Sinéad Lyle

Blocks of cyclotomic Hecke algebras

Abstract This paper classifies the blocks of the cyclotomic Hecke algebras of type G ( r , 1 , n ) over an arbitrary field. Rather than working with the Hecke algebras directly we work instead with the cyclotomic Schur algebras. The advantage of these algebras is that the cyclotomic Jantzen sum formula gives an easy combinatorial characterization of the blocks of the cyclotomic Schur algebras. We obtain an explicit description of the blocks by analyzing the combinatorics of ‘Jantzen equivalence.’ We remark that a proof of the classification of the blocks of the cyclotomic Hecke algebras was announced in 1999. Unfortunately, Cox has discovered that this previous proof is incomplete.

Some q-analogues of the Carter-Payne theorem

Abstract We prove a q-analogue of the Carter-Payne theorem for the two special cases corresponding to moving an arbitrary number of nodes between adjacent rows, or moving one node between an arbitrary number of rows. As a consequence, we show that these homomorphism spaces are one dimensional when q ≠ −1. We apply these results to complete the classification of the reducible Specht modules for the Hecke algebras of the symmetric groups when q ≠ −1. Our methods can also be used to determine certain other pairs of Specht modules between which there is a homomorphism. In particular, we describe the homomorphism space for an arbitrary partition μ.

Some reducible Specht modules

We wish to consider which ordinary irreducible representations of the symmetric group Sn remain irreducible modulo a prime p; this is the same as asking which partitions λ of n have the property that the corresponding Specht module S is reducible over a field of characteristic p. If λ or its conjugate partition λ′ is p-regular then the answer is known [6,8]; it is also known in the case p = 2 [7]. This paper discusses some of the reducible Specht modules in the case that p 3. Throughout,p will be an odd prime and λ a partition of some integer n. We will prove some cases of a conjecture by James and Mathas [9], given below. We begin with some definitions.

Row and column removal theorems for homomorphisms between Specht modules

We prove analogues of (Donkins generalisations of) Jamess row and column removal theorems, in the context of homomorphisms between Specht modules for symmetric groups.

Some reducible Specht modules for Iwahori–Hecke algebras of type A with q=−1

Let p be a prime and F a field of characteristic p, and let Hn denote the Iwahori–Hecke algebra of the symmetric group Sn over F at q = −1. We prove that there are only finitely many partitions λ such that both λ and λ are 2-singular and the Specht module S for H|λ| is irreducible.

Some Results Obtained by Application of the LLT Algorithm

We determine the v-decomposition numbers d μλ(v) for μ a partition with at most three parts. We use this information to compute the composition factors of the Specht modules of the Hecke algebra ℋ0 = ℋℂ, ω(𝔖 n ) which correspond to partitions with at most three parts.

Cyclotomic Carter-Payne homomorphisms

We construct a new family of homomorphisms between (graded) Specht modules of the quiver Hecke algebras of type A. These maps have many similarities with the homomorphisms constructed by Carter and Payne in the special case of the symmetric groups, although the maps that we obtain are both more and less general than these.

On Specht modules of general linear groups

We wish to consider the reduction modulo p of the ordinary irreducible unipotent representations of the general linear groups GLn(q) in the case that p q . For each partition λ of n, James [3] defined a GLn(q)-module S, known as a Specht module. Over fields of characteristic zero, these Specht modules form the complete set of nonisomorphic irreducible unipotent GLn(q)-modules. They are usually defined in terms of the intersection of the kernels of certain homomorphisms, and a generating element can be found for each Specht module. However, in all but a few cases, no explicit basis for S as a vector space has been found. The results presented here demonstrate a property of the Specht modules which may help to determine their bases; it severely restricts the form of any such basis vector.

On bases of some simple modules of symmetric groups and Hecke algebras

We consider simple modules for a Hecke algebra with a parameter of quantum characteristic e. Equivalently, we consider simple modules Dλ, labelled by e-restricted partitions λ of n, for a cyclotomic KLR algebra RnΛ0