Symmetries of tensors

Abstract

This thesis studies the symmetries of a fixed tensors by looking at certain group representations this tensor generates. We are particularly interested in the case that the tensor can be written as v 1 ⊗ · · · ⊗ v n , where the v i are selected from a complex vector space. The general linear group representation generated by such a tensor contains subtle information about the matroid M ( v ) of the vector configuration v 1 , · · ·, v n . To begin, we prove the basic results about representations of this form. We give two useful ways of describing these representations, one in terms of symmetric group representations, the other in terms of degeneracy loci over Grassmannians. Some of these results are equivalent to results that have appeared in the literature. When this is the case, we have given new, short proofs of the known results.

We will prove that the multiplicities of hook shaped irreducibles in the representation generated by v 1 ⊗ · · · ⊗ v n are determined by the no broken circuit complex of M ( v ). To do this, we prove a much stronger result about the structure of vector subspace of Sym V spanned by the products Π i∈S v i , where S ranges over all subsets of [ n ]. The result states that this vector space has a direct sum decomposition that determines the Tutte polynomial of M ( v ). We will use a combinatorial basis of the vector space generated the products of the linear forms to completely describe the representation generated by a decomposable tensor when its matroid M ( v ) has rank two.

Next we consider a representation of the symmetric group associated to every matroid. It is universal in the sense that if v 1 , . . . , v n is a realization of the matroid then the representation for the matroid provides non-trivial restrictions on the decomposition of the representation generated by the tensor product of the vectors. A complete combinatorial characterization of this representation is proven for parallel extensions of Schubert matroids. We also describe the multiplicity of hook shapes in this representation for all matroids.

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