄ . V mn. Then V mn is decomposed as an A m Uq Ž ᒄ .-module, V mn s
[
g⌳ m1 , . . . , m r
Zˆ m V ,
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where Zˆ is an irreducible A-module corresponding to . Since the action of Hn, r on V mn commutes with Uq Ž ᒄ ., we have Ž Hn, r . : A. By Lemma 4.2, Zˆ contains Z as an Hn, r-submodule. By Proposition 4.6, we have dim Zˆ s mult V s dim Z . But it is known by Ariki and Koke wAKx that dim Z s dim Z . Hence Zˆ s Z and we obtain the decomposition Ž4.1.1.. It is then clear that A coincides with Ž Hn, r .. The fact that ŽUq Ž ᒄ .. is the full centralizer algebra of Hn, r also follows from this. The theorem is proved. 4.9. It is known that any finite dimensional representation of Uq Ž ᒄ . is completely reducible Žsee, e.g., wLx.. A finite dimensional representation of Uq Ž ᒄ .is said to be of level n if each irreducible component occurs in V mn. Now the following result is an immediate consequence of Theorem 4.1 if one notices that any irreducible representation of Hn, r is obtained as Z for g ⌳ m 1 , . . . , m r if m i G n for i s 1, . . . , r. COROLLARY 4.10. There exists a functor F from the category of finite dimensional right Hn, r-modules to the category of finite dimensional left Uq Ž ᒄ .-modules of le¨ el n defined as follows. For an Hn, r-module M, let F Ž M . s M mH n , r V mn , equipped with a Uq Ž ᒄ .-module structure induced from that of V mn. If m i G n for i s 1, . . . , r, F gi¨ es rise to an equi¨ alence of categories. 5. SEMISIMPLICITY OF Hn, r 5.1. The semisimplicity of Hn, r was established in wAKx. In that paper, they constructed all the irreducible representations of Hn, r , and showed that dim Hn, r coincides with the sum of squares of their dimensions. The semisimplicity of Hn, r and the fact that dim Hn, r s < Wn, r < are derived from this. In this section, we shall show these two facts by making use of the representation of Hn, r on V mn, without referring to individual representations. In order to avoid the confusion, we want to clarify the setting in this section. The main result in the previous section relies on the results of wAKx. However, the results in this section are only based on the construction of representations of Hn, r on V mn. As explained in Remark 3.10 Ži., this construction is independent from wAKx. ŽStrictly speaking, we use the fact that dim Hn, r is finite, which was shown in wAKx by a direct computation from the definition of Hn, r without using the representation theory.. We also note that the results in Section 4 are not used in this section, except Proposition 4.6, which is derived just by using the representation
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theory of quantum groups and that of Wn, r , independently from the discussion on Hn, r . 5.2. Let R s ⺪w q, qy1 , u1 , . . . , u r x. As discussed in Remark 3.10 Žii., we consider the action of HR on VRmn, which we also denote by . Let : R ª ⺓ be a specialization homomorphism defined by Ž q . s 1, Ž u i . s i for i s 1, . . . , r. Then ⺓ mR, VRmn , V mn, and the action of 1 m a i Ž a i g HR . on ⺓ mR VRmn coincides with the action of si g Wn, r on V mn. For each w g Wn, r , we fix a Žnot necessarily reduced. expression w s si1 si 2 ⭈⭈⭈ si q by generators s1 , . . . , sn , and set a w s a i1 a i 2 ⭈⭈⭈ a i q. Then we have the following proposition. PROPOSITION 5.3. The elements a w g HR N w g Wn, r 4 are linearly independent o¨ er R. In particular, 1 m a w g Hn, r 4 form a K-basis of Hn, r and dim K Hn, r s < Wn, r <. Proof. Let us choose integers m1 , . . . , m r such that m i G n for i s 1, . . . , r. We consider the representation of HR on VRmn and that of Wn, r on V mn. We note that the representation is faithful since all the irreducible representations of Wn, r occur in V mn by Proposition 2.8. In order to show the first assertion of the proposition, we have only to show that Ž a w . are linearly independent operators on VRmn. We show this by using a specialization argument. Let us define R i for i s 1, . . . , r by R i s ⺓w u1 , . . . , u i x. We also put R rq1 s R and R 0 s ⺓. Then we have a sequence of specializations, rq1
r
2
1
R s R rq1 ª R r ª ⭈⭈⭈ ª R1 ª ⺓, where i : R i s R iy1 w u i x ª R iy1 is the R iy1-homomorphism given by i Ž u i . s i for i F r and rq1: R ª R r is given by rq1Ž q . s 1. We denote by aŽwi. the operator on VRmn induced by the specialization VRmn , i i mn Ž rq1. Ž . R i mR VR . Here a w s Ž a w . and aŽ0. s w . Therefore the first w assertion of the proposition follows from the following. Ž5.3.1.. The elements
aŽwk .
w g Wn , r 4
are linearly independent over R k for k s 0, . . . , r q 1. We show Ž5.3.1. by induction on k. This certainly holds for k s 0 since is faithful. Let k G 1 and assume that Ž5.3.1. holds for k y 1. Suppose that aŽwk . are linearly dependent over R k . Then we have a linear relation Ýc w aŽwk . s 0 with c w g R k s R ky1 w u k x. Here we may assume that some c w is not divisible by u k y k . But this relation yields, by the specialization k : R k ª R ky1 , a relation Ýc w Ž k . aŽwky1. s 0. Hence by our assumption, we have c w Ž k . s 0 for all w g Wn, r . It follows that c w is divisible by u k y k which contradicts our choice of c w . The last step R ª R r is done
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similarly. This proves Ž5.3.1. and the first assertion of the proposition follows. We show the second assertion. It is known by wAK, Proposition 3.4x that there exist elements a w N w g Wn, r 4 which generate HR as an R-module. Hence, in view of the above discussion, these elements form an R-free basis of HR . Therefore dim Hn, r s < Wn, r <, and any choice of 1 m a w 4 forms a K-basis of Hn, r . This proves the second assertion. We can now state THEOREM 5.4. Let Uq Ž ᒄ . be as in Section 3. Assume that m i G n for i s 1, . . . , r. Then Hn, r is isomorphic to End Uq Ž ᒄ . V mn. In particular, Hn, r is semisimple. Proof. It follows from Proposition 5.3 that : Hn, r ª End K V mn is injective. It is clear that Ž Hn, r . is contained in End Uq Ž ᒄ . V mn. We shall compute the dimension of End Uq Ž ᒄ . V mn. By Proposition 4.6, we see that dim End Uq Ž ᒄ . V mn s
Ý
g⌳ m1 , . . . , m r
Ž dim Z .
2
.
Ž 5.4.1.
On the other hand, it follows from Proposition 2.8 that Ž⺓Wn, r . s End UŽ ᒄ . V mn, and the dimension of End UŽ ᒄ . V mn is also given by the right hand side of Ž5.4.1.. Since is faithful, we see that dim End Uq Ž ᒄ . V mn s < Wn, r <. Since dim Hn, r s < Wn, r < by Proposition 5.3, we have Hn , r , Ž Hn , r . s End Uq Ž ᒄ . V mn , and so Hn, r is semisimple. This proves the theorem. REFERENCES wAx wAKx
S. Ariki, Cyclotomic q-Schur algebras as quotients of quantum algebras, preprint. S. Ariki and K. Koike, A Hecke algebra of Ž⺪rr⺪. X ᑭ n and construction of its irreducible representations, Ad¨ . Math. 106 Ž1994., 216᎐243. wATYx S. Ariki, T. Terasoma, and H. Yamada, Schur᎐Weyl reciprocity for the Hecke algebra of Ž⺪rr⺪. X ᑭ n , J. Algebra 178 Ž1995., 374᎐390. wDJMx R. Dipper, G. James, and A. Mathas, Cyclotomic q-Schur algebra, preprint. wFx W. Fulton, ‘‘Young Tableaux,’’ London Math. Soc. Student Texts 35, Cambridge Univ. Press, Cambridge, UK, 1997. wJx M. Jimbo, A q-analogue of UŽ gl Ž N q 1.., Hecke algebra and the Yang᎐Baxter equation, Lett. Math. Phys. 11 Ž1986., 247᎐252. wLx G. Lusztig, ‘‘Introduction to Quantum Groups,’’ Progress in Math. Vol. 110, Birkhauser, Boston, 1993. ¨ wUTSx K. Ueno, T. Takebayashi, and Y. Shibukawa, Gelfand᎐Zetlin bases for Uq Ž gl Ž N q 1.. modules, Lett. Math. Phys. 18 Ž1989., 215᎐221. wWx H. Weyl, ‘‘The Classical Groups. Their Invariants and Representations,’’ Princeton Univ. Press, Princeton, NJ, 1939.
