Equivariant Steinberg summands

The Steinberg Representation

Introduction. Group representations occupy a sort of middle ground between abstract groups and transformation groups, i.e., groups acting in concrete ways as permutations of sets, homeomorphisms of topological spaces, diffeomorphisms of manifolds, etc. The requirement that the elements of a group act as linear operators on a vector space limits somewhat the complexity of the action without sacr...

Steinberg groups as amalgams

For any root system and any commutative ring, we give a relatively simple presentation of a group related to its Steinberg group St. This includes the case of infinite root systems used in Kac–Moody theory, for which the Steinberg group was defined by Tits and Morita–Rehmann. In most cases, our group equals St, giving a presentation with many advantages over the usual presentation of St. This e...

The Steinberg Wiring Problem

Plünnecke’s Inequality for Different Summands

The aim of this paper is to prove a general version of Plünnecke’s inequality. Namely, assume that for finite sets A, B1, . . . Bk we have information on the size of the sumsets A + Bi1 + · · · + Bil for all choices of indices i1, . . . il. Then we prove the existence of a non-empty subset X of A such that we have ‘good control’ over the size of the sumset X + B1 + · · · + Bk. As an application...

Indecomposable Summands of Foulkes Modules

In this paper we study the modular structure of the permutation module H ) of the symmetric group S2n acting on set partitions of a set of size 2n into n sets each of size 2, defined over a field of odd characteristic p. In particular we characterize the vertices of the indecomposable summands of H ) and fully describe all of its indecomposable summands that lie in blocks of p-weight at most tw...