On the bricks (Schur representations) of finite dimensional algebras

Kaveh Mousavand; Charles Paquette

Home/Archive/arXiv

arXiv · 2508.11789

On the bricks (Schur representations) of finite dimensional algebras

Kaveh Mousavand·Charles Paquette

View original source ↗Read source document ↗Export metadata ↓

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 $τ$-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.

Explore related subjects

math.RT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Kaveh Mousavand, Charles Paquette. 2025-08-15. On the bricks (Schur representations) of finite dimensional algebras. https://arxiv.org/abs/2508.11789

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A Course on Lie algebras and Chevalley groups

These are expanded notes from graduate courses about Lie algebras and Chevalley groups held at the University of Stuttgart. In the 1950s Chevalley showed how linear groups over arbitrary fields could be obtained~ -- ~by a uniform procedure~ -- ~from the simple Lie algebras over $\C$ occurring in the Cartan--Killing classification. Together with subsequent variations, Chevalley's work had a profound and long-lasting impact on group theory and Lie theory in general. Classical, and widely used references are the lectures notes by Steinberg (1967) and the monograph by Carter (1972). Our aim here is to present a self-contained introduction to the theory of Chevalley groups, based on recent simplifications arising from Lusztig's fundamental theory of ``canonical bases''. A further feature of our text is that we explicitly incorporate algorithmic methods in our treatment, both for the handling of substantial examples and regarding some aspects of the general theory. Eventually, this may turn into a book project.

math.RT↗

arXiv2026

On the p-part of the conductor of a generalised character

We show that the $p$-part of the conductor of a generalised character of a finite group is equal to the conductor of its generalised decomposition numbers. We use this to show that $p$-parts of conductors of irreducible characters are preserved under isotypies and perfect isometries that arise in the context of stable equivalences of Morita type with endopermutation source. We apply this to blocks with abelian defect and Frobenius inertial quotient.

math.RT↗

arXiv2026

On the triviality of inhomogeneous deformations of $\mathfrak{osp}(1|2n)$

We specify a symmetrized mixed-oscillator deformation family of $B(0,n)=\operatorname{osp}(1|2n)$, with even mixed coefficients and one odd square-zero parameter. For every $n\geq1$, we derive its bracket from a faithful oscillator realization and exhibit an odd cochain whose coboundary is the recovered deformation coefficient. The resulting even change of generators is an exact isomorphism over the exterior parameter algebra. For $n=1$, the cochain agrees with the normalization of Bakalov-Sullivan. We give the source relations and the even-central specialization explicitly, together with a Lean 4 formalization.

math.RT↗