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.

Stability for Representations of Hecke Algebras of Type A

In this paper, we introduce the concept of the stability of a sequence of modules over Hecke algebras. We prove that a finitely generated consistent sequence of the representations of Hecke algebras is representation stable.

Equivariant multiplicities of simply-laced type flag minors

Let g \mathfrak {g} be a finite simply-laced type simple Lie algebra. Baumann-Kamnitzer-Knutson recently defined an algebra morphism D ¯ \overline {D} on the coordinate ring C [ N ] \mathbb {C}[N] related to Brion’s equivariant multiplicities via the geometric Satake correspondence. This map is known to take distinguished values on the elements of the MV basis corresponding to smooth MV cycles, as well as on the elements of the dual canonical basis corresponding to Kleshchev-Ram’s strongly homogeneous modules over quiver Hecke algebras. In this paper we show that when g \mathfrak {g} is of type A n A_n or D 4 D_4 , the map D ¯ \overline {D} takes similar distinguished values on the set of all flag minors of C [ N ] \mathbb {C}[N] , raising the question of the smoothness of the corresponding MV cycles. We also exhibit certain relations between the values of D ¯ \overline {D} on flag minors belonging to the same standard seed, and we show that in any A D E ADE type these relations are preserved under cluster mutations from one standard seed to another. The proofs of these results partly rely on Kang-Kashiwara-Kim-Oh’s monoidal categorification of the cluster structure of C [ N ] \mathbb {C}[N] via representations of quiver Hecke algebras.

Path combinatorics and light leaves for quiver Hecke algebras

We recast the classical notion of “standard tableaux" in an alcove-geometric setting and extend these classical ideas to all “reduced paths" in our geometry. This broader path-perspective is essential for implementing the higher categorical ideas of Elias–Williamson in the setting of quiver Hecke algebras. Our first main result is the construction of light leaves bases of quiver Hecke algebras. These bases are richer and encode more structural information than their classical counterparts, even in the case of the symmetric groups. Our second main result provides path-theoretic generators for the “Bott–Samelson truncation" of the quiver Hecke algebra.