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Matthew Fayers

Publications and source records attributed to Matthew Fayers.

34 records · Page 2Linked to original sources
arXiv2015

Dyck tilings and the homogeneous Garnir relations for graded Specht modules

Suppose $λ$ and $μ$ are integer partitions with $λ\supseteqμ$. Kenyon and Wilson have introduced the notion of a cover-inclusive Dyck tiling of the skew Young diagram $λ\setminusμ$, which has applications in the study of double-dimer models. We examine these tilings in more detail, giving various equivalent conditions and then proving a recurrence which we use to show that the entries of the transition matrix between two bases for a certain permutation module for the symmetric group are given by counting cover-inclusive Dyck tilings. We go on to consider the inverse of this matrix, showing that its entries are determined by what we call cover-expansive Dyck tilings. The fact that these two matrices are mutual inverses allows us to recover the main result of Kenyon and Wilson. We then discuss the connections with recent results of Kim et al, who give, a simple expression for the sum, over all $μ$, of the number of cover-inclusive Dyck tilings of $λ\setminusμ$. Our results provide a new proof of this result. Finally, we show how to use our results to obtain simpler expressions for the homogeneous Garnir relations for the universal Specht modules introduced by Kleshchev, Mathas and Ram for the cyclotomic quiver Hecke algebras.

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arXiv2014

Generalised column removal for graded homomorphisms between Specht modules

Let $n$ be a positive integer, and let $\mathscr{H}_n$ denote the affine KLR algebra in type A. Kleshchev, Mathas and Ram have given a homogeneous presentation for graded column Specht modules $\operatorname{S}_λ$ for $\mathscr{H}_n$. Given two multipartitions $λ$ and $μ$, we define the notion of a \emph{dominated} homomorphism $\operatorname{S}_λ\to\operatorname{S}_μ$, and use the KMR presentation to prove a generalised column removal theorem for graded dominated homomorphisms between Specht modules. In the process, we prove some useful properties of $\mathscr{H}_n$-homomorphisms between Specht modules which lead to an immediate corollary that, subject to a few demonstrably necessary conditions, every homomorphism $\operatorname{S}_λ\to\operatorname{S}_μ$ is dominated, and in particular $\operatorname{Hom}_{\mathscr{H}_n}(\operatorname{S}_λ,\operatorname{S}_μ)=0$ unless $λ$ dominates $μ$. Brundan and Kleshchev show that certain cyclotomic quotients of $\mathscr{H}_n$ are isomorphic to (degenerate) cyclotomic Hecke algebras of type A. Via this isomorphism, our results can be seen as a broad generalisation of the column removal results of Fayers and Lyle and of Lyle and Mathas; generalising both into arbitrary level and into the graded setting.

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arXiv2014

The irreducible representations of the alternating group which remain irreducible in characteristic p

Let p be an odd prime, and A_n the alternating group of degree n. We determine which ordinary irreducible representations of A_n remain irreducible in characteristic p, verifying the author's conjecture from [Represent. Theory 14, 601-626]. Given the preparatory work done in [op. cit.], our task is to determine which self-conjugate partitions label Specht modules for the symmetric group in characteristic p having exactly two composition factors. This is accomplished through the use of the Robinson-Brundan-Kleshchev 'i-restriction' functors, together with known results on decomposition numbers for the symmetric group and additional results on the Mullineux map and homomorphisms between Specht modules.

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arXiv2014

A generalisation of core partitions

Suppose $s$ and $t$ are coprime natural numbers. A theorem of Olsson says that the $t$-core of an $s$-core partition is again an $s$-core. We generalise this theorem, showing that the $s$-weight of the $t$-core of a partition $λ$ is at most the $s$-weight of $λ$. Then we consider the set $\mathcal C_{s:t}$ of partitions for which equality holds, which we call $[s{:}t]$-cores; this set has interesting structure, and we expect that it will be the subject of future study. We show that the set of $[s{:}t]$-cores is a union of finitely many orbits for an action of a Coxeter group of type $\tilde A_{s-1}\times\tilde A_{t-1}$ on the set of partitions. We also consider the problem of constructing an $[s{:}t]$-core with specified $s$-core and $t$-core.

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arXiv2013

Some new decomposable Specht modules

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.

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arXiv2012

An algorithm for semistandardising homomorphisms

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 $\mathscr{H}_n$ of the symmetric group $\mathfrak{S}_n$. We give an explicit fast algorithm for expressing a tableau homomorphism $\hatϕ_A:S^μ\to M^λ$ as a linear combination of semistandard homomorphisms. Along the way we provide a utility result related to removing rows from tableaux.

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arXiv2011

Irreducible Specht modules for Iwahori-Hecke algebras of type B

We consider the problem of classifying irreducible Specht modules for the Iwahori-Hecke algebra of type B with parameters Q,q. We solve this problem completely in the case where q is not a root of unity, and in the case q=-1 we reduce the problem to the corresponding problem in type A.

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arXiv2011

The t-core of an s-core

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.

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arXiv2010

On the irreducible Specht modules for Iwahori--Hecke algebras of type A with $q=-1$

Let $p$ be a prime and $\mathbb{F}$ a field of characteristic $p$, and let $\mathcal{H}_n$ denote the Iwahori--Hecke algebra of the symmetric group $\mathfrak{S}_n$ over $\mathbb{F}$ 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 $S^λ$ for $\mathcal{H}_{|\la|}$ is irreducible.

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arXiv2009

General runner removal and the Mullineux map

We prove a new `runner removal theorem' for $q$-decomposition numbers of the level 1 Fock space of type $A^{(1)}_{e-1}$, generalising earlier theorems of James--Mathas and the author. By combining this with another theorem relating to the Mullineux map, we show that the problem of finding all $q$-decomposition numbers indexed by partitions of a given weight is a finite computation.

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arXiv2008

Regularisation and the Mullineux map

We classify the pairs of conjugate partitions whose regularisations are images of each other under the Mullineux map. This classification proves a conjecture of Lyle, answering a question of Bessenrodt, Olsson and Xu.

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arXiv2007

Core blocks of Ariki-Koike algebras II: the weight of a core block

We study combinatorial blocks of multipartitions, exploring further the notions of weight, hub and core block introduced by the author in earlier papers. We answer the question of which pairs (w,theta) occur as the weight and hub of a block, and we examine the action of the affine Weyl group on the set of blocks.

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