Connectedness of quasi-hereditary structures

TL;DR

This paper investigates the relationships between different quasi-hereditary structures of an algebra, providing criteria for when permutations of indices preserve quasi-heredity and proving that all such structures are interconnected.

Contribution

It introduces a homological criterion for adjacent transpositions to preserve quasi-heredity and proves the connectedness of all quasi-hereditary structures for a given algebra.

Findings

Homological conditions characterize when adjacent transpositions yield quasi-hereditary structures.

Any two quasi-hereditary structures are connected through a sequence of permutations.

Permutations between connected structures also produce quasi-hereditary structures.

Abstract

Dlab and Ringel showed that algebras being quasi-hereditary in all orders for indices of primitive idempotents becomes hereditary. So, we are interested in for which orders a given quasi-hereditary algebra is again quasi-hereditary. As a matter of fact, we consider permutations of indices, and if the algebra with permuted indices is quasi-hereditary, then we say that this permutation gives a quasi-hereditary structure. In this article, we first give a criterion for adjacent transpositions giving quasi-hereditary structures, in terms of homological conditions of standard or costandard modules over a given quasi-hereditary algebra. Next, we consider those which we call connectedness of quasihereditary structures. The definition of connectedness can be found in Definition 4.1. We then show that any two quasi-hereditary structures are connected, which is our main result. By this result, once we know that there are two quasi-hereditary structures, then permutations in some sense lying between them give also quasi-hereditary structures.