Schur᎐Weyl Reciprocity for Ariki᎐Koike AlgebrasU Masahiro Sakamoto and Toshiaki Shoji Department of Mathematics, Science Uni¨ ersity of Tokyo, Noda, Chiba 278-8510, Japan Communicated by Michel Broue´ Received May 1, 1998 Let ᒄ s gl m 1 [ ⭈⭈⭈ [ gl m r be a Levi subalgebra of gl m , with m s Ý ris1 m i , and V the natural representation of the quantum group Uq Ž ᒄ .. We construct a representation of the Ariki᎐Koike algebra Hn, r on the n-fold tensor space of V, commuting with the action of Uq Ž ᒄ ., and prove the Schur᎐Weyl reciprocity for the actions of Uq Ž ᒄ . and Hn, r on it. 䊚 1999 Academic Press
1. INTRODUCTION The natural representation of UŽ gl m . on V s ⺓ m induces the action of V , which commutes with the permutation action of the symmetric group ᑭ n . The classical Schur᎐Weyl reciprocity asserts that the images of UŽ gl m . and of ᑭ n in End V mn are mutually fully centralizers of each other. Let ᒄ s gl m 1 [ ⭈⭈⭈ [ gl m r be a Levi subalgebra of gl m , with m s Ý ris1 m i , and Wn, r the complex reflection group ᑭ n h Ž⺪rr⺪. n. The action of ᑭ n on V mn can be extended to an action of Wn, r , commuting with the action of UŽ ᒄ . obtained as the restriction of UŽ gl m .. The Schur᎐Weyl reciprocity also holds between the action of UŽ ᒄ . and Wn, r on V mn. The q-analogue of the Schur᎐Weyl reciprocity, i.e., the reciprocity between the quantum group Uq Ž gl m . and the Hecke algebra Hn of type A ny 1 , was obtained by Jimbo wJx. In this paper, we establish the q-analogue of the latter case; i.e., we consider the quantum group Uq Ž ᒄ . and the Ariki᎐Koike algebra Hn, r which is the Hecke algebra of Wn, r . More precisely, we consider the natural representation of Uq Ž gl m . of an m-dimensional vector space V. It induces an action of the subalgebra Uq Ž ᒄ . of Uq Ž gl m . on V mn. We construct a representation of Hn, r on V mn commuting with the action of Uq Ž ᒄ ., and prove the Schur᎐Weyl reciprocity between mn
U
This paper is a contribution to the Joint Research Project ‘‘Representation Theory of Finite and Algebraic Groups’’ 1997᎐1999 under the Japanese᎐German Cooperative Science Promotion Program supported by JSPS and DFG. 293 0021-8693r99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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the actions of Uq Ž ᒄ . and Hn, r on V mn. In wATYx, Ariki, Terasoma, and Yamada treated a similar problem in the special case where m1 s ⭈⭈⭈ s m r s 1. Our approach was inspired by their arguments. For the proof of the Schur᎐Weyl reciprocity, we need to use the classification of irreducible representations of Hn, r together with some properties of representations obtained by Ariki and Koike wAKx. However, for the construction of the representation of Hn, r on V mn, we only need the generators and relations for Hn, r . This representation provides us an alternative approach to some results given in wAKx. In fact, it is verified that in the case where m i G n for any i G 1, the representation of Wn, r on V mn is faithful. By using a specialization argument, one can show in this case that the representation of Hn, r on V mn is also faithful. It follows from this by a simple argument that we have dim Hn, r s < Wn, r < and Hn, r is semisimple. Note that in wAKx, these results are obtained by constructing all the irreducible representations of Hn, r and by counting their dimensions. Using the result of our paper, Ariki wAx defined certain type of cyclotomic q-Schur algebras as quotients of quantum groups, and studied the relationship with the cyclotomic q-Schur algebras defined by Dipper, James, and Mathas wDJMx. 2. A REVIEW OF KNOWN RESULTS 2.1. Let Uq Ž gl m . be the quantized universal enveloping algebra of gl m over K s ⺡Ž q, u1 , . . . , u r ., a field of rational functions in variables q, u1 , . . . , u r , which is an associative algebra defined by generators e i , f i , Ž1 F i - m., q " i Ž1 F i F m. and by the relations qi qj s qj qi ,
q i qy i s qy i q i s 1,
¡q
y1
i
y i
q ej q
s ~ qe
¢e ¡qf
ej
j
j
q i f j qy i s qy1 f j
¢f
if j s i , otherwise,
j
~
if j s i y 1,
j
ei f j y f j ei s ␦i j
if j s i y 1, if j s i , otherwise, K i y Ky1 i q y qy1
,
e i " 1 e i2 y Ž q q qy1 . e i e i " 1 e i q e i2 e i " 1 s 0, f i " 1 f i2 y Ž q q qy1 . f i f i " 1 f i q f i2 f i " 1 s 0, ei e j s e j ei , fi f j s f j fi if < i y j < G 2,
Ž 2.1.1.
ARIKI ᎐ KOIKE ALGEBRAS
295
where in the above formula, K i stand for q iy iq 1 for i s 1, . . . , m y 1. It is known that Uq Ž gl m . has a Hopf algebra structure with comultiplication ⌬: Uq Ž gl m . ª Uq Ž gl m . m Uq Ž gl m . defined by ⌬Ž q" i . s q" i m q" i , ⌬ Ž ei . s ei m 1 q K i m ei , ⌬ Ž f i . s f i m Ky1 q 1 m fi . i Let V be an m-dimensional vector space over K with a basis E s ¨ 1 , . . . , ¨ m 4 . The natural representation of Uq Ž gl m . on V is defined by
Ž ei . ¨ j s Ž fi . ¨ j s Ž q" i .¨j s
½ ½
½
¨ jy1
if j s i q 1,
0
if j / i q 1,
¨ jq1
if j s i ,
0
if j / i ,
q"1¨j
if j s i ,
0
if j / i.
For a positive integer n, we consider the tensor product space V mn, on which Uq Ž gl m .mn acts naturally. We define inductively an algebra homomorphism ⌬Ž k . : Uq Ž gl m . ª Uq Ž gl m .mk by ⌬Ž k . s Ž ⌬Ž ky1. m id.( ⌬ for each k G 3, where ⌬Ž2. s ⌬. Combined with ⌬Ž n., Uq Ž gl m . acts naturally on V mn. We denote this action also by . It is expressed as follows: ny1
Ž ei . s
Ý
K im p m e i m 1mŽ ny1yp. ,
ps0 ny1
Ž fi . s
Ý
1m p m f i m Ž Ky1 i .
m Ž ny1yp .
,
Ž 2.1.2.
ps0
Ž q " i . s q " i m ⭈⭈⭈ m q " i . 2.2. We fix positive integers m1 , m 2 , . . . , m r such that Ý i m i s m. Let Vi be an m i-dimensional vector space with a basis Ei s ¨ jŽ i. N j s 1, . . . , m i 4 for i s 1, . . . , r. We define a total order on the set @ Ei by the lexicographic order on the set of pairs Ž i, j . such that 1 F i F r, 1 F j F m i , and identify the set @ Ei with E by this total order. Hence we have V s r [is1 Vi . We now consider the Lie algebra ᒄ s gl m 1 [ ⭈⭈⭈ [ gl m r. The Lie algebra gl m i acts naturally on Vi , and by the above identification, ᒄ is regarded as a subalgebra of gl m . Hence Uq Ž ᒄ . is a subalgebra of Uq Ž gl m ., and acts on V mn by the restriction of the action of Uq Ž gl m ..
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For a Young diagram , we denote by < < the number of boxes in , and by l Ž . the number of rows of . Irreducible representations of Uq Ž gl m . occurring in V mn are parametrized by Young diagrams with l Ž . F m. Let V be an irreducible Uq Ž gl m .-module with highest weight vector ¨ corresponding to s Ž 1 G 2 G ⭈⭈⭈ G m G 0.. Then we have q i ¨ s q i ¨ for i s 1, . . . , m. Since Uq Ž ᒄ . s Uq Ž gl m 1 . m ⭈⭈⭈ m Uq Ž gl m r ., irreducible representations of Uq Ž ᒄ . are parametrized by r-tuples s Ž Ž1., . . . , Ž r . . of Young diagrams Ž i. with l Ž Ž i. . F m i . The corresponding irreducible Uq Ž ᒄ .-module V is given as V s VŽ1. m ⭈⭈⭈ m VŽ r . . For an r-tuple , we denote by < < s Ý ris1 < Ž i. < the size of . Let ⌳ m 1 , . . . , m r be the set of r-tuples of Young diagrams of size n such that l Ž Ž i. . F m i . Then it is known that irreducible representations of Uq Ž ᒄ . occurring in V mn are parametrized by ⌳ m 1 , . . . , m r. We define a function b: E ª ⺞ by bŽ ¨ jŽ i. . s i. Note that our definition of the total order on E implies that if bŽ x 1 . - bŽ x 2 . for x 1 , x 2 g E , then we have x 1 - x 2 . 2.3. We recall here the definition and fundamental properties of the Ariki᎐Koike algebra Hn, r . Let Wn, r be the complex reflection group ᑭ n h Ž⺪rr⺪. n. Ariki and Koike introduced in wAKx a Hecke algebra Hn, r associated to Wn, r , which is a q-analogue of the group algebra of Wn, r and is defined by generators and relations as follows: Hn, r is the K-algebra with generators a1 , . . . , a n and relations
Ž a1 y u1 . Ž a1 y u 2 . ⭈⭈⭈ Ž a1 y u r . s 0, Ž ai y q . Ž ai q qy1 . s 0 Ž 2 F i F n . , a1 a2 a1 a2 s a2 a1 a2 a1 , a i a iq1 a i s a iq1 a i a iq1 ai a j s a j ai
Ž 2.3.1.
