Search arXiv⌕ Search

arXiv subjects

Matthew Fayers

Publications and source records attributed to Matthew Fayers.

At least 19 recordsLinked to original sources
arXiv2026

Spin characters of symmetric and alternating groups which are proportional in characteristic 3

Let $G$ be a finite group and $p$ a prime. It is interesting to determine when two ordinary irreducible representations of $G$ have the same $p$-modular reduction; this is the same as saying that the corresponding rows of the decomposition matrix are equal, or that the characters of the two representations agree on $p$-regular conjugacy classes. In fact we consider the more general problem of asking when two rows of the decomposition matrix are proportional. In the case where $G$ is a double cover of the alternating or symmetric group, this problem has been solved except when $p=3$. Here we resolve the missing case for spin characters (i.e. characters which are not lifted from the covered group), which completely solves the problem for the double cover of the symmetric group. There are surprising parallels to our solution to the corresponding problem for $p=2$.

math.RT↗
arXiv2026

Monoids generated by projections

We define and explore semireflection monoids on a finite-dimensional vector space. These are monoids generated by semireflections: linear maps fixing a subspace of codimension 1. We mostly focus on the case of projection monoids (where the generating semireflections are non-invertible). After exploring some general theory, we give some important examples, and give classification results for projection monoids on $\mathbb C^2$ and $\mathbb R^3$. We then briefly introduce affine projection monoids.

math.GR↗
arXiv2025

Ribbon blocks for centraliser algebras of symmetric groups

Suppose $l,m$ are natural numbers with $l\le m$, and $\mathbb{F}$ a field of characteristic $p$, and let $\mathcal{C}_{l,m}^{\mathbb{F}}$ denote the centraliser of the group algebra $\mathbb{F}S_l$ inside $\mathbb{F}S_m$. Ellers and Murray give a conjectured classification of the blocks of $\mathcal{C}_{l,m}^{\mathbb{F}}$, in terms of the $p$-blocks of $S_l$ and $S_m$. We prove this conjecture for a family of blocks that we call ribbon blocks and belt blocks. These are the blocks containing Specht modules labelled by skew partitions having no repeated entries in their $p$-content.

math.RT↗
arXiv2024

Spin characters of the symmetric group which are proportional to linear characters in characteristic 2

For a finite group, it is interesting to determine when two ordinary irreducible representations have the same $p$-modular reduction; that is, when two rows of the decomposition matrix in characteristic $p$ are equal, or equivalently when the corresponding $p$-modular Brauer characters are the same. We complete this task for the double covers of the symmetric group when $p=2$, by determining when the $2$-modular reduction of an irreducible spin representation coincides with a $2$-modular Specht module. In fact, we obtain a more general result: we determine when an irreducible spin representation has $2$-modular Brauer character proportional to that of a Specht module. In the course of the proof, we use induction and restriction functors to construct a function on generalised characters which has the effect of swapping runners in abacus displays for the labelling partitions.

math.RT↗
arXiv2023

Representations of symmetric and alternating groups and their double covers that remain irreducible modulo every prime

We classify globally irreducible representations of alternating groups and double covers of symmetric and alternating groups. In order to achieve this classification we also completely characterise irreducible representations of such groups which reduce almost homogeneously in every characteristic. This also allows us to classify irreducible representations that remain irreducible in every characteristic as well as irreducible representations of these groups that can appear as composition factors of globally irreducible representations of groups containing $\mathfrak{A}_n$ or $\hat{\mathfrak{A}}_n$ as normal subgroups. In particular we show that, apart from finitely many exceptions, for any of these questions such representations are either $1$-dimensional or basic spin representations.

math.RT↗
arXiv2023

Decomposition numbers for abelian defect RoCK blocks of double covers of symmetric groups

We calculate the (super)decomposition matrix for a RoCK block of a double cover of the symmetric group with abelian defect, verifying a conjecture of the first author. To do this, we exploit a theorem of the second author and Livesey that a RoCK block $\mathcal B^{\rho,d}$ is Morita superequivalent to a wreath superproduct of a certain quiver (super)algebra with the symmetric group $\mathfrak S_d$. We develop the representation theory of this wreath superproduct to compute its Cartan invariants. We then directly construct projective characters for $\mathcal B^{\rho,d}$ to calculate its decomposition matrix up to a triangular adjustment, and show that this adjustment is trivial by comparing Cartan invariants.

math.RT↗
arXiv2022

Irreducible spin representations of symmetric and alternating groups which remain irreducible in characteristic 3

For any finite group $G$ and any prime $p$ one can ask which ordinary irreducible representations remain irreducible in characteristic $p$, or more generally, which representations remain homogeneous in characteristic $p$. In this paper we address this question for $p=3$ when $G$ is a proper double cover of the symmetric or alternating group. We obtain a classification except in the case of a certain family of partitions relating to spin RoCK blocks. Our techniques involve induction and restriction, degree calculations, decomposing projective characters and recent results of Kleshchev and Livesey on spin RoCK blocks.

math.RT↗
arXiv2022

Comparing Fock spaces in types $A^{(1)}$ and $A^{(2)}$

We compare the canonical bases of level-$1$ quantised Fock spaces in affine types $A^{(1)}$ and $A^{(2)}$, showing how to derive the canonical basis in type $A^{(2)}_{2n}$ from the the canonical basis in type $A^{(1)}_n$ in certain weight spaces. In particular, we derive an explicit formula for the canonical basis in extremal weight spaces, which correspond to RoCK blocks of double covers of symmetric groups. In a forthcoming paper with Kleshchev and Morotti we will use this formula to find the decomposition numbers for RoCK blocks of double covers with abelian defect.

