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

Trevor Karn

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.

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

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 reflection arrangement acted upon by the type reflection group. This may be interpreted as a presentation for the cohomology of the ``mixed configuration space" of 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.

Paper Structure

This paper contains 2 sections, 2 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Background

Key Result

Theorem 1

For $n \geq 2$, $\overline{\mathsf{VG}}_{n}\cong \mathbb{Q}[z]/\langle z^{n+1}\rangle$.

Theorems & Definitions (8)

  • Theorem 1
  • Example 1
  • Definition 2
  • Theorem 2
  • Definition 3
  • Remark 4
  • Definition 5
  • Example 6