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
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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 )
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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.