Mark Wildon

On types and classes of commuting matrices over finite fields

This paper addresses various questions about pairs of similarity classes of matrices which contain commuting elements. In the case of matrices over finite fields, we show that the problem of determining such pairs reduces to a question about nilpotent classes; this reduction makes use of class types in the sense of Steinberg and Green. We investigate the set of scalars that arise as determinants of elements of the centralizer algebra of a matrix, providing a complete description of this set in terms of the class type of the matrix. Several results are established concerning the commuting of nilpotent classes. Classes which are represented in the centralizer of every nilpotent matrix are classified--this result holds over any field. Nilpotent classes are parametrized by partitions; we find pairs of partitions whose corresponding nilpotent classes commute over some finite fields, but not over others. We conclude by classifying all pairs of classes, parametrized by two-part partitions, that commute. Our results on nilpotent classes complement work of Ko\v{s}ir and Oblak.

The probability that a pair of elements of a finite group are conjugate

Let

A combinatorial method for calculating the moments of Lévy area

G

ON THE DISTRIBUTION OF CONJUGACY CLASSES BETWEEN THE COSETS OF A FINITE GROUP IN A CYCLIC EXTENSION

be a finite group, and let

Surveys in Combinatorics 2013

\kappa(G)

Low-Dimensional Lie Algebras

be the probability that elements

Minimal and maximal constituents of twisted Foulkes characters

g

Searching for knights and spies

,

Commuting conjugacy classes: an application of Hall's marriage theorem to group theory

h\in G

On signed Young permutation modules and signed p-Kostka numbers

are conjugate, when