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July 2020 Mutations and Pointing for Brauer Tree Algebras
Mary Schaps, Zehavit Zvi
Osaka J. Math. 57(3): 689-709 (July 2020).

Abstract

Brauer tree algebras are important and fundamental blocks in the representation theory of finite dimensional algebras. 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 an ordinary Brauer tree algebra and 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 Brauer star algebra. Our main purpose in this research is to combine these two approaches. We first find an algorithm based on centers which are all terminal edges, for which we are able to obtain a tilting complex constructed from irreducible complexes of length two [13], which is obtained from a sequence of mutations. In [1], Aihara gave an algorithm for reducing from tree to star by mutations and showed that it gave a two-term tree-to-star complex. We prove that Aihara's complex is obtained from the corresponding completely folded Rickard tree-to-star complex by a permutation of projectives.

Citation

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Mary Schaps. Zehavit Zvi. "Mutations and Pointing for Brauer Tree Algebras." Osaka J. Math. 57 (3) 689 - 709, July 2020.

Information

Published: July 2020
First available in Project Euclid: 13 July 2020

zbMATH: 07224929
MathSciNet: MR4121784

Subjects:
Primary: 16G10, 18E30

Rights: Copyright © 2020 Osaka University and Osaka City University, Departments of Mathematics

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Vol.57 • No. 3 • July 2020
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