Representation stability for diagram algebras

Abstract

We define the notion of representation stability for sequences of modules over the Temperley–Lieb algebras, the Brauer algebras, and the partition algebras in the semisimple setting analogous to Church and Farb. We also find a condition when such sequences are representation stable analogous to Church, Ellenberg, and Farb. To this end, we introduce stability categories for these diagram algebras—analogs to Randal-Williams and Wahl's homogeneous categories. We prove that finitely presented modules over these stability categories give rise to representation stable sequences as long as the parameters ensure semisimplicity.