A Proof of the Second brick-Brauer-Thrall Conjecture for $E$-infinite algebras

Abstract

For any finite-dimensional algebra $A$ over an algebraically closed field, we prove that if $A$ is $E$-infinite, then the Second brick-Brauer-Thrall Conjecture holds for $A$. In fact, we show that if there is a rational ray outside the $τ$-tilting fan of $A$, then $A$ admits an infinite family of $θ$-stable bricks of dimension $d$, for some positive integer $d$ and weight $θ$. To show our main result, from any $τ$-regular component of $A$ with no dense orbit, we obtain a brick component that also has no dense orbit and whose points in general position are $θ$-stable modules for a given weight $θ$. As an important consequence, we conclude that $A$ is stably-discrete (also called multiplicity-free) if and only if $A$ is $E$-finite. Moreover, we prove that Demonet's (lattice point) Conjecture implies the stable Second brick-Brauer-Thrall Conjecture. For $E$-infinite algebras, our results settle several open conjectures. Together with our earlier work and recent developments, this leads to important reductions in the study of some challenging open problems.