Brick Lodge

A gathering place for recent developments on bricks

Introduction:

This page is devoted to recent developments in the study of bricks in representation theory of algebras and related areas. It aims to serve as a growing repository of important results and developments from recent years, as well as a space for sharing new results, posing open problems and questions, and exchanging ideas. The hope is that it will provide a useful resource for researchers working on bricks and related topics, and help foster further developments and connections within the subject.

• A. Open Conjectures
• B. Open Questions
• C. New Results
• D. Observations

My brick-trajectory:

In recent years, a major focus of my research has been to deepen our understanding of the behavior of bricks and related phenomena in the context of finite-dimensional algebras. For a finite-dimensional algebra A over an algebraically closed field k, a (left) A-module M is a brick if the endomorphism algebra of M is a skew field — that is, every non-zero A-homomorphism f: M → M is invertible. When M is finitely generated, this is equivalent to its endomorphism algebra being k, in which case bricks are also known as Schur representations. Bricks play a central role across stability conditions, wall-chamber structures, τ-tilting theory, the lattice theory of torsion classes, wide subcategories, and the spectrum of algebras. For a summary of recent progress on a series of modern problems on bricks, see this survey.

Building on my doctoral work, and primarily motivated by two open conjectures I first posed in my 2019 preprint (Conjecture 6.6), I have carried out extensive work on a systematic study of bricks. My stronger conjecture — now called the Second brick-Brauer-Thrall conjecture (2nd bBT) — concerns the distribution of bricks over algebras admitting infinitely many non-isomorphic bricks, and can be viewed as the modern analogue of the classical Second Brauer-Thrall conjecture (now a theorem). For remarks on the 2nd bBT and related problems, see Section 2 of this paper.

The 2nd bBT conjecture connects several classical and modern aspects of representation theory: the geometry of representation varieties, families of stable modules, components of Auslander–Reiten quivers, g-vector fans of algebras, and the behavior of infinite-dimensional (particularly generic) modules. It also relates to, and in some cases implies, other open conjectures in the field. We have settled it for some families of algebras, but as of Fall 2025 it remains open. For further details and future directions, see my Research Statement.