Representations and characters of finite groups

Let $p$ be an odd prime and let $n$ be a natural number. In this article we determine the irreducible constituents of the permutation module induced by the action of the symmetric group $\mathfrak{S}_n$ on the cosets of a Sylow $p$-subgroup $P_n$. As a consequence, we determine the number of irreducible representations of the corresponding Hecke algebra $\mathcal{H}(\mathfrak{S}_n, P_n, 1_{P_n})$.

We develop a package using the computer algebra system GAP for computing the decomposition of a representation $\rho$ of a finite group $G$ over $\mathbb{C}$ into irreducibles, as well as the corresponding decomposition of the centraliser ring of $\rho(G)$. Currently, the only open-source programs for decomposing representations are for non-zero characteristic fields. While methods for characteristic zero are known, there are no open-source computer programs that implement th...

We restrict irreducible characters of alternating groups of degree divisible by $p$ to their Sylow $p$-subgroups and study the number of linear constituents.

These are lecture notes for a one semester introductory course I gave at Indiana University. The goal was to make this exposition as clear and elementary as possible. A particular emphasis is given on examples involving SU(1,1). These notes are in part based on lectures given by my graduate advisor Wilfried Schmid at Harvard University and PQR2003 Euroschool in Brussels as well as other sources.

A set of invariants for a finite group is described. These arise naturally from Frobenius' early work on the group determinant and provide an answer to a question of Brauer. Whereas it is well known that the ordinary character table of a group does not determine the group uniquely, it is a consequence of the results presented here that a group is determined uniquely by its ``3-character'' table.

The representation dimension of a finite group $G$ is the minimal dimension of a faithful complex linear representation of $G$. We prove that the representation dimension of any finite group $G$ is at most $\sqrt{|G|}$ except if $G$ is a $2$-group with elementary abelian center of order $8$ and all irreducible characters of $G$ whose kernel does not contain $Z(G)$ are fully ramified with respect to $G/Z(G)$. We also obtain bounds for the representation dimension of quotients ...

These notes provide a concise introduction to the representation theory of reductive algebraic groups in positive characteristic, with an emphasis on Lusztig's character formula and geometric representation theory. They are based on the first author's notes from a lecture series delivered by the second author at the Simons Centre for Geometry and Physics in August 2019. We intend them to complement more detailed treatments.

Schur multiplier $M(G)$ of a finite group $G$ has been studied heavily. To proceed further to the study of projective (or spin) representations of $G$ and their characters (called spin characters), it is necessary to construct explicitly a representation group $R(G)$ of $G$, a certain central extension of $G$ by $M(G)$, since projective representations of $G$ correspond bijectively to linear representations of $R(G)$. We propose here a practical method to construct $R(G)$ by ...

The goal of these notes is to give a self-contained account of the representation theory of $GL_2$ and $SL_2$ over a finite field, and to give some indication of how the theory works for $GL_n$ over a finite field.

We describe an algorithm for splitting permutation representations of finite group over fields of characteristic zero into irreducible components. The algorithm is based on the fact that the components of the invariant inner product in invariant subspaces are operators of projection into these subspaces. An important element of the algorithm is the calculation of Gr\"obner bases of polynomial ideals. A preliminary implementation of the algorithm splits representations up to d...