math.RT↗
arXiv2021

Crystals, regularisation and the Mullineux map

The Mullineux map is a combinatorial function on partitions which describes the effect of tensoring a simple module for the symmetric group in characteristic $p$ with the one-dimensional sign representation. It can also be interpreted as an isomorphism between crystal graphs for $\widehat{\mathfrak{sl}}_p$. We give a new combinatorial description of the Mullineux map by expressing this crystal isomorphism as a composition of isomorphisms between different crystals. These isomorphisms are defined in terms of new generalised regularisation maps introduced by Millan Berdasco. We then given two applications of our new realisation of the Mullineux map, by providing purely combinatorial proofs of a conjecture of Lyle relating the Mullineux map with regularisation, and a theorem of Paget describing the Mullineux map in RoCK blocks of symmetric groups.

math.RT↗
arXiv2020

Minimal partitions with a given $s$-core and $t$-core

Suppose $s$ and $t$ are coprime positive integers, and let $\sigma$ be an $s$-core partition and $\tau$ a $t$-core partition. In this paper we consider the set $\mathcal P_{\sigma,\tau}(n)$ of partitions of $n$ with $s$-core $\sigma$ and $t$-core $\tau$. We find the smallest $n$ for which this set is non-empty, and show that for this value of $n$ the partitions in $\mathcal P_{\sigma,\tau}(n)$ (which we call $(\sigma,\tau)$-minimal partitions) are in bijection with a certain class of $(0,1)$-matrices with $s$ rows and $t$ columns. We then use these results in considering conjugate partitions: we determine exactly when the set $\mathcal P_{\sigma,\tau}(n)$ consists of a conjugate pair of partitions, and when $\mathcal P_{\sigma,\tau}(n)$ contains a unique self-conjugate partition.

math.CO↗
arXiv2020

Irreducible projective representations of the alternating group which remain irreducible in characteristic 2

For any finite group G it is an interesting question to ask which ordinary irreducible representations of G remain irreducible in a given characteristic p. We answer this question for p=2 when G is the proper double cover of the alternating group. As a key ingredient in the proof, we prove a formula for the decomposition numbers in Rouquier blocks of double covers of symmetric groups, in terms of Schur P-functions.

math.RT↗
arXiv2019

Defect 2 spin blocks of symmetric groups and canonical basis coefficients

This paper addresses the decomposition number problem for spin representations of symmetric groups in odd characteristic. Our main aim is to find a combinatorial formula for decomposition numbers in blocks of defect $2$, analogous to Richards's formula for defect $2$ blocks of symmetric groups. By developing a suitable analogue of the combinatorics used by Richards, we find a formula for the corresponding "$q$-decomposition numbers", i.e.\ the canonical basis coefficients in the level-$1$ $q$-deformed Fock space of type $A^{(2)}_{2n}$; a special case of a conjecture of Leclerc and Thibon asserts that these coefficients yield the spin decomposition numbers in characteristic $2n+1$. Along the way, we prove some general results on $q$-decomposition numbers. This paper represents the first substantial progress on canonical bases in type $A^{(2)}_{2n}$.

math.RT↗
arXiv2019

A note on Kostka numbers

We give an elementary proof of a well-known result on Kostka numbers, following a question from Mark Wildon on MathOverflow. Namely, we show that given partitions $\lambda,\mu,\nu$ of $n$ with $\mu\trianglerighteq\nu$, we have $K_{\lambda\nu}\geqslant K_{\lambda\mu}$.

math.CO↗
arXiv2018

2-chains: an interesting family of posets

We introduce a new family of finite posets which we call 2-chains. These first arose in the study of 0-Hecke algebras, but they admit a variety of different characterisations. We give these characterisations, prove that they are equivalent and derive some numerical results concerning 2-chains.

math.CO↗
arXiv2017

Simultaneous core multipartitions

We initiate the study of simultaneous core multipartitions, generalising simultaneous core partitions, which have been studied extensively in the recent literature. Given a multipartition datum (s|c), which consists of a non-negative integer s and an l-tuple c of integers, we introduce the notion of an (s|c)-core multipartition. Given an arbitrary set of multicore data, we give necessary and sufficient conditions for the corresponding set of simultaneous core multipartitions to be finite. We then study the special case of simultaneous core bipartitions, giving exact enumerative results in some special subcases.

math.CO↗
arXiv2016

Irreducible projective representations of the symmetric group which remain irreducible in characteristic $2$

For any finite group $G$ and any prime $p$ one can ask which ordinary irreducible representations remain irreducible in characteristic $p$. We answer this question for $p=2$ when $G$ is a proper double cover of the symmetric group. Our techniques involve constructing part of the decomposition matrix for a Rouquier block of a double cover, restricting to subgroups using the Brundan--Kleshchev modular branching rules and comparing the dimensions of irreducible representations via the bar-length formula.

math.RT↗
arXiv2015

$(s,t)$-cores: a weighted version of Armstrong's conjecture

The study of core partitions has been very active in recent years, with the study of $(s,t)$-cores - partitions which are both $s$- and $t$-cores - playing a prominent role. A conjecture of Armstrong, proved recently by Johnson, says that the average size of an $(s,t)$-core, when $s$ and $t$ are coprime positive integers, is $\frac1{24}(s-1)(t-1)(s+t-1)$. Armstrong also conjectured that the same formula gives the average size of a self-conjugate $(s,t)$-core; this was proved by Chen, Huang and Wang. In the present paper, we develop the ideas from the author's paper [J. Combin. Theory Ser. A 118 (2011) 1525-1539] studying actions of affine symmetric groups on the set of $s$-cores in order to give variants of Armstrong's conjectures in which each $(s,t)$-core is weighted by the reciprocal of the order of its stabiliser under a certain group action. Informally, this weighted average gives the expected size of the $t$-core of a random $s$-core.

math.CO↗