On semisimplicity criteria and non-semisimple representation theory for the Kadar-Yu algebras

TL;DR

This paper explores the non-semisimple representation theory of Kadar-Yu algebras, providing new determinant formulas for standard modules that connect the semisimple cases of Brauer and Temperley-Lieb algebras.

Contribution

It introduces generalized Chebyshev-like formulas for gram determinants, advancing understanding of the non-semisimple structure of KY algebras and extending geometric frameworks.

Findings

Derived generalized Chebyshev-like forms for gram determinants.

Connected non-semisimple KY algebra representations to geometric frameworks.

Enhanced understanding of morphisms between standard modules.

Abstract

The Kadar--Yu algebras are a physically motivated sequence of towers of algebras interpolating between the Brauer algebras and Temperley--Lieb algebras. The complex representation theory of the Brauer and Temperley--Lieb algebras is now fairly well understood, with each connecting in a different way to Kazhdan--Lusztig theory. The semisimple representation theory of the KY algebras is also understood, and thus interpolates, for example, between the double-factorial and Catalan combinatorial realms. However the non-semisimple representation theory has remained largely open, being harder overall than the (already challenging) Brauer case. In this paper we determine generalised Chebyshev-like forms for the determinants of gram matrices of contravariant forms for standard modules. This generalises the root-of-unity paradigm for Temperley--Lieb algebras (and many related algebras); interpolating in various ways between this and the `integral paradigm' for Brauer algebras. The standard module gram determinants give a huge amount of information about morphisms between standard modules, making thorough use of the powerful homological machinery of towers of recollement (ToR), with appropriate gram determinants providing the ToR `bootstrap'. As for the Brauer and TL cases the representation theory has a strongly alcove-geometric flavour, but the KY cases guide an intriguing generalisation of the overall geometric framework.