Efficiency of mutation reduction for Brauer trees

Abstract

Brauer tree algebras are important and fundamental blocks in the modular representation theory of groups. Aihara develped an algorithm, 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. We give a modification of Aihara's algorithm and tested it against the original algorithm for computational efficiency. We prove that all versions of Aihara's algorithm are the fastest possible in the sense of requiring the least number steps to reach the Brauer star, when compared to all possible complete mutation reduction algorithms.