Finite groups acting linearly: Hochschild cohomology and the cup product

TL;DR

This paper explores the Hochschild cohomology of skew group algebras formed by finite groups acting linearly on complex vector spaces, revealing the algebraic structure of the cohomology and its relation to group actions and combinatorics.

Contribution

It demonstrates that the cup product in Hochschild cohomology aligns with a smash product and describes the algebraic structure using a partial order related to fixed point codimensions.

Findings

Cup product coincides with a smash product.

Hochschild cohomology structure expressed via a partial order.

Explicit description for Coxeter and complex reflection groups.

Abstract

When a finite group acts linearly on a complex vector space, the natural semi-direct product of the group and the polynomial ring over the space forms a skew group algebra. This algebra plays the role of the coordinate ring of the resulting orbifold and serves as a substitute for the ring of invariant polynomials from the viewpoint of geometry and physics. Its Hochschild cohomology predicts various Hecke algebras and deformations of the orbifold. In this article, we investigate the ring structure of the Hochschild cohomology of the skew group algebra. We show that the cup product coincides with a natural smash product, transferring the cohomology of a group action into a group action on cohomology. We express the algebraic structure of Hochschild cohomology in terms of a partial order on the group (modulo the kernel of the action). This partial order arises after assigning to each group element the codimension of its fixed point space. We describe the algebraic structure for Coxeter groups, where this partial order is given by the reflection length function; a similar combinatorial description holds for an infinite family of complex reflection groups.