arXiv subjects

R. B. Howlett

Publications and source records attributed to R. B. Howlett.

3 recordsLinked to original sources

arXiv2014

Extraspecial towers and Weil representations

This paper was motivated by a remarkable group, the maximal subgroup $M=S_3\ltimes 2^{2+1}_{-}\ltimes3^{2+1}\ltimes2^{6+1}_{-}$ of the sporadic simple group ${\rm Fi}_{23}$, where $S_3$ is the symmetric group of degree 3, and $2^{2+1}_{-}$, $3^{2+1}$ and $2^{6+1}_{-}$ denote extraspecial groups. The representation $3^{2+1}\to{\rm GL}(3,\mathbb{F}_4)\to{\rm GL}(6,\mathbb{F}_2)$ extends (remarkably) to $S_3\ltimes 2^{2+1}_{-}\ltimes3^{2+1}$ and preserves a quadratic form (of minus type) which allows the construction of $M$. The paper describes certain (Weil) representations of extraspecial groups which extend, and preserve various forms. Incidentally, $M$ is a remarkable solvable group with derived length 10, and composition length 24.

math.RT↗

arXiv2014

Writing representations over minimal fields

The chief aim of this paper is to describe a procedure which, given a $d$-dimensional absolutely irreducible matrix representation of a finite group over a finite field $\mathbb{E}$, produces an equivalent representation such that all matrix entries lie in a subfield $\mathbb{F}$ of $\mathbb{E}$ which is as small as possible. The algorithm relies on a matrix version of Hilbert's Theorem 90, and is probabilistic with expected running time ${\rm O}(|\mathbb{E}:\mathbb{F}|d^3)$ when $|\mathbb{F}|$ is bounded. Using similar methods we then describe an algorithm which takes as input a prime number and a power-conjugate presentation for a finite soluble group, and as output produces a full set of absolutely irreducible representations of the group over fields whose characteristic is the specified prime, each representation being written over its minimal field.

math.RT↗

arXiv2005

Reflections in abstract Coxeter groups

Let $W$ be a Coxeter group and $r\in W$ a reflection. If the group of order 2 generated by $r$ is the intersection of all the maximal finite subgroups of $W$ that contain it, then any isomorphism from $W$ to a Coxeter group $W'$ must take $r$ to a reflection in $W'$. The aim of this paper is to show how to determine, by inspection of the Coxeter graph, the intersection of the maximal finite sugroups containing $r$. In particular we show that the condition above is satisfied whenever $W$ is infinite and irreducible, and has the property that all rank two parabolic subgroups are finite. So in this case all isomorphisms map reflections to reflections.

math.GR↗