On invariant subrings of Orlik--Solomon and Varchenko--Gel'fand algebras in type A

TL;DR

This paper presents explicit generators and relations for the invariant subrings of Orlik--Solomon and Varchenko--Gel'fand algebras in type A, linking algebraic structures to the topology of mixed configuration spaces.

Contribution

It provides the first simple, explicit presentations of these invariant subrings for type A reflection arrangements, enhancing understanding of their algebraic and topological properties.

Findings

Explicit generators and relations for invariant subrings

Refined descriptions of invariant rings including dimension and presentations

Connection to cohomology of mixed configuration spaces

Abstract

We provide simple presentations in terms of generators and relations for the invariant subring of both the Orlik--Solomon algebra and Varchenko--Gel'fand ring of the type $A_n$ reflection arrangement acted upon by the type $A_{n-1}$ reflection group. This may be interpreted as a presentation for the cohomology of the ``mixed configuration space" of $n$ red points and one blue point. We provide increasingly refined descriptions of the invariant ring starting with the total dimension and ending with the simple presentation in terms of generators and relations.