Ž 2 F i - n. ,
Ž < i y j < G 2. .
It is known by wAKx that Hn, r is a semisimple algebra with dim K Hn, r s < Wn, r < s n!r n. We define a subalgebra Tn, r of Hn, r as follows: Let us define t 1 , t 2 , . . . , t n recursively by t 1 s a1 , t j s a j t jy1 a j for j G 2, and let Tn, r be the subalgebra of Hn, r generated by t 1 , . . . , t n . It is shown in wAKx that Tn, r is a commutative semisimple algebra. Irreducible representations of Hn, r and Tn, r are determined in wAKx. In order to describe their results, we prepare some notations. Let s Ž Ž1., . . . , Ž r . . be an r-tuple of Young diagrams with size n. An r-tuple ⺣ s Ž S Ž1., . . . , S Ž r . . of tableaus is called a standard tableau of shape if each jŽ1 F j F n. occurs exactly once as an entry in one tableau of ⺣, and each S Ž i. is a standard tableau of shape Ž i.. If the
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number i appears in the intersection of the jth row and the kth column of S Ž p., we write Ž⺣; i . s p and cŽ⺣; i . s k y j. For each standard tableau ⺣ of size n, one can construct a linear character ⺣ of Tn, r by
⺣ Ž t i . s u Ž⺣ ; i. q 2 cŽ⺣ ; i. PROPOSITION 2.4 ŽwAKx..
Ž 1 F i F n. .
Ž 2.3.2.
Let Hn, r and Tn, r be as abo¨ e. Then
Ži. The complete set of irreducible representations of Tn, r is gi¨ en by ⺣ 4 , where ⺣ runs o¨ er all the r-tuple standard tableaus of size n. Žii. Irreducible representations of Hn, r are parametrized by r-tuple Young diagrams of size n. We denote by Z the irreducible Hn, r-module corresponding to . The decomposition of Z into Tn, r-modules is described as follows. PROPOSITION 2.5 ŽwAK, Proposition 3.16x.. Let Z be an irreducible Hn, r-module. Then Z s [⺣ ⺣ , where ⺣ runs o¨ er all the standard tableaus of shape . As a corollary to the above propositions, we have COROLLARY 2.6. Ži. Let Z be an irreducible Hn, r-module. Then for each standard tableau ⺣ of shape , there exists a common eigen¨ ector ¨ ⺣ in Z for t 1 , . . . , t n with eigen¨ alues gi¨ en in the right hand side of Ž2.3.2.. Žii. Con¨ ersely let Z be an Hn, r-module. Assume that Z contains a common eigen¨ ector ¨ ⺣ for t 1 , . . . , t n with eigen¨ alues gi¨ en as in Ž2.3.2. for some ⺣ of shape . Then the Hn, r-submodule of Z generated by ¨ ⺣ is a sum of copies of Z . 2.7. Here we recall the classical Schur᎐Weyl reciprocity for Wn, r and UŽ ᒄ .. So, let UŽ ᒄ . be the universal enveloping algebra of ᒄ. As in the case of Uq Ž ᒄ ., we have the natural representation of UŽ ᒄ . on an m-dimensional vector space V over ⺓, with basis E s ¨ 1 , . . . , ¨ m 4 . Hence UŽ ᒄ . acts naturally on V mn. We denote this action by . On the other hand, Wn, r acts on V mn as follows: Wn, r is generated by s1 , s2 , . . . , sn , where s2 , . . . , sn are generators of ᑭ n corresponding to transpositions, Ž1, 2., . . . , Ž n y 1, n., and s1 satisfies the relation s1r s 1, s1 s2 s1 s2 s s2 s1 s2 s1 , s1 si s si s1 for i G 3. The group ᑭ n acts on V mn by permuting the components of the tensor product, while s1 acts on V mn by s1 Ž x 1 m ⭈⭈⭈ m x n . s bŽ x 1 . x 1 m ⭈⭈⭈ m x n , for x i g E , where is a fixed primitive r th root of unity. We denote this action by . The action of Wn, r on V mn commutes with that of UŽ ᒄ .. Irreducible representations of UŽ ᒄ . and Wn, r are parametrized by r-tuple
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Young diagrams as before. We denote by V Žresp. Z . the irreducible UŽ ᒄ .-module Žresp. the irreducible Wn, r-module. corresponding to . Let ⺓Wn, r by the group algebra of Wn, r over ⺓. It is well known that the following Schur᎐Weyl reciprocity holds for the ⺓Wn, r m UŽ ᒄ .-module V mn. PROPOSITION 2.8. ŽUŽ ᒄ .. and Ž⺓Wn, r . are mutually the full centralizer algebras of each other; i.e., we ha¨ e
Ž U Ž ᒄ . . s End W n , r V mn ,
Ž ⺓Wn , r . s End UŽ ᒄ . V mn .
More precisely, we ha¨ e the decomposition of the ⺓Wn, r m UŽ ᒄ .-module V mn as V mn s
[
g⌳ m1 , . . . , m r
Z m V .
Since we could not find a reference for this result, and since this proposition plays an essential role in our later discussion, we give here a brief outline of the proof. For the first statement, we may consider the action of GLm 1Ž⺓. = ⭈⭈⭈ = GLm rŽ⺓. on V mn instead of UŽ ᒄ .. Then Weyl’s original proof wWx can be applied in our case with minor modifications, which gives the first assertion. For the second assertion, we note the following. First consider the ⺓ ᑭ n m UŽ gl m . module V mn. Let s Ž 1 G ⭈⭈⭈ G k . be a partition of n, and T the standard tableau of shape obtained by inserting the numbers 1, 2, . . . , n in order along consecutive rows. Let wX s ¨ i1 m ⭈⭈⭈ m ¨ i n be a vector of V mn such that the first 1 components are equal to ¨ 1 , and the next 2 components are equal to ¨ 2 , and so on. Let c g ⺓ ᑭ n be the Young symmetrizer with respect to T Žfor the definition, see, e.g., wF, Section 7x., and set w s cwX . Then w is a highest weight vector with highest weight in V mn, and w generates the Specht module S of ᑭ n . Now we consider the general case. For a given s Ž Ž1., . . . , Ž r . . of size n, we choose wi g Vimn i with n i s < Ž i. <, a highest weight vector for UŽ gl m i . with highest weight Ž i. as above. Set w s wr m ⭈⭈⭈ m w 1 g V mn. Then w is a highest weight vector with highest weight in V mn, and the Wn, r-module generated by w coincides with Z . The second assertion follows from this. 3. AN ACTION OF Hn, r ON V mn 3.1. In this section we shall construct a representation of Hn, r on V mn commuting with the action of Uq Ž ᒄ .. First we recall a result of Jimbo, who constructed a representation of the Hecke algebra of type A ny 1 on V mn. Let Hn be the subalgebra of Hn, r generated by a2 , . . . , a n . Then by wAKx, Hn is isomorphic to the Hecke algebra associated to the symmetric group ᑭ n .
ARIKI ᎐ KOIKE ALGEBRAS
299
It is known by Jimbo wJx that Hn acts on V mn by nyk . ky2. Ž a k . s idmŽ m Rˇ m idmŽ , V V
where Rˇ g End K Ž V m V . is defined by
¡q¨ m ¨ j
~
if i s j,
i
RˇŽ ¨ i m ¨ j . s ¨ j m ¨ i
¢¨ m ¨ q Ž q y q j
i
y1
if i ) j,
. ¨i m ¨ j
Ž 3.1.1.
if i - j.
This action of Hn commutes with the action of Uq Ž gl m . on V mn. We set Ti s Ž a i . for i s 2, . . . , n. Let i be the action of ᑭ n on V mn corresponding to the transposition Ž i y 1, i . as in 2.4. We define, for i s 2, . . . , n, an endomorphism Si g End K Ž V mn . as follows; Let x s x 1 m x 2 m ⭈⭈⭈ m x n be a basis element in V mn with x i g E . Then we define Si Ž x . s
½
Ti Ž x .
if b Ž x iy1 . s b Ž x i . ,
i Ž x .
if b Ž x iy1 . / b Ž x i . .
