Representations Generated by Decomposable Tensors

We form the smallest general linear or symmetric group representation containing a decomposable tensor. The vectors forming this tensor give rise to a linear matroid which, to some extent, controls the irreducible decomposition of these representations. The multiplicities of hook shaped irreducibles are determined by the characteristic polynomial of the matroid, but the multiplicities of other shapes remain a mystery,in general. After some examples and discussion of multiplicities, some interesting open problems will be presented.

No familiarity with matroids will be assumed.