Induced modules of support $\tau$-tilting modules and extending modules of semibricks over blocks of finite groups

TL;DR

This paper investigates how support τ-tilting modules and semibricks over blocks of group algebras behave under induction and extension, revealing conditions under which these modules are preserved across related blocks in finite group algebras.

Contribution

It establishes conditions for the preservation of support τ-tilting modules and semibricks under induction and extension between blocks of group algebras.

Findings

Induced modules of support τ-tilting modules are preserved under certain conditions.

Extending modules of semibricks maintain their properties across blocks.

Results connect module theory with block extension in finite groups.

Abstract

In this article we study support $\tau$-tilting modules, semibricks and more over blocks of group algebras. Let $k$ be an algebraically closed field of characteristic $p>0$, $\tilde{G}$ a finite group and $G$ a normal subgroup of $\tilde{G}$. Moreover, let $\tilde{B}$ be a block of $k\tilde{G}$ and $B$ a block of $kG$ covered by $\tilde{B}$. We show that, under certain conditions for the factor group $\tilde{G}/G$ and $B$, induced modules and extending modules of support $\tau$-tilting modules and semibricks over $B$ are also the ones over $\tilde{B}$, respectively.