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ORIENTED FLIP GRAPHS, NONCROSSING TREE PARTITIONS, AND REPRESENTATION THEORY OF TILING ALGEBRAS

Part of: Algebraic combinatorics Abelian categories Representation theory of rings and algebras

Published online by Cambridge University Press:  07 February 2019

ALEXANDER GARVER Open the ORCID record for ALEXANDER GARVER [Opens in a new window]  and
THOMAS MCCONVILLE
ALEXANDER GARVER*
Affiliation:
Université du Québec à Montréal, Montréal, Canada e-mail: alexander.garver@gmail.com
THOMAS MCCONVILLE
Affiliation:
Mathematical Sciences Research Institute, Berkeley, CA, USA e-mail: thomasmcconvillea@gmail.com
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Abstract

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The purpose of this paper is to understand lattices of certain subcategories in module categories of representation-finite gentle algebras called tiling algebras, as introduced by Coelho Simões and Parsons. We present combinatorial models for torsion pairs and wide subcategories in the module category of tiling algebras. Our models use the oriented flip graphs and noncrossing tree partitions, previously introduced by the authors, and a description of the extension spaces between indecomposable modules over tiling algebras. In addition, we classify two-term simple-minded collections in bounded derived categories of tiling algebras. As a consequence, we obtain a characterization of c-matrices for any quiver mutation-equivalent to a type A Dynkin quiver.

MSC classification

Primary: 16G20: Representations of quivers and partially ordered sets
Secondary: 05E10: Combinatorial aspects of representation theory 18E40: Torsion theories, radicals
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 62 , Issue 1 , January 2020 , pp. 147 - 182
Copyright
Copyright © Glasgow Mathematical Journal Trust 2019 

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