We consider representations of the Ariki-Koike algebra, a q-deformation of the group algebra of the complex reflection group C r S n. The representations of this algebra are naturally indexed by multipartitions of n, and for each multipartition λ we define a nonnegative integer called the weight of λ. We prove some basic properties of this weight function, and examine blocks of small weight.
We study the 0-Hecke algebra of an arbitrary finite Coxeter group, building on work of Norton (J. Austral. Math. Soc. Ser. A 27 (1979) 337). We examine the correspondence between injective and projective modules, extensions between simple modules and (in type A) the structure of induced simple modules.
Let F be a field, n a non-negative integer, a partition of n and S the corresponding Specht module for the Iwahori-Hecke algebra H F,q (S n ). James and Mathas conjecture a necessary and sufficient condition on for S to be irreducible. We prove the sufficiency of this condition in the case where F has infinite characteristic and also in the case where q = 1.
We consider the $t$-core of an $s$-core partition, when $s$ and $t$ are
coprime positive integers. Olsson has shown that the $t$-core of an $s$-core is
again an $s$-core, and we examine certain actions of the affine symmetric group
on $s$-cores which preserve the $t$-core of an $s$-core. Along the way, we give
a new proof of Olsson's result. We also give a new proof of a result of
Vandehey, showing that there is a simultaneous $s$- and $t$-core which contains
all others
Abstract. Let F be a field, q a non-zero element of F and H n = H F,q (S n ) the Iwahori-Hecke algebra of the symmetric group S n . If B is a block of H n of e-weight 3 and the characteristic of F is at least 5, we prove that the decomposition numbers for B are all at most 1. In particular, the decomposition numbers for a p-block of S n of defect 3 are all at most 1.
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