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

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.

PRINCIPAL RADICAL SYSTEMS, LEFSCHETZ PROPERTIES, AND PERFECTION OF SPECHT IDEALS OF TWO-ROWED PARTITIONS

We show that the Specht ideal of a two-rowed partition is perfect over an arbitrary field, provided that the characteristic is either zero or bounded below by the size of the second row of the partition, and we show this lower bound is tight. We also establish perfection and other properties of certain variants of Specht ideals, and find a surprising connection to the weak Lefschetz property. Our results, in particular, give a self-contained proof of Cohen–Macaulayness of certain h-equals sets, a result previously obtained by Etingof–Gorsky–Losev over the complex numbers using rational Cherednik algebras.

Characterization of quasi-Yetter–Drinfeld modules

In this paper, we characterize quasi-Yetter–Drinfeld modules over a Hopf algebra [Formula: see text], which was introduced in [Y. Bazlov and A. Berenstein, Braided doubles and rational Cherednik algebras, Adv. Math. 220 (2009), 1466–1530]. We first show that the quasi-Drinfeld center of the category of [Formula: see text]-modules is equivalent to the category [Formula: see text] of quasi-Yetter–Drinfeld modules. Next, we prove that [Formula: see text] is equivalent to the category of generalized Hopf bimodules. Finally, we show that [Formula: see text] is also equivalent to the category of quasi-coactions over some Majid’s braided group if [Formula: see text] is quasi-triangular.

Towards a classification of finite-dimensional representations of rational Cherednik algebras of type D

Using a combinatorial description due to Jacon and Lecouvey of the wall crossing bijections for cyclotomic rational Cherednik algebras, we show that the irreducible representations [Formula: see text] of the rational Cherednik algebra [Formula: see text] of type [Formula: see text] for symmetric bipartitions [Formula: see text] are infinite dimensional for all parameters [Formula: see text]. In particular, all finite dimensional irreducible representations of rational Cherednik algebras of type [Formula: see text] arise as restrictions of finite-dimensional irreducible representations of rational Cherednik algebras of type [Formula: see text].