Ž 3.1.2.
Note that the two conditions on x in Ž3.1.2. remain unchanged by the Ž x . is given by Tiy1 Ž x . Žresp i Ž x .. action of Si . Hence Si is invertible; Sy1 i if x satisfies the former Žresp. the latter . condition in Ž3.1.2.. We define an endomorphism g End K Ž V mn . by Ž x . s u bŽ x 1 . x for a basis element x s x 1 m ⭈⭈⭈ m x n . We have the following theorem. THEOREM 3.2.
Let Ž a i . g End K Ž V mn . be the elements gi¨ en by
Ž a1 . s Ty1 ⭈⭈⭈ Tny1 Sn ⭈⭈⭈ S2 , 2 Ž a i . s Ti
for i s 2, . . . , n.
Then the action of a i gi¨ es rise to a representation of Hn, r on V mn commuting with the action of Uq Ž ᒄ .. Remark 3.3. In wATYx, Ariki, Terasoma, and Yamada constructed a representation of Hn, r on V mn satisfying a similar property, in the special case where m1 s ⭈⭈⭈ s m r s 1, i.e., in the case where ᒄ is a Cartan subalgebra ᒅ of gl m . In this case our definition of Ž a1 . coincides with theirs. In fact, in this case, the first case in Ž3.1.2. occurs only when x iy1 s x i , and so S i Ž x . s qx by Ž3.1.1.. Hence Sn ⭈⭈⭈ S2 Ž x . s u bŽ x 1 . q ␣ x 2 m ⭈⭈⭈ m x n m x 1 ,
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where ␣ is the number of x i such that x i s x 1 for 2 F i F n. This agrees with Ž x . in wATY, Prop. 2.1x. The remainder of this section is devoted to the proof of the theorem. Before proving the existence of the representation, we shall show that the thus obtained representation of Hn, r actually commutes with . LEMMA 3.4. The actions of Hn, r and Uq Ž ᒄ . on V mn commute with each other. Proof. Since Uq Ž ᒄ . commutes with Ti for i s 2, . . . , n Žcf. 3.1., it is enough to show that Uq Ž ᒄ . commutes with Ž a1 .. Clearly commutes with Uq Ž ᒄ .. So the lemma is reduced to showing the commutativity of Si with e j , f j , q " j g ᒄ. Since S i preserves the weights of the vectors in V mn, Si commutes with q " j. We show that S i commutes with e j . Let x s x 1 m ⭈⭈⭈ m x n be a basis element. First we note that Ž e j . preserves the condition in Ž3.1.2.. If bŽ x iy1 . s bŽ x i . we have S i Ž e j .Ž x . s Ž e j . S i Ž x . since S i s Ti on x and on Ž e j .Ž x .. Now assume that bŽ x iy1 . / bŽ x i .. We p show that i Ž e j .Ž x . s Ž e j . i Ž x .. Set X p s K m m e j m 1mŽ ny1yp. for j p s 0, . . . , n y 1. Then i commutes with X p unless p s i y 2, i y 1. Hence in view of Ž2.1.2., we have only to verify the following equation:
i Ž X iy2 q X iy1 . Ž x . s Ž X iy2 q X iy1 . Ž i Ž x . . .
Ž 3.4.1.
Since i affects only the i y 1 and ith components of the tensor product, and since the remaining parts are the same for X iy2 Ž x . and X iy1Ž x ., we may only consider the parts corresponding to i y 1 and ith components. The corresponding part of the left hand side is equal to e j x i m K j x iy1 q x i m e j x iy1 . The corresponding part of the right hand side is equal to K j x i m e j x iy1 q e j x i m x iy1 . We must show that these two expressions are equal. Now assume that e j g gl m k for some k. If bŽ x i . / k, bŽ x iy1 . / k, we have e j x iy1 s e j x i s 0 and so both expressions are equal to 0. If bŽ x i . s k, then bŽ x iy1 . / k, and so e j x iy1 s 0, K j x iy1 s x iy1. Again we can check that expressions are equal. The case where bŽ x iy1 . s k is done similarly. Hence we have verified the formula Ž3.4.1. and the commutativity for e j follows. The case for f j is dealt with similarly. 3.5. We put T1 s Ž a1 .. In order to prove the theorem, we have to show that T1 , . . . , Tn satisfy the same relations as in Ž2.3.1.. By a result of Jimbo wJx Žsee 3.1., we know already that T2 , . . . , Tn satisfy the relations in
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Ž2.3.1.. So, we have only to show the following three relations:
Ž T1 y u1 . Ž T1 y u 2 . ⭈⭈⭈ Ž T1 y u r . s 0, T1T2 T1T2 s T2 T1T2 T1 , T1Tj s Tj T1
Ž 3.5.1.
for j s 3, . . . , n.
We shall verify these relations by subsequent lemmas. Note that the outline of the proof is quite similar to that of wATYx, although our situation is more complicated than theirs. First we show the following lemma. LEMMA 3.6. Let Yj, p be the subspace of V mn generated by basis elements x s x 1 m ⭈⭈⭈ m x n such that bŽ x p . G j for j G 1. Then y1 Ži. If x g Yj, p , then Tpq1 ⭈⭈⭈ Tny1 Sn ⭈⭈⭈ S pq1Ž x . g x q Yjq1, p . Žii. ŽT1 y u j .Yj, 1 ; Yjq1, 1. Žiii. ŽT1 y u1 .ŽT1 y u 2 . ⭈⭈⭈ ŽT1 y u r . s 0.
In particular, the first relation in Ž3.5.1. holds. Proof. We shall prove the first assertion by backward induction on p. The assertion is clear for p s n. Assume that it holds for p, and take x g Yj, py1. It is easily checked that Tp Ž Yj, py1 . s Yj, p for any j G 1. Then we have S p Ž x . g Yj, p . In fact, if S p Ž x . s Tp Ž x ., the above relation is applied. If S p Ž x . s p Ž x ., this is also clear. Hence by applying the induction hypothesis to S p Ž x ., we have y1 Tpy1 Tpq1 ⭈⭈⭈ Tny1 Sn ⭈⭈⭈ S pq1 S p Ž x . g Tpy1 Ž S p Ž x . q Yjq1, p . ,
s Tpy1 Ž S p Ž x . . q Yjq1, py1 . Ž 3.6.1. Assume now that bŽ x p . / bŽ x py1 .. Then S p Ž x . s p Ž x . and Tpy1 Ž p Ž x . . s
½
x y Ž q y qy1 . p Ž x .
if b Ž x p . ) b Ž x py1 . ,
x
if b Ž x p . - b Ž x py1 . .
Note in the first case of the above formula, we have p Ž x . g Yjq1, py1 and so Tpy1 Ž p Ž x .. is contained in x q Yjq1, py1. It follows that the right hand side of Ž3.6.1. is contained in x q Yjq1, py1. Next assume that bŽ x p . s bŽ x py1 .. Then we have Tpy1 S p Ž x . s x, and again the right hand side of Ž3.6.1. is contained in x q Yjq1, py1. Hence Ži. follows. We show the second assertion of the lemma. By applying Ži. to the case p s 1, we have for any x g Yj, 1 , T1 Ž x . s Ty1 ⭈⭈⭈ Tny1 Sn ⭈⭈⭈ S2 Ž x . g u bŽ x 1 . x q Yjq1, 1 . 2
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It follows that
Ž T1 y u j . Ž x . g Ž u bŽ x . y u j . x q Yjq1, 1 . 1
Now since x g Yj, 1 , we have bŽ x 1 . G j. Then Ž u bŽ x 1 . y u j . x s 0 if bŽ x 1 . s j, and x g Yjq1, 1 if bŽ x 1 . ) j. Hence the right hand side of the above formula is contained in Yjq1, 1. This proves the second assertion. The third assertion is immediate from Žii. if we notice that Y1, 1 s V mn and Yrq1, 1 s 04 . Before verifying the second relation in Ž3.5.1., we prove some properties of Si . LEMMA 3.7. The following formulae hold for j G 3. Ži. S j S jy1Tj s Tjy1 S j S jy1. y1 Žii. S j S jy1 S j Sy1 jy1Tjy1 s Tj S j S jy1 S j S jy1 . Žiii. S j S jy1 S j S jy1Tjy1 s Tj S j S jy1 S j S jy1. Proof. It is enough to show the lemma in the case where n s 3 and j s 3. The formula is verified by evaluating both sides at basis elements x s x 1 m x 2 m x 3 according to the patterns of bŽ x i .. We define Rˇi j g End K Ž V mn . as an endomorphism which acts as Rˇ in Ž3.1.1. on the i and jth component of x and leaves other factors invariant. Then for g ᑭ n , we have Rˇi jy1 s Rˇ Ž i. Ž j. , where the left hand side stands for the action of on V mn as in 2.7. First we consider the formula Ži.. In the case where bŽ x i . are all the same, the formula follows from the relation T3 T2 T3 s T2 T3 T2 , while in the case where all the bŽ x i . are distinct, we have S3 S2 T3 Ž x . s 3 2 Rˇ23 Ž x . s Rˇ12 3 2 Ž x . s T2 S 3 S 2 Ž x . . A similar computation works also for the case where bŽ x 2 . s bŽ x 3 ., and bŽ x 1 . / bŽ x 2 .. Next consider the case where bŽ x 1 . s bŽ x 2 ., and bŽ x 3 . / bŽ x 1 .. We use the following notation. Set w q x s q y qy1 , and define
w qxij s qi j s
½ ½
w qx
if x i - x j ,
0
otherwise,
q
if x i s x j ,
1
otherwise,
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where x i - x j denotes the total order on E . Then we have S 3 S 2 T3 Ž x . s
½
Rˇ23 Ž x 3 m x 1 m x 2 .
