Lattice structure of Weyl groups via representation theory of preprojective algebras. (16th May 2018)

• Record Type: Journal Article
• Authors: Iyama, Osamu; Reading, Nathan; Reiten, Idun; Thomas, Hugh
• Is Part Of: Compositio mathematica. Volume 154:Number 6(2018)
• Page Start: 1269
• Page End: 1305
• Publication Date: 2018-05-16
• Subjects: 16G10 (primary), -- 05E10, -- 06B10, -- 18E40, -- 20F55 (secondary); preprojective algebra, -- Weyl group, -- weak order, -- 𝜏-tilting theory, -- brick, -- join-irreducible element, -- lattice congruence; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X18007078
• Languages: English
• ISSNs: 0010-437X

Abstract

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 $D$ .