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Brent Everitt

Publications and source records attributed to Brent Everitt.

At least 19 recordsLinked to original sources
arXiv2026

Clique complexes of groups and order sheaves

We study the clique complex $X_G$ of a finite group: a simplicial complex whose faces are in one-to-one correspondence with the elements of the group, and arise via a decomposition of group elements into $p$-elements for primes $p$. The $1$-skeleton of this complex then has the classical Gruenberg-Kegel graph $\Gamma_G$ as a quotient and a finite simple group is determined up to isomorphism by its clique complex. The clique complex can also be endowed with a combinatorial sheaf --- called the order sheaf --- and the resulting homology can be computed for an arbitrary simplicial complex in terms of the homology of the links of vertices. The homology of the clique complex with coefficients in this order sheaf is then computed for some special classes of groups, in particular those for which the centralizers of elements are nilpotent.

math.GR↗
arXiv2023

Homology of matching complexes and representations of symmetric groups

We compute the homology of the matching complex $M(\Gamma)$, where $\Gamma$ is the complete hypergraph on $n\geq 2$ vertices, and analyse the $S_n$-representations carried by this homology. These results are achieved using standard techniques in combinatorial topology, such as the theory of shellings. We then broaden the scope to the larger class of fibre-closed families of simplicial complexes and consider these through the lens of representation stability. This allows us to prove a number of results of an asymptotic nature, such as an analysis of the growth of Betti numbers and the kinds of irreducible $S_n$-representations that appear.

math.GR↗
arXiv2022

Sheaf homology of hyperplane arrangements, Boolean covers and exterior powers

We compute the sheaf homology of the intersection lattice of a hyperplane arrangement with coefficients in the graded exterior sheaf of the natural sheaf. This builds on the results of our previous paper, where this homology was computed for the natural sheaf, itself a generalisation of an old result of Lusztig. The computational machinery we develop in this paper is quite different though: sheaf homology is lifted to what we call Boolean covers, where we instead compute homology cellularly. A number of tools are given for the cellular homology of these Boolean covers, including a deletion-restriction long exact sequence.

math.AT↗
arXiv2021

The sympathetic sceptics guide to semigroup representations

This is an elementary introduction to the representation theory of finite semigroups. We illustrate the Clifford-Munn correspondence between the representations of a semigroup and the representations of its maximal subgroups. The emphasis throughout is on naturally occurring examples.

math.GR↗
arXiv2020

Deletion-restriction for sheaf homology of graded atomic lattices

We give a long exact sequence for the homology of a graded atomic lattice equipped with a sheaf of modules, in terms of the deleted and restricted lattices. This is then used to compute the homology of the arrangement lattice of a hyperplane arrangement equipped with the natural sheaf. This generalises an old result of Lusztig.

math.AT↗
arXiv2018

Galois Theory - a first course

These notes are a self-contained introduction to Galois theory, designed for the student who has done a first course in abstract algebra.

math.GR↗
arXiv2015

Cellular cohomology of posets with local coefficients

We describe a "cellular" approach to the computation of the cohomology of a poset with coefficients in a presheaf. A cellular cochain complex is constructed, described explicitly and shown to compute the cohomology under certain circumstances. The descriptions are refined further for certain classes of posets including the cell posets of regular CW-complexes and geometric lattices.

math.AT↗
arXiv2014

Metrical Diophantine approximation for quaternions

Analogues of the classical theorems of Khintchine, Jarnik and Jarnik-Besicovitch in the metrical theory of Diophantine approximation are established for quaternions by applying results on the measure of general `lim sup' sets.

math.NT↗
arXiv2013

The homotopy theory of Khovanov homology

We show that the unnormalised Khovanov homology of an oriented link can be identified with the derived functors of the inverse limit. This leads to a homotopy theoretic interpretation of Khovanov homology.

math.GT↗
arXiv2013

A (very short) introduction to buildings

These lectures are an informal elementary introduction to buildings. They are written for, and by, a non-expert. The aim is to get to the definition of a building and feel that it is an entirely natural thing. To maintain the lecture style examples have replaced proofs. The notes at the end indicate where these proofs can be found. Most of what we say has its origins in the work of Jacques Tits, and our account borrows heavily from the books of Abramenko and Brown and of Ronan. Lecture 1 illustrates all the essential features of a building in the context of an example, but without mentioning any building terminology. In principle anyone could read this. Lectures 2-4 firm-up and generalize these specifics: Coxeter groups appear in Lecture 2, chambers systems in Lecture 3 and the definition of a building in Lecture 4. Lecture 5 addresses where buildings come from by describing the first important example: the spherical building of an algebraic group.

math.GR↗
arXiv2012

Khovanov homotopy types and the Dold-Thom functor

We show that the spectrum constructed by Everitt and Turner as a possible Khovanov homotopy type is a product of Eilenberg-MacLane spaces and is thus determined by Khovanov homology. By using the Dold-Thom functor it can therefore be obtained from the Khovanov homotopy type constructed by Lipshitz and Sarkar.

math.GT↗
arXiv2011

Partial mirror symmetry, lattice presentations and algebraic monoids

This is the second in a series of papers that develops the theory of reflection monoids, motivated by the theory of reflection groups. Reflection monoids were first introduced in arXiv:0812.2789. In this paper we study their presentations as abstract monoids. Along the way we also find general presentations for certain join-semilattices (as monoids under join) which we interpret for two special classes of examples: the face lattices of convex polytopes and the geometric lattices, particularly the intersection lattices of hyperplane arrangements. Another spin-off is a general presentation for the Renner monoid of an algebraic monoid, which we illustrate in the special case of the "classical" algebraic monoids.

math.GR↗
arXiv2010

Bundles of coloured posets and a Leray-Serre spectral sequence for Khovanov homology

The decorated hypercube found in the construction of Khovanov homology for links is an example of a Boolean lattice equipped with a presheaf of modules. One can place this in a wider setting as an example of a coloured poset, that is to say a poset with a unique maximal element equipped with a presheaf of modules. In this paper we initiate the study of a bundle theory for coloured posets, producing for a certain class of base posets a Leray-Serre type spectral sequence. We then show how this theory finds application in Khovanov homology by producing a new spectral sequence converging to the Khovanov homology of a given link.

math.GT↗
arXiv2009

The Combinatorial Topology of Groups

This is the first installment of a book on combinatorial and geometric group theory from the topological point of view. This is a classical subject. The installment contains Chapters 1, 3 and 4, and there are nine chapters in total: 1. Combinatorial Complexes 2. Topological Invariants 3. Coverings 4. Galois Theory 5. Generators and Relations 6. The Topological Dictionary 7. Amalgams 8. The Arboreal Dictionary 9. Ends.

math.GR↗
arXiv2008

Partial symmetry, reflection monoids and Coxeter groups

This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection monoids, introduce new examples, and determine their orders.

math.GR↗
arXiv2008

Weyl groups, lattices and geometric manifolds

By studying the action of the Weyl group of a simple Lie algebra on its root lattice, we construct torsion free subgroups of small and explicitly determined index in a large infinite class of Coxeter groups. One spin-off is the construction of hyperbolic manifolds of very small volume in up to 8 dimensions.

math.GT↗