if x 2 ) x 3 ,
Rˇ23 Ž x 3 m x 1 m x 2 . q w q x Rˇ13 Ž x 1 m x 3 m x 2 .
if x 2 - x 3 .
It follows that S3 S2 T3 Ž x . s q12 x 3 m x 2 m x 1 q w q x 12 x 3 m x 1 m x 2 q w q x 23 Ž q12 x 2 m x 3 m x 1 q w q x 12 x 1 m x 3 m x 2 . . On the other hand, since T2 S3 S2 Ž x . s Rˇ12 3 Rˇ12 Ž x . s Rˇ12 Rˇ13 Ž x 1 m x 3 m x 2 . , we have T2 S3 S2 Ž x . s q12 Ž x 3 m x 2 m x 1 q w q x 23 x 2 m x 3 m x 1 . q w q x 12 Ž x 3 m x 1 m x 2 q w q x 23 x 1 m x 3 m x 2 . . Hence we get the desired equality. Finally the case where bŽ x 1 . s bŽ x 3 . and bŽ x 2 . / Ž x 1 . is verified in a similar way. Thus the formula Ži. is proved. Next we consider the formula Žii.. In the case where all the bŽ x i . are the y1 same, this follows from the relation T3T2 T3Ty1 2 T2 s T3 T3 T2 T3 T2 . In the case where all the bŽ x i . are distinct, this follows from the relation 3 2 3 2 Rˇ12 s Rˇ23 3 2 3 2 . We assume that bŽ x 1 . s bŽ x 2 . and bŽ x 3 . / bŽ x 1 .. Then we have S3 S2 S3 Sy1 2 T2 Ž x . s S 3 S 2 S 3 Ž x . s T3 2 3 Ž x . . On the other hand, we have y1 T3 S3 S2 S3 Sy1 2 Ž x . s T3 T3 2 3 T2 Ž x . s T3 2 3 Ž x . .
Hence we get the desired equality. Next assume that bŽ x 1 . s bŽ x 3 . and bŽ x 2 . / bŽ x 1 .. Then we have T3 S3 S2 S3 Sy1 2 Ž x . s T3 3 2 T3 Ž x 2 m x 1 m x 3 . s T3 T2 Ž x 1 m x 3 m x 2 . . On the other hand, we have S3 S2 S3 Sy1 2 T2 Ž x . s S3 S2 S3 Ž x 1 m x 2 m x 3 q w q x 12 x 2 m x 1 m x 3 . s S3 S2 Ž x 1 m x 3 m x 2 q w q x 12 T3 Ž x 2 m x 1 m x 3 . . s 3T2 Ž x 1 m x 3 m x 2 . q w q x 12 3 2 T3 Ž x 2 m x 1 m x 3 . s 3T2 Ž x 1 m x 3 m x 2 . q w q x 12 T2 Ž x 1 m x 3 m x 2 . s T3 T2 Ž x 1 m x 3 m x 2 . .
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Hence we get the desired equality. Finally we assume that bŽ x 2 . s bŽ x 3 . and bŽ x 1 . / bŽ x 2 ..l Then in a similar way as above, we can show that both Ž x . and S3 S2 S3 Sy1 Ž . Ž . T3 S3 S2 S3 Sy1 2 2 T2 x coincides with T3 3 T2 x 2 m x 3 m x 1 . Hence the formula Žii. holds. Finally we consider the formula Žiii.. This case is dealt with almost similarly as in the case Žii.. In fact, the same argument is applied to the case where all the bŽ x i . are the same, or are distinct. Moreover, it is the same for the last two cases of the above computation. In the case where bŽ x 1 . s bŽ x 2 . and bŽ x 3 . / bŽ x 1 ., we see that both of S3 S2 S3 S2 T2 Ž x . and T3 S3 S2 S3 S2 Ž x . coincide with T33 2 3 Ž x .. Thus the formula Žiii. holds, and the lemma is proved. We can now prove the second relation of Ž3.5.1.. LEMMA 3.8. Ži. Žii. Žiii. Živ. Žv.
Set s Sn ⭈⭈⭈ S2 . Then the following formulae hold.
Tj s Tjy1 for j s 3, . . . , n. 2 T2 s Tn 2 . y1 T2 s T2 y1 . Ž . 2 T2 s TnŽ . 2 . T1T2 T1T2 s T2 T1T2 T1.
In particular the second relation of Ž3.5.1. holds. Proof. First we show Ži.. Note that Si Tj s Tj Si for < i y j < G 2. Hence by using the relation Ži. in Lemma 3.7, we have
Tj s Sn ⭈⭈⭈ S jq1 Ž S j S jy1Tj . S jy2 ⭈⭈⭈ S2 s S n ⭈⭈⭈ S jq1 Ž Tjy1 S j S jy1 . S jy2 ⭈⭈⭈ S2 s Tjy1 . This proves Ži.. We show Žii.. Again we note that Si S j s S j Si for < i y j < G 2. Then we have
2 T2 s Ž Sn ⭈⭈⭈ S2 . Ž Sn ⭈⭈⭈ S2 . T2 s Sn Ž Sny1 Sn . ⭈⭈⭈ Ž S3 S4 . Ž S2 S3 . S2 T2 y1 s Ž Sn Sny1 Sn Sy1 ny1 .Ž S ny1 S ny2 S ny1 S ny2 . ⭈⭈⭈
⭈⭈⭈ Ž S4 S3 S4 Sy1 3 . Ž S 3 S 2 S 3 S 2 . T2 . Now we have Ž S3 S2 S3 S2 .T2 s T3 Ž S3 S2 S3 S2 . by applying Lemma 3.7 Žiii. with j s 3. Then by applying Lemma 3.7 Žii., for j s 4, . . . , n successively,
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305
we see that the last formula is equal to y1 2 Tn Ž Sn Sny1 Sn Sy1 ny1 . ⭈⭈⭈ Ž S4 S3 S4 S3 . Ž S3 S2 S3 S2 . s Tn .
This proves Žii.. We show Žiii.. First we note that commutes with S3 , . . . , Sn by definition. Then we see easily that y1 T2 s Sy1 2 S2 T2 . Similarly we have T2 y1 s T2 Sy1 S . Hence in order to prove the state2 2 ment, it is enough to show that y1 Sy1 2 S 2 T2 s T2 S 2 S 2 .
Ž 3.8.1.
We show Ž3.8.1.. We may consider the case n s 2. Let x s x 1 m x 2 be a basis element in V m V. First assume that bŽ x 1 . s bŽ x 2 .. Then clearly we have 2 y1 Sy1 2 S 2 T2 Ž x . s T2 Ž x . s T2 S 2 S 2 Ž x . .
Next assume that bŽ x 1 . / bŽ x 2 .. We write bŽ x 1 . s ␣ and bŽ x 2 . s  . Then we have Sy1 2 S2 T2 Ž x . s 2 2 T2 Ž x . s u␣ u T2 Ž x . , T2 Sy1 2 S2 Ž x . s T2 2 2 Ž x . s T2 Ž u␣ u x . . Thus Ž3.8.1. holds and the assertion Žiii. is proved. We show Živ.. By Žiii. and Žii., we have 2 2 Ž . T2 s 2 Ž y1 . T2 s 2 T2 Ž y1 . s Tn Ž . .
