James' Submodule Theorem and the Steinberg Module

James' submodule theorem is a fundamental result in the representation theory of the symmetric groups and the finite general linear groups. In this note we consider a version of that theorem for a general finite group with a split $BN$-pair. This gives rise to a distinguished composition factor of the Steinberg module, first described by Hiss via a somewhat different method. It is a major open problem to determine the dimension of this composition factor

Computing Kazhdan--Lusztig cells for unequal parameters

Following Lusztig, we consider a Coxeter group $W$ together with a weight function $L$. This gives rise to the pre-order relation $\leq_{L}$ and the corresponding partition of $W$ into left cells. We introduce an equivalence relation on weight functions such that, in particular, $\leq_{L}$ is constant on equivalent classes. We shall work this out explicitly for $W$ of type $F_4$ and check that several of Lusztig's conjectures concerning left cells with unequal parameters hold in this case, even for those parameters which do not admit a geometric interpretation. The proofs involve some explicit computations using {\sf CHEVIE}

Eigenvalues of real symmetric matrices

We present a proof of the existence of real eigenvalues of real symmetric matrices which does not rely on any limit or compactness arguments, but only uses the notions of "sup", "inf".Comment: 2 pages; appears in the Amer. Math. Monthly (2015