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.
Related
- Representation embeddings and the third Brauer--Thrall conjecture
- A Proof of the Jung-Kim Conjecture on the Centers of Walled Brauer Algebras
- The Brauer A(∞) condition and Navarro's conjecture on Brauer characters under coprime actions
- The associated graded algebras of biserial fractional Brauer graph algebras
- Brauer graph algebras are closed under stable equivalence of Morita type
BibTeX
@misc{mousavand2026proof,
title = {{A Proof of the Second brick-Brauer-Thrall Conjecture for $E$-infinite algebras}},
author = {Kaveh Mousavand and Charles Paquette},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31417},
eprint = {2609.31417},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31417},
}
RIS
TY - UNPB
TI - A Proof of the Second brick-Brauer-Thrall Conjecture for E-infinite algebras
AU - Kaveh Mousavand
AU - Charles Paquette
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.31417
UR - https://arxiv.org/abs/2609.31417
AB - 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.
ER -