RSOS models and Jantzen-Seitz representations of Hecke algeb(4)

Hecke algebras at roots of unity.

Theorem 2.4 9] Let 0 j< n and= s+ t with 0 s t< n. In addition, let C be i the Cartan matrix of sln, ei be the (n? 1)-dimensional unit vector (0;; 0; 1; 0; 0)t andX qmt C?1 m?mt C?1 es?t+n+st=n; b j 0 (q)= Qn?1 (q) i=1 mi m Pn? where the sum is taken over all m 2 (Z 0) (n?1) satisfying t+ i=11 imi= 0 ( mod n), and (q)k= (1? q)(1? q2 ) (1? qk ):we set en= 0. Then

3 Modular representationsThe set S partitions FOW (n; j; k) has a de nite meaning in modular representation theory. of Indeed, j;k FOW (n; j; k) labels the modular representations of Sm in characteristic n that remain irreducible under reduction to Sm?1 14, 20]. However, in the case of FOW (n; j; k), n can be any positive integer, whereas in the case of Sm, it has to be a prime number. This di erence can be removed by working in the context of Hecke algebras at an nth root of unity, where the Jantzen-Seitz problem still makes sense. The Hecke algebra Hm (v) of type Am?1, is the C (v )-algebra generated by the elements T1;:::; Tm?1 subject to the relations

Ti Ti+1 Ti= Ti+1 Ti

Ti+1; Ti Tj= Tj Ti ji? j j> 1; Ti2= (v? 1)Ti+ v:is semisimple, and its irreducible representations are parametrised by (m). A convenient realisation of the representation labelled by is the Specht module S in which the entries of the representation matrices of the generators are elements of Z v]. In the non-generic case when v is a primitive nth root of unity, Hm (v) is no longer semisimple in general, and its representations are not necessarily completely reducible. The full set of irreducible representations is indexed by n (m) (see 4]). The irreducible module labelled by 2 n (m) is denoted by D . De ne JS (n) to be the set of all partitions that label the irreducible representations of p p Hm ( n 1) which remain irreducible under reduction to Hm?1 ( n 1). One of the aims of this work is to show that the partitions in JS (n) are de ned by the same conditions as in the case of symmetric groups, and to explain why

Sm, through identifying Ti with the simple transposition (i; i+1) 2 Sm. In fact, for generic values of v, Hm (v) is isomorphic to the group algebra of the symmetric group Sm . Hence, it

In the case v= 1, Hm (v) may be identi ed with the group algebra of the symmetric group

JS (n)=

j;k

FOW (n; j; k):

To tackle this problem, it is convenient to consider, as was done in 22, 2], the Grothenp p dieck group G0 (Hmp n 1)) of the category of nitely generated Hm ( n 1)-modules. The el( ements of G0 (Hm ( n 1)) are classes M] of modules, where M1]= M2] if and only if the simple composition factors of M1 and M2 occur with identical multiplicities (the order of the composition factors in the series is disregarded). The sum is de ned by M]+ N]= M N]. It is known that this is a free abelian group with as basis the set f D L of classes of irre]g p p ducible Hm ( n 1)-modules. Then de ne G (n) as the direct sum G (n)= m G0 (Hm ( n 1)). b As observed in 22], G (n) can be identi ed with the basic representation V ( 0 ) of sln, the action of the Chevalley generators ei, fi being given by the i-restriction and i-induction operators, as de ned in the fties by G. de B. Robinson in the case of symmetric groups. 4

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