We show Žv.. We note that commutes with T3 , . . . , Tn . By using Ži., we have T1T2 T1 s Ž Ty1 ⭈⭈⭈ Tny1 . Ž Ty1 ⭈⭈⭈ Tny1 . 2 3 y1 s Ž Ty1 ⭈⭈⭈ Tny1 .Ž Ty1 ⭈⭈⭈ Tny1 . Ž . . 2 2 2
Hene by Živ., we have y1 T1T2 T1T2 s Ž Ty1 ⭈⭈⭈ Tny1 .Ž Ty1 ⭈⭈⭈ Tny1 . Tn Ž . . 2 2 2
Ž 3.8.2.
y1 s Ž Ty1 ⭈⭈⭈ Tny1 .Ž Ty1 ⭈⭈⭈ Tny1 . Ž . . 3 2
Ž 3.8.3.
On the other hand, again by using Ži., we have T2 T1T2 T1 s Ž Ty1 ⭈⭈⭈ Tny1 . Ž Ty1 ⭈⭈⭈ Tny1 . 3 3 2
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In view of Ž3.8.2. and Ž3.8.3., the proof of Žv. is reduced to showing the following formula, which is easily obtained by using the standard argument on the generators of Hn Žsee wATY, Lemma 2.3x.:
Ž Ty1 2
y1 y1 ⭈⭈⭈ Tny1 .Ž Ty1 ⭈⭈⭈ Tny1 ⭈⭈⭈ Tny1 .Ž Ty1 ⭈⭈⭈ Tny1 . Tn s Ž Ty1 .. 2 3 2
This shows Žv., and the lemma is now proved. Finally we show LEMMA 3.9. For each j G 3, we ha¨ e T1Tj s Tj T1. Hence the third relation of Ž3.5.1. holds. Proof. By using Ži., we have T1Tj s Ty1 ⭈⭈⭈ Tny1 Tj 2 s Ty1 ⭈⭈⭈ Tny1 Tjy1 2 y1 y1 y1 s Ty1 ⭈⭈⭈ Ž Tjy1 Tj Tjy1 . Tjq1 ⭈⭈⭈ Tny1 2 y1 y1 y1 s Ty1 ⭈⭈⭈ Tjy2 Tj . ⭈⭈⭈ Tny1 Ž TjTjy1 2
s Tj T1 . The lemma is proved. Theorem 3.2 is now proved by Lemma 3.6, Lemma 3.8, and Lemma 3.9. Remark 3.10. Ži. For the construction of the representation of Hn, r on V mn, no properties established in wAKx are required. We do not need to assume that Hn, r is semisimple nor dim Hn, r is finite. In fact, the construction proceeds by just using the description of Hn, r in terms of generators and relations. In the beginning of 3.1, it was used that Hn is isomorphic to the Hecke algebra of type A ny 1. This was proved in wAKx. But for the construction of our representation, this is not essential. One can define Ti s Ž a i . for i G 2 as in 3.1. Then it satisfies the last two conditions in Ž2.3.1. by Jimbo wJx, which is enough for our later discussion. Žii. The construction of the representation of Hn, r works in a more general setting. Let R s ⺪w q, qy1 , u1 , . . . , u r x. We denote by VR the Rlattice of V spanned by the basis E s ¨ 1 , . . . , ¨ m 4 . The algebra Hn, r has an R-structure, Hn, r s K mR HR , where HR is the R-algebra generated by a1 , . . . , a n subject to the conditions in Ž2.3.1.. The quantum group Uq Ž ᒄ . also has the Kostant form Uq, R which acts on VRmn. Then the previous arguments can be applied to this setting, and one gets a representation of HR on VRmn commuting with the actions of Uq, R . We now consider an integral domain RX , with elements uX1 , . . . , uXr g RX and an invertible ele-
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307
ment qX g RX . By the specialization, u i ¬ uXi , q ¬ qX , we can consider the algebras HRX s RX mR HR , Uq, RX s RX mR Uq, R , and the representation of X Uq, RX on VRX s RX mR VR . Then clearly we have an action of HRX on VRmn commuting with the action of Uq, RX . This is the situation considered in wAx. 4. SCHUR᎐WEYL RECIPROCITY FOR Hn, r AND Uq Ž ᒄ . In this section, we prove the Schur᎐Weyl reciprocity for the Hn, r m Uq Ž ᒄ .-module V mn. THEOREM 4.1. Ž Hn, r . and ŽUq Ž ᒄ .. are mutually the full centralizer algebras of each other. Moreo¨ er, the Hn, r m Uq Ž ᒄ .-module V mn is decomposed as V mn s
[
g⌳ m1 , . . . , m r
Z m V .
Ž 4.1.1.
The theorem will be proved in 4.8 after some preliminaries. First we prepare some lemmas. LEMMA 4.2. Let g ⌳ m 1 , . . . , m r. Then V mn contains an irreducible Hn, rmodule isomorphic to Z , consisting of highest weight ¨ ectors of Uq Ž ᒄ . with highest weight . Proof. First we consider the case where r s 1. Note that in this case Hn,1 is identified with Hn since a1 s u1 ⭈ 1. Then the subalgebra Tn, 1 is nothing but the subalgebra of Hn generated by tX2 s a22 , tX3 s a3 a22 a3 , . . . , tXn s a n tXny1 a n . By Jimbo ŽwJx., it is known that the Hn m Uq Ž gl m .-module V mn is decomposed as V mn s
[Z
g⌳ m
m V ,
Ž 4.2.1.
where ⌳ m is the set of Young diagrams of size n with l Ž . F m, and V Žresp. Z . is the irreducible Uq Ž gl m .-module with highest weight Žresp. the irreducible Hn-module corresponding to ., respectively. Now by the decomposition Ž4.2.1., for each g ⌳ m one can find an irreducible Hn, 1-module Z consisting of highest weight vectors for V . Let S be a standard tableau of type . Then by applying Corollary 2.6 Ži., we can find a common eigenvector ¨ S for tX2 , . . . , tXn satisfying Ž2.3.2. which is a highest weight vector for V. We now return to the general setting. In view of Corollary 2.6 Žii., to prove the lemma it is enough to show the existence of a common
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eigenvector ¨ ⺣ for some ⺣ of shape , which is also a highest weight vector for V . We choose ⺣ s Ž S Ž1., . . . , S Ž r . . as follows: set n i s < Ž i. < and pi s n i q ⭈⭈⭈ n r for i s 1, . . . , r. Also we set prq1 s 0. We define S Ž i. by filling n i numbers piq1 q 1 - piq1 q 2 - ⭈⭈⭈ - pi in the boxes in Ž i., first to all the entries of the first row, and then to all the entries of the second row, and so on, in increasing order. We consider the action of Hn i m Uq Ž gl m i . on Vimn i . The standard tableau S Ž i. obviously determines a standard tableau S of type Ž i. with numbers 1, . . . , n i 4 . Hence by the previous argument, one can find a common eigenvector wi g Vimn i for tX2 , . . . , tXn i g Tn i , 1 satisfying Ž2.3.2. with respect to S , which is also a highest weight vector for VŽ i. . We now set mn ry 1 ¨ ⺣ s wr m wry1 m ⭈⭈⭈ m w 1 g Vrmn r m Vry1 m ⭈⭈⭈ m V1mn1 .
Since wi is a highest weight vector of VŽ i. , ¨ ⺣ is a highest weight vector of V s VŽ r . m ⭈⭈⭈ m VŽ1. . We shall show that ¨ ⺣ is a common eigenvector for t 1 , . . . , t n with eigenvalues as in the right hand side of Ž2.3.2., which is equivalent to the following statement. Ž4.2.2.. Assume that pkq 1 - j F pk . Then we have t j¨ ⺣ s u k q 2 c ¨ ⺣
with c s c Ž S Ž k . ; j y pkq1 . .
By Corollary 2.6 Žii. and by the commutativity of the actions of Hn, r and Uq Ž ᒄ ., the proof of the lemma is reduced to showing Ž4.2.2.. We show Ž4.2.2.. Assume that pkq 1 - j F pk , and set pkq1 s p, pk s q. Now t j ¨ ⺣ can be written as y1 ⭈⭈⭈ Tny1 S n ⭈⭈⭈ S2 T2 ⭈⭈⭈ Tj ¨ ⺣ . t j ¨ ⺣ s Tjq1
First we note that Tpq 1 ⭈⭈⭈ Tj ¨ ⺣ g Vrmn r m ⭈⭈⭈ m V1mn1 .
Ž 4.2.3.
Let x s x 1 m ⭈⭈⭈ m x n be a basis element of V mn occurring in the expression of Tpq 1 ⭈⭈⭈ Tj ¨ ⺣ . Then x p g Vk , and x i ) x p for all i - p. Moreover we have x i f Vk for i - p. This implies that T2 ⭈⭈⭈ Tp Ž x . s x p m x 1 m ⭈⭈⭈ m x py1 m x pq1 m ⭈⭈⭈ m x n . and we see that S p ⭈⭈⭈ S2 T2 ⭈⭈⭈ Tp Ž x . s u k x. It follows that y1 t j ¨ ⺣ s u k Tjq1 ⭈⭈⭈ Tny1 Sn ⭈⭈⭈ S pq1Tpq1 ⭈⭈⭈ Tj ¨ ⺣ .
