University of Sydney Algebra Seminar

Richard Dipper (University of Stuttgart)

Friday 25 November, 12:05-12:55pm, Carslaw 273

Unipotent Specht modules of finite general linear groups and some conjectures of Higman, Lehrer and Isaacs

Let \(G=GL(n,q)\), \(q\) some prime power, and let \(U\) be the subgroup of unipotent lower triangular matrices of \(G\). We determine the precise \(U\)-module structure of the unipotent Specht type modules of \(G\) associated with two part partitions over finite fields of characteristic not dividing \(q\). This provides a new construction and a representation-theoretic interpretation for the standard basis of these modules which was obtained by Brandt, James, Lyle and myself in previous work. It is conceivable that this construction is accessible to generalisation to arbitrary partitions of \(n\). It is explained how this new approach relates to longstanding conjectures of Higman, Lehrer and a more recent one of Isaacs on the number of ordinary irreducible characters of \(U\).