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On the Combinatorics of Gentle Algebras

Part of: Representation theory of rings and algebras

Published online by Cambridge University Press:  29 July 2019

Thomas Brüstle ,
Guillaume Douville ,
Kaveh Mousavand ,
Hugh Thomas  and
Emine Yıldırım
Thomas Brüstle
Affiliation:
Département de Mathématiques, Université de Sherbrooke, 2500, boul. de l’Université Sherbrooke,QC J1K 2R1, Canada, Email: thomas.brustle@usherbrooke.ca
Guillaume Douville
Affiliation:
Département de mathématiques, Université du Québec à Montréal, Montréal, QC H3C 3P8, Canada, Email: douvilleg@gmail.comk.mousavand@lacim.cahugh.ross.thomas@gmail.comemineyyildirim@gmail.com
Kaveh Mousavand
Affiliation:
Département de mathématiques, Université du Québec à Montréal, Montréal, QC H3C 3P8, Canada, Email: douvilleg@gmail.comk.mousavand@lacim.cahugh.ross.thomas@gmail.comemineyyildirim@gmail.com
Hugh Thomas
Affiliation:
Département de mathématiques, Université du Québec à Montréal, Montréal, QC H3C 3P8, Canada, Email: douvilleg@gmail.comk.mousavand@lacim.cahugh.ross.thomas@gmail.comemineyyildirim@gmail.com
Emine Yıldırım
Affiliation:
Département de mathématiques, Université du Québec à Montréal, Montréal, QC H3C 3P8, Canada, Email: douvilleg@gmail.comk.mousavand@lacim.cahugh.ross.thomas@gmail.comemineyyildirim@gmail.com
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Abstract

For $A$ a gentle algebra, and $X$ and $Y$ string modules, we construct a combinatorial basis for $\operatorname{Hom}(X,\unicode[STIX]{x1D70F}Y)$. We use this to describe support $\unicode[STIX]{x1D70F}$-tilting modules for $A$. We give a combinatorial realization of maps in both directions realizing the bijection between support $\unicode[STIX]{x1D70F}$-tilting modules and functorially finite torsion classes. We give an explicit basis of $\operatorname{Ext}^{1}(Y,X)$ as short exact sequences. We analyze several constructions given in a more restricted, combinatorial setting by McConville, showing that many but not all of them can be extended to general gentle algebras.

Keywords

gentle algebras𝜏-tilting theory

MSC classification

Primary: 16G20: Representations of quivers and partially ordered sets
Type
Article
Information
Canadian Journal of Mathematics , Volume 72 , Issue 6 , December 2020 , pp. 1551 - 1580
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Copyright
© Canadian Mathematical Society 2019

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Footnotes

T. B. was partially supported by an NSERC Discovery Grant. G. D. was partially supported by an NSERC Alexander Graham Bell scholarship. K. M. and E. Y. were partially supported by ISM scholarships. H. T. was partially supported by NSERC and the Canada Research Chairs program.

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