Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras

Enomoto, Haruhisa

doi:10.48550/arxiv.2002.09205

Cited by 2 publications

(2 citation statements)

References 12 publications

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“…A similar kind of correspondences between bricks and positive Schur roots for Langlands dual datum has considered in the work of [GLS20] for 𝐻 (𝐶, 𝐷, Ω) (see Definition 2.6). So, the combinatorial structure about torsion-free classes are expected to be described in terms of the root system of 𝔤(𝐶 T ) as in Asai [Asa17] or Enomoto [Eno20].…”

Section: G-vectors and Wall-chamber Structuresmentioning

confidence: 99%

PBW parametrizations and generalized preprojective algebras

Murakami

2020

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KOTA MURAKAMI A. Geiß-Leclerc-Schröer [Invent. Math. 209 (2017)] has introduced a notion of generalized preprojective algebras associated with generalized Cartan matrices 𝐶 and their symmetrizers 𝐷. They realize crystal structures on the set of maximal dimensional irreducible components of nilpotent varieties [Selecta Math. (N.S.) 24 ( 2018)]. For general finite types, we give stratifications of these components via partial orders of torsion classes in module categories of generalized preprojective algebras in terms of Weyl groups. In addition, we realize Mirković-Vilonen polytopes from generic modules of these components, and give a identification as crystals between the set of Mirković-Vilonen polytopes and the set of maximal dimensional irreducible components except for type G 2 . This generalizes the works of Baumann-Kamnitzer [Represent. Theory 16 ( 2012)] and Baumann-Kamnitzer-Tingley [Publ. Math. Inst. Hautes Études Sci. 120 (2014)].

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Section: G-vectors and Wall-chamber Structuresmentioning

confidence: 99%

PBW parametrizations and generalized preprojective algebras

Murakami

2020

Preprint

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“…Then the assertion inductively follows from this. See also [Eno1,Theorem 3.5] for the related argument.…”

Section: • Choose a Torsion-free Classmentioning

confidence: 99%

ICE-closed subcategories and wide $τ$-tilting modules

Enomoto,

Sakai

2020

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In this paper, we study ICE-closed (= Image-Cokernel-Extension-closed) subcategories of an abelian length category using torsion classes. To each interval [U , T ] in the lattice of torsion classes, we associate a subcategory T ∩ U ⊥ called the heart. We show that every ICE-closed subcategory can be realized as a heart of some interval of torsion classes, and give a lattice-theoretic characterization of intervals whose hearts are ICE-closed. In particular, we prove that ICE-closed subcategories are precisely torsion classes in some wide subcategories. For an artin algebra, we introduce the notion of wide τ -tilting modules as a generalization of support τ -tilting modules. Then we establish a bijection between wide τ -tilting modules and ICE-closed subcategories satisfying certain finiteness conditions, which extends Adachi-Iyama-Reiten's bijection on torsion classes. Furthermore, we discuss the Hasse quiver of the poset of ICE-closed subcategories for the hereditary case by introducing a mutation of rigid modules.Contents 24 6.1. Some properties on the lattice of ICE-closed subcategories 24 6.2. Examples of computations on ICE-closed subcategories 26 References 29

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Bruhat inversions in Weyl groups and torsion-free classes over preprojective algebras

Cited by 2 publications

References 12 publications

PBW parametrizations and generalized preprojective algebras

PBW parametrizations and generalized preprojective algebras

ICE-closed subcategories and wide $τ$-tilting modules

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