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CLUSTER CATEGORIES FROM GRASSMANNIANS AND ROOT COMBINATORICS

Part of: Algebraic combinatorics Lie algebras and Lie superalgebras Representation theory of rings and algebras

Published online by Cambridge University Press:  03 June 2019

KARIN BAUR Open the ORCID record for KARIN BAUR [Opens in a new window] ,
DUSKO BOGDANIC Open the ORCID record for DUSKO BOGDANIC [Opens in a new window]  and
ANA GARCIA ELSENER Open the ORCID record for ANA GARCIA ELSENER [Opens in a new window]
KARIN BAUR
Affiliation:
Karl-Franzens-Universitat Graz, Mathematics, Heinrichstraße 36, Graz 8010, Austria School of Mathematics, University of Leeds, Leeds, LS2 9JT, UK email [email protected]
DUSKO BOGDANIC
Affiliation:
Karl-Franzens-Universitat Graz, Mathematics, Heinrichstraße 36, Graz 8010, Austria email [email protected]
ANA GARCIA ELSENER
Affiliation:
Karl-Franzens-Universitat Graz, Mathematics, Heinrichstraße 36, Graz 8010, Austria email [email protected]
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Abstract

The category of Cohen–Macaulay modules of an algebra $B_{k,n}$ is used in Jensen et al. (A categorification of Grassmannian cluster algebras, Proc. Lond. Math. Soc. (3) 113(2) (2016), 185–212) to give an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of $k$-planes in $n$-space. In this paper, we find canonical Auslander–Reiten sequences and study the Auslander–Reiten translation periodicity for this category. Furthermore, we give an explicit construction of Cohen–Macaulay modules of arbitrary rank. We then use our results to establish a correspondence between rigid indecomposable modules of rank 2 and real roots of degree 2 for the associated Kac–Moody algebra in the tame cases.

MSC classification

Primary: 05E10: Combinatorial aspects of representation theory 16G50: Cohen-Macaulay modules 17B22: Root systems
Type
Article
Information
Nagoya Mathematical Journal , Volume 240 , December 2020 , pp. 322 - 354
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Copyright
© 2019 Foundation Nagoya Mathematical Journal

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