On the graded center of the stable category of a finite $p$-group

TL;DR

This paper demonstrates that the graded center of the stable module category of a finite $p$-group has infinite dimension in all degrees for groups of rank at least 2, revealing new properties of symmetric algebras.

Contribution

It establishes the infinite dimensionality of the graded center in all degrees for certain $p$-groups, answering a previously open question.

Findings

The graded center has infinite dimension in each odd degree for $p$-groups of rank ≥ 2.

For $p=2$, the graded center also has infinite dimension in each even degree.

Provides examples of symmetric algebras with non-finite-dimensional degree zero center.

Abstract

We show that for any finite $p$-group $P$ of rank at least 2 and any algebraically closed field $k$ of characteristic $p$ the graded center $Z^*(\modbar(kP))$ of the stable module category of finite-dimensional $kP$-modules has infinite dimension in each odd degree, and if $p=2$ also in each even degree. In particular, this provides examples of symmetric algebras $A$ for which $Z^0(\modbar(A))$ is not finite-dimensional, answering a question raised in [10]