Kaveh Mousavand, OIST

Title: Some applications of bricks in classical and modern problems in representation theory

Abstract: Bricks (also known as Schur representations) form a special subfamily of indecomposable modules, and they are used in the algebraic and geometric study of representation theory of algebras. We start by looking at some classical results on bricks, including a characterization of locally representation-directed algebras (due to Dräxler). Then, we consider some new directions of research in which bricks have played crucial roles. More specifically, we briefly recall an elegant correspondence between bricks and indecomposable \(\tau\)-rigid-modules (due to Demonet-Iyama-Jasso), which has many applications in \(\tau\)-tilting theory. We use the notion of \(\tau\)-rigidity to give a new characterization of locally representation-directed algebras, and to further generalize this family. If time permits, we also report on some new results on an open conjecture (so-called the 2nd brick-Brauer-Thrall conjecture) which I posed in 2019. Part of this talk is based on my recent joint work with Charles Paquette.