Essential orders on stratified algebras with duality and $\mathcal{S}$-subcategories in $\mathcal{O}$

TL;DR

This paper establishes the uniqueness of the essential order for certain stratified algebras with duality, and applies this to classify and analyze homological properties of blocks in category b5.

Contribution

It generalizes previous results to a broader class of stratified algebras and classifies blocks in category b5 up to equivalence.

Findings

Uniqueness of the essential order for stratified algebras with duality.

Classification of regular integral blocks in category b5.

Description of homological invariants of these blocks.

Abstract

We prove uniqueness of the essential order for stratified algebras having simple preserving duality, generalizing a recent result of Coulembier for quasi-hereditary algebras. We apply this to classify, up to equivalence, regular integral blocks of $\mathcal{S}$-subcategories in the BGG category $\mathcal{O}$. We also describe various homological invariants of these blocks.