Matthew Fayers

Decomposition numbers for weight three blocks of symmetric groups and Iwahori–Hecke algebras

Let F be a field, q a non-zero element of F and H n = H F,q (G n ) the Iwahori-Hecke algebra of the symmetric group G 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 G n of defect 3 are all at most 1.

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.

An LLT-type algorithm for computing higher-level canonical bases

Abstract We give a fast algorithm for computing the canonical basis of an irreducible highest-weight module for U q ( s l e ) , generalising the LLT algorithm.

An algorithm for semistandardising homomorphisms

Abstract Suppose μ is a partition of n and λ a composition of n, and let S μ , M λ denote the Specht module and permutation module defined by Dipper and James for the Iwahori–Hecke algebra H n of the symmetric group W n . We give an explicit fast algorithm for expressing a tableau homomorphism ϕ ˆ A : S μ → M λ as a linear combination of semistandard homomorphisms. Along the way we provide a utility result related to removing rows from tableaux.

Some new decomposable Specht modules

Abstract We present (with proof ) a new family of decomposable Specht modules for the symmetric group in characteristic 2. These Specht modules are labelled by partitions of the form ( a , 3 , 1 b ) , and are the first new examples found for thirty years. Our method of proof is to exhibit summands isomorphic to irreducible Specht modules, by constructing explicit homomorphisms between Specht modules.

Weight two blocks of Iwahori–Hecke algebras in characteristic two

We study blocks of the Iwahori–Hecke algebraHq(Sn) of weight two over a field of characteristic two. Using techniques and notation developed by Scopes, Richards, Chuang and Tan for the case of odd characteristic, we find the decomposition numbers and classify extensions between simple modules for these blocks.

Generalised column removal for graded homomorphisms between Specht modules

Let n be a positive integer, and let

A generalisation of core partitions

\mathscr {H}_n