On the bricks (Schur representations) of finite dimensional algebras

Abstract: This manuscript treats the diverse applications of bricks within modern representation theory and several related domains, and reviews the recent developments and new results on bricks (a.k.a Schur representations). The current survey is an extended version of a mini-course by the second-named author, delivered in the research school on ``New Developments in Representation Theory of Algebras", held in November of 2024, at Okinawa Institute of Science and Technology (OIST), Japan. The review is mainly oriented towards the direction of research developed by the authors, which has evolved around the algebraic and geometric properties of bricks. More specifically, we discuss the emergence of bricks in $\tau$-tilting theory, torsion theory, geometric representation theory and invariant theory, while providing some links between those. Although we review the applications and properties of bricks from many different areas, the article is not meant to be an exhaustive survey on bricks in representation theory. In the setting of finite dimensional algebras over an algebraically closed field, this manuscript (and many of the recent works of the authors) is strongly motivated by an open conjecture originally posed by the first-named author in 2019, the so-called \emph{second brick Brauer-Thrall conjecture}. In the later sections, where the main focus is on the tame algebras and some other new notions of tameness, we prove some new results on the aforementioned conjecture.