Hochschild cohomology of polynomial representations of $GL_2(\bar{\mathbb{F}}_p)$

TL;DR

This paper explicitly computes the Hochschild cohomology algebras of certain polynomial representation blocks of GL_2 over algebraically closed fields of characteristic p, revealing their structure and multiplication rules.

Contribution

It provides the first explicit description of Hochschild cohomology algebras for these specific polynomial representation blocks of GL_2 in characteristic p.

Findings

Hochschild cohomology algebras are finite-dimensional for these blocks.

Explicit bases and multiplication rules are established.

Results apply to blocks with a number of simple modules as a power of p.

Abstract

We compute the Hochschild cohomology algebras of Ringel-self-dual blocks of polynomial representations of $\GL_2$ over an algebraically closed field of characteristic $p>2$, that is, of any block whose number of simple modules is a power of $p$. These algebras are finite-dimensional and we provide an explicit description of their bases and multiplications.