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On the other hand we have S q ⭈⭈⭈ S pq1Tpq1 ⭈⭈⭈ Tj ¨ ⺣ g Vrmn r m ⭈⭈⭈ m V1mn1 . Let y s y 1 m ⭈⭈⭈ m yn be a basis element of V mn occurring in the expression of S q ⭈⭈⭈ S pq1Tpq1 ⭈⭈⭈ Tj ¨ ⺣ . Then we see that yq g Vk , yi f Vk for all i ) q. Moreover, yi - yq for all i ) q. This implies, by a similar argument y1 y1 Ž . as above, that we have Tqq 1 ⭈⭈⭈ Tn S n ⭈⭈⭈ S qq1 y s y. Hence y1 t j ¨ ⺣ s u k Tjq1 ⭈⭈⭈ Tqy1 S q ⭈⭈⭈ S pq1Tpq1 ⭈⭈⭈ Tj ¨ ⺣ .
Now by Ž4.2.3. we see that S pq 1 , . . . , S q acts successively on Tpq1 ⭈⭈⭈ Tj ¨ ⺣ by Tpq 1 ⭈⭈⭈ Tq . Hence we have t j ¨ ⺣ s u k Tj ⭈⭈⭈ Tpq1Tpq1 ⭈⭈⭈ Tj ¨ ⺣ . But Tj ⭈⭈⭈ Tpq1Tpq1 ⭈⭈⭈ Tj ¨ ⺣ coincides with wr m ⭈⭈⭈ m wXk m ⭈⭈⭈ m w 1 , where wXk s tXjyp w k . Now by the definition of w k , we have wXk s q 2 c w k with c s cŽ S Ž k . ; j y p .. It follows that t j ¨ ⺣ s u k q 2 c ¨ ⺣ . This implies Ž4.2.2., and so the lemma is proved. 4.3. Next we shall compute the multiplicity of V in V mn. We perform this by considering the restriction of Uq Ž ᒄ . to a smaller subalgebra. So, assume that m r ) 1, and let ᒄX s gl m 1 [ ⭈⭈⭈ [ Ž gl m ry1 [ gl 1 .. We regard ᒄX as a subalgebra of ᒄ by letting gl m ry1 be a subalgebra of gl m r corresponding to the basis ¨ 1Ž r ., . . . , ¨ mŽ r r.y1 , and gl 1 to be that corresponding to ¨ mŽ r r.. For g ⌳ m 1 , . . . , m r and g ⌳ m 1 , . . . , m ry1, 1 , we define a relation $ by the following condition: let s Ž Ž1., . . . , Ž r . . and s Ž Ž1., . . . , Ž r ., Ž rq1. .. Then Ž i. s Ž i. for i s 1, . . . , r y 1 and
Ž1r . G Ž1r . G ⭈⭈⭈ G Žmr .ry1 G Žmr .ry1 G Žmr .r ,
Ž 4.3.1.
where Ž r . s Ž Ž1r . G Ž2r . G ⭈⭈⭈ G Žmr .r G 0. Žresp. Ž r . s Ž Ž1r . G Ž2r . G ⭈⭈⭈ G Žmr .ry1 G 0.. is a partition with l Ž Ž r . . F m r Žresp. a partition with l Ž Ž r . . F m r y 1.. Note that we have < < s < < s n, and l Ž Ž rq1. . F 1. Moreover the condition Ž4.3.1. implies that < Ž r . < F < Ž r . <. Hence Ž rq1. is determined uniquely once Ž1., . . . , Ž r . is determined as above. Now Uq Ž ᒄX . is regarded as a subalgebra of Uq Ž ᒄ .. The following lemma describes the restriction of V to Uq Ž ᒄX .-submodules. LEMMA 4.4.
Let g ⌳ m 1 , . . . , m r. Then we ha¨ e V < Uq Ž ᒄ X . s
[V . $
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Proof. This lemma seems to be well known. But since we could not find an appropriate reference for it, we give a proof below. The lemma is easily reduced to the case where r s 1. So we consider the subalgebra gl my 1 [ gl 1 of gl m . Then Uq Ž gl my1 . is a subalgebra of Uq Ž gl m .. For a partition s Ž 1 G ⭈⭈⭈ G m G 0. with l Ž . F m and for X s Ž X1 G ⭈⭈⭈ G Xmy1 G 0. with l Ž X . F m y 1, we define a relation X $ by a similar formula as Ž4.3.1.. Let V be an irreducible Uq Ž gl m .-module. Then the restriction of V to Uq Ž gl my 1 . is described by the branching rule due to Ueno᎐Takebayashi᎐Shibukawa wUTSx as follows: V < Uq Ž g l my 1 . s
[V . X
X$
Hence the restriction to Uq Ž gl my1 [ gl 1 . s Uq Ž gl my1 . m Uq Ž gl 1 . is given as V < Uq Ž g l my 1[ g l 1 . s
[V X$
X
m VY ,
where VY , K is an irreducible Uq Ž gl 1 .-module with weight Y which is determined uniquely from X . Now assume that g ⌳ m , i.e., < < s n and l Ž . F m. Then V occurs in V mn. It follows that the highest weight s Ž X , Y . of V s VX m VY should satisfy < X < q < Y < s n. Since l Ž Y . F 1, this condition determines uniquely Y , and actually the condition X $ is equivalent to the condition $ , where s Ž ., 1-tuple partition. Thus the lemma is proved. For g ⌳ m 1 , . . . , m r Žresp. g ⌳ m 1 , . . . , m ry1, 1 ., we denote by mult V Žresp. mult V . the multiplicity of V Žresp. V . in V mn. As a corollary to Lemma 4.4, we have the following. COROLLARY 4.5.
For g ⌳ m 1 , . . . , m ry1, 1 , we ha¨ e mult V s
Ý
%
mult V .
Recall that Z is an irreducible Wn, r-module corresponding to . We now prove PROPOSITION 4.6.
For each g ⌳ m 1 , . . . , m r, we ha¨ e mult V s dim ⺓ Z .
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Proof. First we note the following formula. For g ⌳ m 1 , . . . , m ry1, 1 , let Z be an irreducible Wn, rq1-module. Then we have dim Z s
Ý
%
dim Z .
Ž 4.6.1.
In fact, under the setting in 2.7, we can consider the restriction of UŽ ᒄ .-module V to UŽ ᒄX .. Then the discussion in Lemma 4.4 can be applied also for this case, and we have a similar formula as Corollary 4.5 with respect to mult V and mult V , where mult V stands for the multiplicity of V in V mn. But by Proposition 2.8, we know that mult V s dim Z , and mult V s dim Z . This implies Ž4.6.1.. We prove the proposition by induction on the semisimple rank of ᒄ, i.e., on M s Ý ris1Ž m i y 1.. First assume that M s 0. Then m1 s ⭈⭈⭈ s m r s 1 and s Ž Ž1., . . . , Ž r . ., with l Ž Ž i. . F 1. Hence each Ž i. can be identified with a non-negative integer. Now dim V s 1 and mult V coincides with the dimension of the weight space in V mn for the weight . It follows that r mult V s n!rŁ is1 Ž i.!, which is equal to dim Z . Next we consider the case where ᒄ is not a Cartan subalgebra of gl m . Then there exists some k such that m k ) 1. Clearly we may assume that k s r. Let ᒄX be as in 4.3. Now for a given g ⌳ m 1 , . . . , m r, choose g ⌳ m 1 , . . . , m ry1, 1 such that $ by Ži r . s Ži r . for i s 1, . . . , m r y 1 and Ž rq1. s Ž Žmr .r ., one part partition. Then for X g ⌳ m 1 , . . . , m r, the condition $ X implies that Ž r . F Ž X .Ž r . with respect to the usual partial order of weights, and Ž i. s Ž X .Ž i. for i s 1, . . . , r y 1, where X s ŽŽ X .Ž1., . . . , Ž X .Ž r . .. Now by induction hypothesis, we may assume that mult V s dim Z for g ⌳ m 1 , . . . , m ry1, 1. By Corollary 4.5 and Ž4.6.1. together with the above remark, we have
Ý
X
$
mult VX s
Ý
X
$
dim Z X ,
Ž 4.6.2.
