Quasi-hereditary structure of twisted split category algebras revisited

TL;DR

This paper revisits the quasi-hereditary structure of twisted split category algebras, introducing a new coarser partial order that simplifies the proof of quasi-heredity and extends results to related algebras like biset functors and the Brauer algebra.

Contribution

It introduces a new partial order on irreducible modules that simplifies the proof of quasi-heredity and extends these results to condensed algebras related to biset functors.

Findings

The new partial order $
lhd$ is explicitly evaluated for biset functors and the Brauer algebra.

The coarser order $
lhd$ allows passing quasi-heredity to the condensed algebra $
A
varepsilon
A
varepsilon$.

The partial order restricts possible composition factors of standard modules.

Abstract

Let $k$ be a field of characteristic $0$, let $\mathsf{C}$ be a finite split category, let $\alpha$ be a 2-cocycle of $\mathsf{C}$ with values in the multiplicative group of $k$, and consider the resulting twisted category algebra $A:=k_\alpha\mathsf{C}$. Several interesting algebras arise that way, for instance, the Brauer algebra. Moreover, the category of biset functors over $k$ is equivalent to a module category over a condensed algebra $\varepsilon A\varepsilon$, for an idempotent $\varepsilon$ of $A$. In [2] the authors proved that $A$ is quasi-hereditary (with respect to an explicit partial order $\le$ on the set of irreducible modules), and standard modules were given explicitly. Here, we improve the partial order $\le$ by introducing a coarser order $\unlhd$ leading to the same results on $A$, but which allows to pass the quasi-heredity result to the condensed algebra $\varepsilon A\varepsilon$ describing biset functors, thereby giving a different proof of a quasi-heredity result of Webb, see [26]. The new partial order $\unlhd$ has not been considered before, even in the special cases, and we evaluate it explicitly for the case of biset functors and the Brauer algebra. It also puts further restrictions on the possible composition factors of standard modules.