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

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.

Unitary representations of type B rational Cherednik algebras and crystal combinatorics

“…Unitary representations stand out from the rest of the representations in O c (W ) by their relative tractability. In many cases they are known to possess closed-form, multiplicity-free character formulas in terms of a saturated subset of the poset of lowest weights [30], [7], [13]. These formulas are known in many cases to admit a combinatorial interpretation via affine type A alcove geometry [7].…”

“…In many cases they are known to possess closed-form, multiplicity-free character formulas in terms of a saturated subset of the poset of lowest weights [30], [7], [13]. These formulas are known in many cases to admit a combinatorial interpretation via affine type A alcove geometry [7]. It is often possible to write explicit bases for the unitary representations or for the corresponding representations of closely related algebras such as the Morita equivalent diagrammatic Cherednik algebra or the Hecke algebra.…”

“…It is often possible to write explicit bases for the unitary representations or for the corresponding representations of closely related algebras such as the Morita equivalent diagrammatic Cherednik algebra or the Hecke algebra. 1 At the categorical level, the unitary irreducible representations in type A, and for certain parameters, type G(ℓ, 1, n), have been proven to possess resolutions by complexes of direct sums of standard modules [4], [7], [6]. Such resolutions, known in the literature as BGG resolutions, categorify character formulas via the Euler characteristic of the complex.…”

“…In type A, the BGG resolutions of unitary modules are invariant under "runner removal Morita equivalences" [9], equivalently, under dilation of e-alcoves in the affine type A alcove geometry determining the maps in the complex [2]. Consequently, the BGG resolutions and hence the character formulas of all unitary irreducible modules for type A Cherednik algebras at all parameters are actually classified by a subset of them, the unitary modules L1 e (e m ) labeled by rectangles with e columns, where c = 1 e is the parameter for the Cherednik algebra H c (S n ) and n = em for some m ≥ 1 [7]. All other characters and resolutions of unitary irreducible representations in type A are simply obtained by relabeling partitions via an easy combinatorial procedure.…”

“…Similarly, using these methods in type A the recent paper of [19] shows how these bases arise from parabolic Hilbert schemes. In [7], we construct bases of the unitary modules of Hecke and diagrammatic Cherednik algebras of types A and, for certain parameters, G(ℓ, 1, n). 2 The BGG resolution of L(triv) ∈ O 1 h (W ) for W a real reflection group, where triv is the trivial representation of W and h is the Coxeter number of W , was constructed in [4].…”