Hochschild cohomology of symmetric groups and generating functions,II

TL;DR

This paper explores the Hochschild cohomology of symmetric groups, establishing formulas for their dimensions, demonstrating non-trivial first cohomology in certain blocks, and connecting these to generating functions and block centers.

Contribution

It provides new formulas relating Hochschild cohomology dimensions to block centers and confirms the non-vanishing of first cohomology in positive defect blocks.

Findings

First Hochschild cohomology of positive defect blocks is non-zero.

Derived formulas linking cohomology dimensions to block centers.

Connected generating functions of cohomology to block structures.

Abstract

We relate the generating functions of the dimensions of the Hochschild cohomology in any fixed degree of the symmetric groups with those of blocks of the symmetric groups. We show that the first Hochschild cohomology of a positive defect block of a symmetric group is non-zero, answering in the affirmative a question of the third author. To do this, we prove a formula expressing the dimension of degree one Hochschild cohomology as a sum of dimensions of centres of blocks of smaller symmetric groups. This in turn is a consequence of a general formula that makes more precise a theorem of our previous paper describing the generating functions for the dimensions of Hochschild cohomology of symmetric groups.