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Verma Howe duality and LKB representations

Abel Lacabanne, Daniel Tubbenhauer, Pedro Vaz

TL;DR

The paper introduces Verma Howe duality, replacing symmetric powers with Verma modules and establishing a commuting-action framework between $U_q( rak{gl}_2)$ and $U_q( rak{gl}_n)$. It constructs dense, non-highest/lowest weight multiplicity spaces and, via a GT-pattern/Casimir analysis and flatness arguments, lifts the duality to the quantum setting. As a key application, colored higher LKB representations arise from this duality and are proven to be simple modules for braid groups and relevant subgroups, including the pure braid group, over general fields and parameters. The results broaden Howe duality to infinite-dimensional, nonsemisimple contexts and pave the way for further categorification and connections to handlebody braids and symmetric web calculi.

Abstract

We establish a version of Howe duality that involves a tensor product of Verma modules. Surprisingly, this duality leaves the realm of lowest and highest weight modules. We quantize this duality, and as an application, we prove that the (colored higher) LKB representations arise from this duality and use this description to show that they are simple as modules for the braid group and for various of its subgroups, including the pure braid group.

Verma Howe duality and LKB representations

TL;DR

The paper introduces Verma Howe duality, replacing symmetric powers with Verma modules and establishing a commuting-action framework between and . It constructs dense, non-highest/lowest weight multiplicity spaces and, via a GT-pattern/Casimir analysis and flatness arguments, lifts the duality to the quantum setting. As a key application, colored higher LKB representations arise from this duality and are proven to be simple modules for braid groups and relevant subgroups, including the pure braid group, over general fields and parameters. The results broaden Howe duality to infinite-dimensional, nonsemisimple contexts and pave the way for further categorification and connections to handlebody braids and symmetric web calculi.

Abstract

We establish a version of Howe duality that involves a tensor product of Verma modules. Surprisingly, this duality leaves the realm of lowest and highest weight modules. We quantize this duality, and as an application, we prove that the (colored higher) LKB representations arise from this duality and use this description to show that they are simple as modules for the braid group and for various of its subgroups, including the pure braid group.
Paper Structure (16 sections, 27 theorems, 50 equations)

This paper contains 16 sections, 27 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Schur--Weyl(--Brauer) and Howe dualities
  3. What this paper does
  4. Outlook
  5. A duality involving Verma modules
  6. Verma and dense modules
  7. Verma Howe duality
  8. The proof of Verma Howe duality
  9. The classical case
  10. The quantum case
  11. The colored higher LKB representations are simple
  12. Pure and colored braids
  13. LKB representations
  14. The proof of simplicity
  15. The classical case
  16. ...and 1 more sections

Key Result

Lemma 1

All quantum numbers are nonzero and thus invertible.

Theorems & Definitions (104)

  • Remark 1
  • Remark 2
  • Remark 3
  • Definition 1
  • Lemma 1
  • proof
  • Remark 4
  • Definition 2
  • Remark 5
  • Example 1
  • ...and 94 more