Criterion for Quasi-Heredity

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 quasi-heredity. In this article, we give a criterion for adjacent transpositions giving quasi-heredity, in terms of homological conditions of standard or costandard modules over a given quasi-hereditary algebra.