Second brick–Brauer–Thrall (2nd bBT) conjecture

Establish that a finite-dimensional algebra A over an algebraically closed field is brick-infinite if and only if there exists an infinite family of non-isomorphic bricks all having the same dimension. Equivalently, determine whether brick-infinite algebras always admit infinitely many bricks of a fixed dimension in their module category.

Background

Bricks (Schur representations) are modules M with End_A(M) a division algebra; their geometry inside representation varieties links to torsion theory and τ-tilting theory. The classical Brauer–Thrall theorems concern indecomposables; the authors propose a conceptual brick analogue, refining earlier geometric formulations in terms of orbits in representation varieties.

This formulation equates brick-infinite (equivalently, τ-tilting infinite) algebras with the existence of infinite families of bricks of a fixed dimension, making the conjecture accessible in purely module-theoretic terms and connecting it to several other brick-centric conjectures reviewed in the paper.