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Lattice structure of Weyl groups via representation theory of preprojective algebras

Part of: Representation theory of rings and algebras Special aspects of infinite or finite groups Algebraic combinatorics Abelian categories Lattices

Published online by Cambridge University Press:  16 May 2018

Osamu Iyama ,
Nathan Reading ,
Idun Reiten  and
Hugh Thomas
Osamu Iyama
Affiliation:
Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan email iyama@math.nagoya-u.ac.jphttp://www.math.nagoya-u.ac.jp/∼iyama/
Nathan Reading
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA email reading@math.ncsu.eduhttp://www4.ncsu.edu/∼nreadin/
Idun Reiten
Affiliation:
Department of Mathematical Sciences, Norges teknisk-naturvitenskapelige universitet, 7491 Trondheim, Norway email idun.reiten@math.ntnu.nohttp://www.ntnu.edu/employees/idun.reiten
Hugh Thomas
Affiliation:
Département de mathématiques, Université du Québec à Montréal, CP 8888, Succursale Centre-Ville, Montréal, QC, H3C 3P8, Canada email hugh.ross.thomas@gmail.comhttp://www.lacim.uqam.ca/∼hugh
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Abstract

This paper studies the combinatorics of lattice congruences of the weak order on a finite Weyl group $W$, using representation theory of the corresponding preprojective algebra $\unicode[STIX]{x1D6F1}$. Natural bijections are constructed between important objects including join-irreducible congruences, join-irreducible (respectively, meet-irreducible) elements of $W$, indecomposable $\unicode[STIX]{x1D70F}$-rigid (respectively, $\unicode[STIX]{x1D70F}^{-}$-rigid) modules and layers of $\unicode[STIX]{x1D6F1}$. The lattice-theoretically natural labelling of the Hasse quiver by join-irreducible elements of $W$ is shown to coincide with the algebraically natural labelling by layers of $\unicode[STIX]{x1D6F1}$. We show that layers of $\unicode[STIX]{x1D6F1}$ are nothing but bricks (or equivalently stones, or 2-spherical modules). The forcing order on join-irreducible elements of $W$ (arising from the study of lattice congruences) is described algebraically in terms of the doubleton extension order. We give a combinatorial description of indecomposable $\unicode[STIX]{x1D70F}^{-}$-rigid modules for type $A$ and $D$.

Keywords

preprojective algebraWeyl groupweak order𝜏-tilting theorybrickjoin-irreducible elementlattice congruence

MSC classification

Primary: 16G10: Representations of Artinian rings
Secondary: 05E10: Combinatorial aspects of representation theory 06B10: Ideals, congruence relations 18E40: Torsion theories, radicals 20F55: Reflection and Coxeter groups
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 6 , June 2018 , pp. 1269 - 1305
Copyright
© The Authors 2018 

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