Two point stabilisers of partition actions of linear groups

Abstract

A partition of a kl-dimensional vector space V is a set of l subspaces each of dimension k such that their direct sum is the original space V. In this paper we show that, unless l = 2 , the action of a group L ¯ such that PSL ( V ) ▪ L ¯ ⩽ PGL ( V ) on the set of partitions of V into l subspaces of dimension k is base two: there exist two partitions V and W such that L ¯ V , W = 1 . We will also show that, given any finite group G, there exist k, l and partitions V , W such that L ¯ V , W ≅ G . These results complement work the author has done with partition actions of the symmetric groups.