Equivalence between module categories over quiver Hecke algebras and Hernandez–Leclerc's categories in general types

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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Rouquier blocks of the cyclotomic Hecke algebras of G(de, e, r)

Nagoya Mathematical Journal ◽

10.1017/s0027763000009880 ◽

2010 ◽

Vol 197 ◽

pp. 175-212

Author(s):

Maria Chlouveraki

Keyword(s):

Infinite Series ◽

Hecke Algebras ◽

Weyl Groups ◽

Reflection Groups ◽

Complex Reflection ◽

Complex Reflection Groups ◽

Cyclotomic Hecke Algebras

The Rouquier blocks of the cyclotomic Hecke algebras, introduced by Rouquier, are a substitute for the families of characters defined by Lusztig for Weyl groups, which can be applied to all complex reflection groups. In this article, we determine them for the cyclotomic Hecke algebras of the groups of the infinite seriesG(de, e, r), thus completing their calculation for all complex reflection groups.

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