Connectedness of 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 first 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. Next, we consider those which we call connectedness of quasi-heredity. The definition of connectedness can be found in \Cref{def of connected}. We then show that any two quasi-heredity are connected, which is our main result. By this result, once we know that there are two quasi-heredity, then permutations in some sense lying between them give also quasi-heredity. MSC Classification: 16G20 , 06A05 , 20B30