Products of Linear Forms and Tutte Polynomials

TL;DR

This paper establishes a connection between the Hilbert series of a subspace spanned by products of vectors from a sequence and the Tutte polynomial, unifying several previous results in combinatorics and algebra.

Contribution

It proves that the Hilbert series of the space spanned by products of vectors corresponds exactly to a Tutte polynomial evaluation, providing a new algebraic-combinatorial link.

Findings

Hilbert series equals Tutte polynomial evaluation T(elta;1+x,y)

Unifies previous results by Ardila, Postnikov, Orlik, Terao, and Wagner

Provides a new algebraic perspective on Tutte polynomials

Abstract

Let \Delta be a finite sequence of n vectors from a vector space over any field. We consider the subspace of \operatorname{Sym}(V) spanned by \prod_{v \in S} v, where S is a subsequence of \Delta. A result of Orlik and Terao provides a doubly indexed direct sum of this space. The main theorem is that the resulting Hilbert series is the Tutte polynomial evaluation T(\Delta;1+x,y). Results of Ardila and Postnikov, Orlik and Terao, Terao, and Wagner are obtained as corollaries.