A deformation of the Orlik-Solomon algebra

TL;DR

This paper introduces a deformation of the Orlik-Solomon algebra for matroids, demonstrating its algebraic properties and establishing conditions for Koszulness, with implications for algebraic and combinatorial structures.

Contribution

It defines a new deformation of the Orlik-Solomon algebra and proves it has a Groebner basis, showing its Hilbert series matches the original after homogenization, especially for supersolvable matroids.

Findings

Generators form a Groebner basis

Deformation and original have same Hilbert series after homogenization

For supersolvable matroids, the algebra is Koszul

Abstract

A deformation of the Orlik-Solomon algebra of a matroid M is defined as a quotient of the free associative algebra over a commutative ring R with 1. It is shown that the given generators form a Groebner basis and that after suitable homogenization the deformation and the Orlik-Solomon have the same Hilbert series as R-algebras. For supersolvable matroids, equivalently fiber type arrangements, there is a quadratic Groebner basis and hence the algebra is Koszul