Borelic pairs for stratified algebras

TL;DR

This paper develops a general theory of algebras with Borelic pairs to analyze cellular diagram algebras, determining parameter conditions for cell modules to form a standard system and providing new proofs of their properties.

Contribution

It introduces a unified framework for algebras with Borelic pairs, applies it to various diagram algebras, and constructs quasi-hereditary 1-covers with exact Borel subalgebras.

Findings

Identified parameter conditions for cell modules to form a standard system.

Provided new proofs of cellular and quasi-hereditary properties of diagram algebras.

Constructed quasi-hereditary 1-covers with exact Borel subalgebras.

Abstract

We determine all values of the parameters for which the cell modules form a standard system, for a class of cellular diagram algebras including partition, Brauer, walled Brauer, Temperley-Lieb and Jones algebras. For this, we develop and apply a general theory of algebras with Borelic pairs. The theory is also applied to give new uniform proofs of the cellular and quasi-hereditary properties of the diagram algebras and to construct quasi-hereditary 1-covers, in the sense of Rouquier, with exact Borel subalgebras, in the sense of K\"onig. Another application of the theory leads to a proof that Auslander-Dlab-Ringel algebras admit exact Borel subalgebras.