Schur–Weyl duality, Verma modules, and row quotients of Ariki–Koike algebras

Abstract

We study the geometry and topology of ∆-Springer varieties associated with tworow partitions. These varieties were introduced in recent work by Griffin-Levinson-Woo to give a geometric realization of a symmetric function appearing in the Delta conjecture by Haglund-Remmel-Wilson. We provide an explicit and combinatorial description of the irreducible components of the two-row ∆-Springer variety and compare it to the ordinary two-row Springer fiber as well as Kato's exotic Springer fiber corresponding to a one-row bipartition. In addition to that, we extend the action of the symmetric group on the homology of the two-row ∆-Springer variety to an action of a degenerate affine Hecke algebra and relate this action to a gl 2 -tensor space. CONTENTS 1. Introduction 1 2. ∆-Springer varieties 5 3. Irreducible components for two-row ∆-Springer varieties 9 4. Comparison with Springer fibers and exotic Springer fibers 14 5. A topological model and the action of the symmetric group 16 6. Action of the degenerate affine Hecke algebra 20 References 30

Handlebody diagram algebras

“…Some work for higher genus on this representation theoretical aspect is done, e.g. in relation to Verma modules [ILZ21], [LV21], [DR18] or complex reflection groups [MaSt16], [SS99]. Following this track for higher genus seems to be a worthwhile goal.…”

On a symplectic quantum Howe duality

“…Sometimes, however, Howe's dualities quantize nicely (meaning with the standard Drinfeld-Jimbo quantum groups). In type A, see for example [LZZ11] or [CKM14] for exterior, [RT16] for symmetric, [TVW17] for exterior-symmetric and [LTV22] for Verma versions of nicely quantized Howe dualities. But again, proving (a) might still be tricky, in particular outside of type A, as e.g.…”

Verma Howe duality and LKB representations

“…Theorem 2B.3 captures the tensor product of Verma modules, while several papers discuss dualities involving one Verma and tensor products of finite dimensional modules, see e.g. [ILZ21] or [LV21]. It would be interesting to combine and compare these, also with an eye on categorification of the story as in [LNV21].…”