Mutations and Pointing for Brauer Tree Algebras

TL;DR

This paper combines two approaches to tilting theory for Brauer tree algebras, providing an algorithm to construct tilting complexes from mutations and pointing, linking different algebraic structures.

Contribution

It introduces a unified method to derive tilting complexes for Brauer tree algebras by connecting Rickard's mutation approach with Aihara's pointing theory.

Findings

An algorithm for constructing tilting complexes from mutations.

Proof that Aihara's tilting complex derives from the folded Rickard complex.

Connection between pointing, mutations, and cyclic orderings in Brauer tree algebras.

Abstract

Brauer tree algebras are important and fundamental blocks in the modular representation theory of groups. In this research, we present a combination of two main approaches to the tilting theory of Brauer tree algebras. The first approach is the theory initiated by Rickard, providing a direct link between the ordinary Brauer tree algebra and a particular algebra called the Brauer star algebra. This approach was continued by Schaps-Zakay with their theory of pointing the tree. The second approach is the theory developed by Aihara, relating to the sequence of mutations from the ordinary Brauer tree algebra to the star-algebra of the Brauer tree. Our main purpose in this research is to combine these two approaches: We find an algorithm for which we are able to obtain a tilting complex constructed from irreducible complexes of length two {[}SZ1{]}, which is obtained from a sequence of mutations and corresponds to the star-to-tree complex for the pointing given by a reversed Green's walk. For the algorithm given by Aihara in \cite{Ai}, we prove that Aihara's tilting complex can be obtained from the completely folded Rickard tree-to-star complex with left alternating pointing by a permutation of projectives corresponding to the cyclic ordering of edges at vertices of non-zero even distance from the exceptional vertex. The natural numbering of the Aihara algorithm can be optained from the left alternating pinting by the inverse of this permutation.