Normal subgroups and support $\tau$-tilting modules

TL;DR

This paper explores the relationship between support τ-tilting modules over a finite group and its normal subgroup, establishing conditions for their correspondence and invariance, with applications to group actions and block theory.

Contribution

It provides new criteria for support τ-tilting modules to be preserved under subgroup restriction and shows their invariance under group actions, extending the understanding of module structures in representation theory.

Findings

Support τ-tilting modules over the group algebra relate to those over the subgroup.

Conditions for support τ-tilting modules to satisfy properties are characterized.

The set of invariant support τ-tilting modules forms a partially ordered set isomorphic to that over the larger group.

Abstract

Let $\tilde{G}$ be a finite group, $G$ a normal subgroup of $\tilde{G}$ and $k$ an algebraically closed field of characteristic $p>0$. The first main result in this paper is to show that support $\tau$-tilting $k\tilde{G}$-modules satisfying some properties are support $\tau$-tilting modules as $kG$-modules too. As the second main result, we give equivalent conditions for support $\tau$-tilting $k\tilde{G}$-modules to satisfy the above properties, and show that the set of the support $\tau$-tilting $k\tilde{G}$-modules with the properties is isomorphic to the set of $\tilde{G}$-invariant support $\tau$-tilting $kG$-modules as partially ordered sets. As an application, we show that the set of $\tilde{G}$-invariant support $\tau$-tilting $kG$-modules is isomorphic to the set of support $\tau$-tilting $k\tilde{G}$-modules in the case that the index $G$ in $\tilde{G}$ is a $p$-power. As a further application, we give a feature of vertices of indecomposable $\tau$-rigid $k\tilde{G}$-modules. Finally, we give the block versions of the above results.