A remark of the number of quasi-hereditary structures

TL;DR

This paper investigates the number of quasi-hereditary structures on certain algebras by analyzing permutations of indices and their equivalence classes, providing a counting method.

Contribution

It introduces a method to count quasi-hereditary structures based on permutation equivalence classes for specific algebras.

Findings

Established a counting method for quasi-hereditary structures.

Connected permutation equivalence to standard modules.

Applied the method to particular algebra classes.

Abstract

Dlab and Ringel showed that algebras being quasi-hereditary in all total 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 instead of total orders. If the standard modules defined by two permutations coincide, we say that the permutations are equivalent. Moreover if the algebra with permuted indices is quasi-hereditary, then this equivalence class of the permutation is called a quasi-hereditary structure. In this article, we give a method of counting the number of quasi-hereditary structures for certain algebras.