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Math & Stats Algebra & Geometry Seminar
Monday, September 8, 2025

Time: 4:45 p.m.  Place: Jeffery Hall, Room 422

Speaker: Kaveh Mousavand (Okinawa Institute of Science and Technology)

Title: Pairwise Hom-orthogonal Modules and Some Open Conjectures

Abstract: Let A be an associative algebra and S={X_1,X_2,…,X_m} be a set of finitely generated (left) A-modules. We say that S is a set of pairwise Hom-orthogonal modules of size m if, for every distinct pair of i and j, there is no nonzero A-module homomorphism from X_i to X_j; that is, Hom_A(X_i,X_j)=0. This naturally leads to the following (still open) question: Given an algebra A, is there an upper bound on the size of such sets of pairwise Hom-orthogonal A-modules?

Motivated by some challenging conjectures in modern representation theory, we recently investigated the above problem in the setting of finite-dimensional associative algebras. In particular, we approached the question through the study of bricks, namely, those modules whose endomorphism algebras are division algebras (also known as Schur representations). In fact, using a range of algebraic and geometric techniques, we show that a full answer to the question follows directly from an open conjecture I proposed in 2019 -- now referred to as theSecond Brick–Brauer–Thrall Conjecture (2nd bBT). Our perspective not only connects this question to several central themes in contemporary research -- such as stability conditions and \tau-tilting theory -- but also leads to reformulations of stronger versions of both the 2nd bBT and our earlier results. In particular, we achieve a significant reduction of several open conjectures, now expressible in the more elementary language of pairwise Hom-orthogonal modules. To make the key problems and results in this talk more accessible, I will, for the most part, assume only a basic familiarity with elementary module theory. This talk is based on joint work with Charles Paquette.