Hopf–Hecke algebras, infinitesimal Cherednik algebras, and Dirac cohomology

AbstractWe provide a homological construction of unitary simple modules of Cherednik and Hecke algebras of type A via BGG resolutions, solving a conjecture of Berkesch–Griffeth–Sam. We vastly generalize the conjecture and its solution to cyclotomic Cherednik and Hecke algebras over arbitrary ground fields, and calculate the Betti numbers and Castelnuovo–Mumford regularity of certain symmetric linear subspace arrangements.

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Dirac cohomology, the projective supermodules of the symmetric group and the Vogan morphism

The Quarterly Journal of Mathematics ◽

10.1093/qmath/hay057 ◽

2018 ◽

Vol 70 (2) ◽

pp. 535-563 ◽

Cited By ~ 1

Author(s):

Kieran Calvert

Keyword(s):

Symmetric Group ◽

Hecke Algebras ◽

Explicit Description ◽

Weyl Groups ◽

Explicit Model ◽

Dirac Theory ◽

Dirac Cohomology ◽

Graded Modules ◽

Projective Representations ◽

Spectrum Data

Abstract We derive an explicit description of the genuine projective representations of the symmetric group Sn using Dirac cohomology and the branching graph for the irreducible genuine projective representations of Sn. Ciubotaru and He [D. Ciubotaru and X. He, Green polynomials of Weyl groups, elliptic pairings, and the extended index. Adv. Math., 283:1–50, 2015], using the extended Dirac index, showed that the characters of the projective representations of Sn are related to the characters of elliptic-graded modules. We derive the branching graph using Dirac theory and combinatorics relating to the cohomology of Borel varieties ℬe of g and are able to use Dirac cohomology to construct an explicit model for the projective representations. We also describe Vogan’s morphism for Hecke algebras in type A using spectrum data of the Jucys–Murphy elements.

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Double Affine Hecke Algebras

10.1017/cbo9780511546501 ◽

2005 ◽

Cited By ~ 76

Author(s):

Ivan Cherednik

Keyword(s):

Hecke Algebras ◽

Double Affine Hecke Algebras ◽

Affine Hecke Algebras

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Dominant and global dimension of blocks of quantised Schur algebras

Mathematische Zeitschrift ◽

10.1007/s00209-021-02792-w ◽

2021 ◽

Author(s):

Ming Fang ◽

Wei Hu ◽

Steffen Koenig

Keyword(s):

Hecke Algebras ◽

Symmetric Groups ◽

Global Dimension ◽

Group Algebras ◽

Dominant Dimension ◽

Schur Algebras ◽

Homological Dimensions ◽

Weyl Duality

AbstractGroup algebras of symmetric groups and their Hecke algebras are in Schur-Weyl duality with classical and quantised Schur algebras, respectively. Two homological dimensions, the dominant dimension and the global dimension, of the indecomposable summands (blocks) of these Schur algebras S(n, r) and $$S_q(n,r)$$ S q ( n , r ) with $$n \geqslant r$$ n ⩾ r are determined explicitly, using a result on derived invariance in Fang, Hu and Koenig (J Reine Angew Math 770:59–85, 2021).

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Laurent phenomenon and simple modules of quiver Hecke algebras

Compositio Mathematica ◽

10.1112/s0010437x19007565 ◽

2019 ◽

Vol 155 (12) ◽

pp. 2263-2295 ◽

Cited By ~ 1

Author(s):

Masaki Kashiwara ◽

Myungho Kim

Keyword(s):

Simple Module ◽

Hecke Algebras ◽

Duke Math ◽

Cluster Algebras ◽

Coordinate Ring ◽

Simple Modules ◽

Laurent Phenomenon ◽

Cluster Variable ◽

Quantum Cluster

In this paper we study consequences of the results of Kang et al. [Monoidal categorification of cluster algebras, J. Amer. Math. Soc. 31 (2018), 349–426] on a monoidal categorification of the unipotent quantum coordinate ring $A_{q}(\mathfrak{n}(w))$ together with the Laurent phenomenon of cluster algebras. We show that if a simple module $S$ in the category ${\mathcal{C}}_{w}$ strongly commutes with all the cluster variables in a cluster $[\mathscr{C}]$, then $[S]$ is a cluster monomial in $[\mathscr{C}]$. If $S$ strongly commutes with cluster variables except for exactly one cluster variable $[M_{k}]$, then $[S]$ is either a cluster monomial in $[\mathscr{C}]$ or a cluster monomial in $\unicode[STIX]{x1D707}_{k}([\mathscr{C}])$. We give a new proof of the fact that the upper global basis is a common triangular basis (in the sense of Qin [Triangular bases in quantum cluster algebras and monoidal categorification conjectures, Duke Math. 166 (2017), 2337–2442]) of the localization $\widetilde{A}_{q}(\mathfrak{n}(w))$ of $A_{q}(\mathfrak{n}(w))$ at the frozen variables. A characterization on the commutativity of a simple module $S$ with cluster variables in a cluster $[\mathscr{C}]$ is given in terms of the denominator vector of $[S]$ with respect to the cluster $[\mathscr{C}]$.

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