Permutation of edges in mutation reduction of pointed Brauer trees

Abstract

Aihara developed an algorithm for Brauer tree algebras, which we call a mutation reduction, for getting from a Brauer tree algebra to the simpler Brauer star algebra using a sequence of mutations centered on edges. Schaps and Zvi, using the Schaps-Zakay theory of pointing the tree, showed that different algorithms for the sequence of mutations give permutations of the edges. Kozakai gave a new algorithm for a mutation reduction that depends on a given pointing and describes the evolution of the pointing under the mutation reduction. In this paper, we define a pointed generalized Aihara algorithm and show that its permutation is the identity. We give a general form for the permutations resulting from Kozakai's algorithm, which we illustrate with examples from uni-branch binary trees.