Induction and restriction functors for cellular categories

TL;DR

This paper develops an axiomatic framework for cellular categories related to towers of algebras, studying their representations via induction and restriction, and providing criteria for semi-simplicity and diagrammatic computations.

Contribution

It introduces a new axiomatic approach to cellular categories associated with quasi-hereditary towers, enhancing understanding of their representation theory and algebraic structures.

Findings

Criteria for semi-simplicity using cohomology groups

Diagrammatic methods for Grothendieck group multiplication

Analysis of algebraic structures on Grothendieck groups

Abstract

Cellular categories are a generalization of cellular algebras, which include a number of important categories such as (affine)Temperley-Lieb categories, Brauer diagram categories, partition categories, the categories of invariant tensors for certain quantised enveloping algebras and their highest weight representations, Hecke categories and so on. The common feather is that, for most of the examples, the endomorphism algebras of the categories form a tower of algebras. In this paper, we give an axiomatic framework for the cellular categories related to the quasi-hereditary tower and then study the representations in terms of induction and restriction. In particular, a criteria for the semi-simplicity of cellular categories is given by using the cohomology groups of cell modules. Moreover, we investigate the algebraic structures on Grothendieck groups of cellular categories and provide a diagrammatic approach to compute the multiplication in the Grothendieck groups of Temperley-Lieb categories.