Abstract

We 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.

Highlights

  • In [1], Bernstein–Gelfand–Gelfand utilise resolutions of simple modules by Verma modules to prove certain beautiful properties of finite-dimensional Lie algebras

  • For Lie groups, this ongoing project draws on techniques from Dirac cohomology [35], Kazhdan–Lusztig theory [64], and the Langlands Program [58], and has provided profound insights into relativistic quantum mechanics [61]

  • We show that our resolutions for unitary simples remain stable under reduction modulo p — in other words the beautiful properties of unitary modules extend beyond the confines of characteristic zero to for arbitrary fields

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Summary

Introduction

In [1], Bernstein–Gelfand–Gelfand utilise resolutions of simple modules by Verma modules to prove certain beautiful properties of finite-dimensional Lie algebras Such resolutions ( known as BGG resolutions) have had spectacular applications in the study of the Laplacian on Euclidean space [19], complex representation theory and homology of Kac–Moody algebras [28], statistical mechanics and conformal field theories [27,46,47], and they provide graded free resolutions (in the sense of commutative algebra) for determinantal varieties [20,40]. In our BGG resolution, (ν) appears in homological degree d if and only if ν is obtained from λ by reflecting across d walls

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Combinatorics
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The quiver Hecke and Cherednik algebras
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Induction and restriction
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Alcove geometries and path-bases of diagrammatic algebras
The alcove geometry
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Standard tableaux as paths
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The skeleton of our BGG resolutions
One column homomorphisms
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Diamonds formed by pairs of one-column morphisms
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Compositions of one-column homomorphisms in diamonds
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The BGG-resolutions for quiver Hecke algebras
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The abacus of a partition
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The affine and extended affine symmetric group actions
The homological degree statistic
Homological degree produced recursively by elements of Se
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Homological degree via rimhooks of minimal leg-length
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The Cherednik algebra of the symmetric group
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Connection to diagrammatic algebra
5.10 Unitary modules and the BGS conjecture
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Berkesch–Griffeth–Sam’s conjecture and beyond
Changing quantum characteristics
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Ringel duality and more BGG resolutions
Computation of Lie algebra and Dirac cohomology
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The Mullineux map on unitary simple modules
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Commutative algebra
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The e-equals ideal
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