scholarly journals On BGG resolutions of unitary modules for quiver Hecke and Cherednik algebras

2022 ◽  
Vol 28 (2) ◽  
Author(s):  
C. Bowman ◽  
E. Norton ◽  
J. Simental
Keyword(s):  
Linear Subspace ◽  
Betti Numbers ◽  
Hecke Algebras ◽  
Simple Modules ◽  
Cherednik Algebras ◽  
Subspace Arrangements

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.

scholarly journals Laurent phenomenon and simple modules of quiver Hecke algebras

2019 ◽  
Vol 155 (12) ◽  
pp. 2263-2295 ◽  
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 $[\mathscr{C}]$.

scholarly journals ON BASES OF SOME SIMPLE MODULES OF SYMMETRIC GROUPS AND HECKE ALGEBRAS

2017 ◽  
Vol 23 (3) ◽  
pp. 631-669 ◽  
Author(s):  
M. DE BOECK ◽  
A. EVSEEV ◽  
S. LYLE ◽  
L. SPEYER
Keyword(s):  
Hecke Algebras ◽  
Symmetric Groups ◽  
Simple Modules

The number of simple modules of the Hecke algebras of type G ( r ,1, n )

2000 ◽  
Vol 233 (3) ◽  
pp. 601-623 ◽  
Author(s):  
Susumu Ariki ◽  
Andrew Mathas
Keyword(s):  
Hecke Algebras ◽  
Simple Modules
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