RSOS models and Jantzen-Seitz representations of Hecke algeb(3)

Hecke algebras at roots of unity.

Now de ne the set JS (n;; d) to be the subset of JS (n) comprising those partitions with n-core and n-weight d (see 13, 7] for de nitions of n-core and n-weight). Then de ne the generating seriesn; (z )=

Xd 0

#JS (n;; d)z d:

Theorem 5.3 (i) If 2 JS (n) then the n-core of is a rectangular partition= (kl )such that k+ l n (it is assumed here that if either k= 0 or l= 0 then (kl ) means the empty partition). (ii) If k 6= 0 6= l then?s k+?l n;(kl ) (z )= z b k?l 0 (z ); where s= min(k; l). (iii) !n;; (z )= n?1 X k=0

b

k+ 0 (z ) k 0

? (n? 1):

Proof: In the tensor product V ( i ) V ( 0 ), all highest weights are of the form k+?l? e with k? l= i mod n (for later convenience we use?l and not l here). We take 0 k; l< n here and can also assume that k?l mod n whereupon k+ l n, and l= 0 only if k= 0. By de nition, the multiplicity of V ( k+?l? e ) is given by the coe cient of z e in b ik;+ 0?l (z ) and hence, by Corollary 2.3, by the number of partitions 2 JS (n) for which wt ( )= k+?l? i? e . We claim that the n-core of such a is the rectangular partition= (kl ), for which we calculate wt ( )= k+?l? k?l? s= k+?l? i? s, where s= min(k; l) is the multiplicity of the colour charge 0 in . This follows from the fact that b for every string of weights;?;? 2;:::; of the sln -module V ( 0 ) (where+ is not a weight of V ( 0 )), those partitions having these weights have the same n-core, and this n-core has weight . Then since= (kl ) w

ith k+ l n is manifestly an n-core, it follows that it is the n-core of, thence proving part (i). Part (ii) follows immediately since in the case k 6= 0 (so that l 6= 0), the partitions k+ enumerated by the branching function b k?l?l are precisely those elements 2 JS (n) 0 having weight wt ( )= k+?l? k?l? e, for some e, and hence n-core (kl ). 0 Finally, for k= 0 and arbitrary l, each partition counted by b?+ 0?l has empty n-core, l and hence contributes to n;;. However, the empty partition occurs for each l, hence an adjustment of n? 1 is needed after summing over all l. No other partition is repeated since, 0 as indicated by Theorem 2.2, the b?+ 0?l to which it contributes is uniquely determined by l?l mod n= ( 1? a1 ) mod n. (The summation over?l is replaced by one over k to give the nal result).

Hecke algebras at roots of unity.

*

12

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20

2

1

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012010

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11202110021022010210120120

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1120021102021

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02021

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01020021102200211022102101021021010220211021002102102Figure 1.

Hecke algebras at roots of unity.

Example 5.4 To illustrate this result, consider again the case n= 3, where we have thefollowing branching functions (to three terms):

b2 0;0 0 1 b 0;+ 02 1 b 1;+ 00 b2 1;2 0 2 b 2;+ 00 2 1 b 2; 0

======

1+ q2+; q+ 2q2+ 2q3+ 1+ q+ 2q2+ q+ q2+ 2q3+ 1+ q+ 2q2+ q+ q2+ 2q3+

;;

;; (3)

:

These are calculated using Corollary 2.3 which leads to the enumeration of the nodes of Fig. 1 labelled by asterisks. The only rectangular 3-cores are;, (1), (2) and (12 ). Using Theorem 5.3, we thus obtain:n;; n;(1) n;(2) n;(12 )

================

1+ 2q+ 5q2+; 1+ 2q+ 2q2+; 1+ q+ 2q2+; 1+ q+ 2q2+:

(4)

These correspond to the following sets:

JS (3;;; 0) JS (3;;; 1) JS (3;;; 2) JS (3; (1); 0) JS (3; (1); 1) JS (3; (1); 2) JS (3; (2); 0) JS (3; (2); 1) JS (3; (2); 2) JS (3; (12); 0) JS (3; (12); 1) JS (3; (12); 2)

f;g; f(3); (21)g; f(6); (51); (32); (412); (321)g; f(1)g; f(4); (22)g; f(7); (43)g; f(2)g; f(5)g; f(8); (3212 )g; f(12)g; f(32)g; f(62); (44)g:

(5)

6 DiscussionWe posed and solved the Jantzen-Seitz problem for Hecke algebras of type A. The solution is obtained by mapping the problem to a problem in exactly solvable lattice models, namely that of characterising the space of states of a certain class of restricted solid-on-solid model in terms of the states of the corresponding (unrestricted) solid-on-solid models. The latter was solved in 9]. The relationship

between the two problems is based on the fact that both can be formulated in the same language of representation theory of q-a ne algebra. The background to these two problems is discussed in greater detail in 7]. In a forthcoming paper 8], we plan to discuss the Jantzen-Seitz problem in the context of type-B Hecke algebras, and more generally, of Ariki-Koike (cyclotomic) Hecke algebras.Aknowledgements| We wish to thank Prof. Christine Bessenrodt for discussions that initiated this work, and Dr. Ole Warnaar for an earlier collaboration on which it is partly based. This work was done while B. Leclerc, M. Okado, and J.-Y. Thibon were visiting the Department of Mathematics, The University of Melbourne. These visits were made possible by nancial support of the Australian Research Council (ARC). The work of O. Foda and T. A. Welsh is also supported by the ARC. Note added| A forthcoming preprint, 25], contains (among other things) an elementary purely combinatorial proof of part (i) of 5.3.

Hecke algebras at roots of unity.

References

1] G.E. Andrews, R.J. Baxter and P.J. Forrester, Eight-vertex SOS model and generalized Rogers-Ramanujan-type identities, J. Stat. Phys. 35 (1984), 193{266. 2] S. Ariki, On the decomposition numbers of the Hecke algebra of G(m; 1; n), preprint, 1996. 3] R.J. Baxter, Exactly solved models in statistical mechanics, (1984) Academic Press. 4] R. Dipper and G.D. James, Representations of Hecke algebras of general linear groups, Proc. London Math. Soc. 52 (1986), 20{52. 5] E. Date, M. Jimbo, A. Kuniba, T. Miwa and …… 此处隐藏：6503字，全部文档内容请下载后查看。喜欢就下载吧 ……