Here we may assume, by backward induction on the partial order of weights Ž X .Ž r ., that mult VX s dim Z X for any X / occurring in the expression of Ž4.6.2.. This implies that mult V s dim Z as asserted. The proposition is now proved. Remark 4.7. In the above proof, the formula Ž4.6.1. was deduced from Proposition 2.8. However, Ž4.6.1. can be formulated in terms of the combinatorics of Young diagrams. Then it is possible to prove Ž4.6.1. directly in a framework of colored Young diagrams. 4.8. We shall prove Theorem 4.1. Let A s End Uq Ž ᒄ . V mn. Then V mn is decomposed as an A m Uq Ž ᒄ .-module, V mn s
[
g⌳ m1 , . . . , m r
Zˆ m V ,
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where Zˆ is an irreducible A-module corresponding to . Since the action of Hn, r on V mn commutes with Uq Ž ᒄ ., we have Ž Hn, r . : A. By Lemma 4.2, Zˆ contains Z as an Hn, r-submodule. By Proposition 4.6, we have dim Zˆ s mult V s dim Z . But it is known by Ariki and Koke wAKx that dim Z s dim Z . Hence Zˆ s Z and we obtain the decomposition Ž4.1.1.. It is then clear that A coincides with Ž Hn, r .. The fact that ŽUq Ž ᒄ .. is the full centralizer algebra of Hn, r also follows from this. The theorem is proved. 4.9. It is known that any finite dimensional representation of Uq Ž ᒄ . is completely reducible Žsee, e.g., wLx.. A finite dimensional representation of Uq Ž ᒄ .is said to be of level n if each irreducible component occurs in V mn. Now the following result is an immediate consequence of Theorem 4.1 if one notices that any irreducible representation of Hn, r is obtained as Z for g ⌳ m 1 , . . . , m r if m i G n for i s 1, . . . , r. COROLLARY 4.10. There exists a functor F from the category of finite dimensional right Hn, r-modules to the category of finite dimensional left Uq Ž ᒄ .-modules of le¨ el n defined as follows. For an Hn, r-module M, let F Ž M . s M mH n , r V mn , equipped with a Uq Ž ᒄ .-module structure induced from that of V mn. If m i G n for i s 1, . . . , r, F gi¨ es rise to an equi¨ alence of categories. 5. SEMISIMPLICITY OF Hn, r 5.1. The semisimplicity of Hn, r was established in wAKx. In that paper, they constructed all the irreducible representations of Hn, r , and showed that dim Hn, r coincides with the sum of squares of their dimensions. The semisimplicity of Hn, r and the fact that dim Hn, r s < Wn, r < are derived from this. In this section, we shall show these two facts by making use of the representation of Hn, r on V mn, without referring to individual representations. In order to avoid the confusion, we want to clarify the setting in this section. The main result in the previous section relies on the results of wAKx. However, the results in this section are only based on the construction of representations of Hn, r on V mn. As explained in Remark 3.10 Ži., this construction is independent from wAKx. ŽStrictly speaking, we use the fact that dim Hn, r is finite, which was shown in wAKx by a direct computation from the definition of Hn, r without using the representation theory.. We also note that the results in Section 4 are not used in this section, except Proposition 4.6, which is derived just by using the representation
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theory of quantum groups and that of Wn, r , independently from the discussion on Hn, r . 5.2. Let R s ⺪w q, qy1 , u1 , . . . , u r x. As discussed in Remark 3.10 Žii., we consider the action of HR on VRmn, which we also denote by . Let : R ª ⺓ be a specialization homomorphism defined by Ž q . s 1, Ž u i . s i for i s 1, . . . , r. Then ⺓ mR, VRmn , V mn, and the action of 1 m a i Ž a i g HR . on ⺓ mR VRmn coincides with the action of si g Wn, r on V mn. For each w g Wn, r , we fix a Žnot necessarily reduced. expression w s si1 si 2 ⭈⭈⭈ si q by generators s1 , . . . , sn , and set a w s a i1 a i 2 ⭈⭈⭈ a i q. Then we have the following proposition. PROPOSITION 5.3. The elements a w g HR N w g Wn, r 4 are linearly independent o¨ er R. In particular, 1 m a w g Hn, r 4 form a K-basis of Hn, r and dim K Hn, r s < Wn, r <. Proof. Let us choose integers m1 , . . . , m r such that m i G n for i s 1, . . . , r. We consider the representation of HR on VRmn and that of Wn, r on V mn. We note that the representation is faithful since all the irreducible representations of Wn, r occur in V mn by Proposition 2.8. In order to show the first assertion of the proposition, we have only to show that Ž a w . are linearly independent operators on VRmn. We show this by using a specialization argument. Let us define R i for i s 1, . . . , r by R i s ⺓w u1 , . . . , u i x. We also put R rq1 s R and R 0 s ⺓. Then we have a sequence of specializations, rq1
r
2
1
R s R rq1 ª R r ª ⭈⭈⭈ ª R1 ª ⺓, where i : R i s R iy1 w u i x ª R iy1 is the R iy1-homomorphism given by i Ž u i . s i for i F r and rq1: R ª R r is given by rq1Ž q . s 1. We denote by aŽwi. the operator on VRmn induced by the specialization VRmn , i i mn Ž rq1. Ž . R i mR VR . Here a w s Ž a w . and aŽ0. s w . Therefore the first w assertion of the proposition follows from the following. Ž5.3.1.. The elements
aŽwk .
w g Wn , r 4
are linearly independent over R k for k s 0, . . . , r q 1. We show Ž5.3.1. by induction on k. This certainly holds for k s 0 since is faithful. Let k G 1 and assume that Ž5.3.1. holds for k y 1. Suppose that aŽwk . are linearly dependent over R k . Then we have a linear relation Ýc w aŽwk . s 0 with c w g R k s R ky1 w u k x. Here we may assume that some c w is not divisible by u k y k . But this relation yields, by the specialization k : R k ª R ky1 , a relation Ýc w Ž k . aŽwky1. s 0. Hence by our assumption, we have c w Ž k . s 0 for all w g Wn, r . It follows that c w is divisible by u k y k which contradicts our choice of c w . The last step R ª R r is done
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similarly. This proves Ž5.3.1. and the first assertion of the proposition follows. We show the second assertion. It is known by wAK, Proposition 3.4x that there exist elements a w N w g Wn, r 4 which generate HR as an R-module. Hence, in view of the above discussion, these elements form an R-free basis of HR . Therefore dim Hn, r s < Wn, r <, and any choice of 1 m a w 4 forms a K-basis of Hn, r . This proves the second assertion. We can now state THEOREM 5.4. Let Uq Ž ᒄ . be as in Section 3. Assume that m i G n for i s 1, . . . , r. Then Hn, r is isomorphic to End Uq Ž ᒄ . V mn. In particular, Hn, r is semisimple. Proof. It follows from Proposition 5.3 that : Hn, r ª End K V mn is injective. It is clear that Ž Hn, r . is contained in End Uq Ž ᒄ . V mn. We shall compute the dimension of End Uq Ž ᒄ . V mn. By Proposition 4.6, we see that dim End Uq Ž ᒄ . V mn s
Ý
g⌳ m1 , . . . , m r
Ž dim Z .
2
.
Ž 5.4.1.
On the other hand, it follows from Proposition 2.8 that Ž⺓Wn, r . s End UŽ ᒄ . V mn, and the dimension of End UŽ ᒄ . V mn is also given by the right hand side of Ž5.4.1.. Since is faithful, we see that dim End Uq Ž ᒄ . V mn s < Wn, r <. Since dim Hn, r s < Wn, r < by Proposition 5.3, we have Hn , r , Ž Hn , r . s End Uq Ž ᒄ . V mn , and so Hn, r is semisimple. This proves the theorem. REFERENCES wAx wAKx
S. Ariki, Cyclotomic q-Schur algebras as quotients of quantum algebras, preprint. S. Ariki and K. Koike, A Hecke algebra of Ž⺪rr⺪. X ᑭ n and construction of its irreducible representations, Ad¨ . Math. 106 Ž1994., 216᎐243. wATYx S. Ariki, T. Terasoma, and H. Yamada, Schur᎐Weyl reciprocity for the Hecke algebra of Ž⺪rr⺪. X ᑭ n , J. Algebra 178 Ž1995., 374᎐390. wDJMx R. Dipper, G. James, and A. Mathas, Cyclotomic q-Schur algebra, preprint. wFx W. Fulton, ‘‘Young Tableaux,’’ London Math. Soc. Student Texts 35, Cambridge Univ. Press, Cambridge, UK, 1997. wJx M. Jimbo, A q-analogue of UŽ gl Ž N q 1.., Hecke algebra and the Yang᎐Baxter equation, Lett. Math. Phys. 11 Ž1986., 247᎐252. wLx G. Lusztig, ‘‘Introduction to Quantum Groups,’’ Progress in Math. Vol. 110, Birkhauser, Boston, 1993. ¨ wUTSx K. Ueno, T. Takebayashi, and Y. Shibukawa, Gelfand᎐Zetlin bases for Uq Ž gl Ž N q 1.. modules, Lett. Math. Phys. 18 Ž1989., 215᎐221. wWx H. Weyl, ‘‘The Classical Groups. Their Invariants and Representations,’’ Princeton Univ. Press, Princeton, NJ, 1939.