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

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.