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On interpolation categories for the hyperoctahedral group

Thorsten Heidersdorf, George Tyriard

Abstract

Two different types of Deligne categories have been defined to interpolate the finite dimensional complex representations of the hyperoctahedral group. The first one, initially defined by Knop and then further studied by Likeng and Savage, uses a categorical analogue of the permutation representation as a tensor generator. The second one, due to Flake and Maassen, is tensor generated by a categorical analogue of the reflection representation. We construct a symmetric monoidal functor between the two and show that it is an equivalence of symmetric monoidal categories.

On interpolation categories for the hyperoctahedral group

Abstract

Two different types of Deligne categories have been defined to interpolate the finite dimensional complex representations of the hyperoctahedral group. The first one, initially defined by Knop and then further studied by Likeng and Savage, uses a categorical analogue of the permutation representation as a tensor generator. The second one, due to Flake and Maassen, is tensor generated by a categorical analogue of the reflection representation. We construct a symmetric monoidal functor between the two and show that it is an equivalence of symmetric monoidal categories.
Paper Structure (21 sections, 26 theorems, 102 equations)

This paper contains 21 sections, 26 theorems, 102 equations.

Table of Contents

  1. Introduction
  2. The hyperoctahedral groups
  3. Background
  4. Conventions about monoidal categories
  5. Additive idempotent completion
  6. The hyperoctahedral group
  7. The Deligne categories attached to the permutation and reflection representation
  8. Partition diagrams
  9. The permutation category of Knop and Likeng-Savage
  10. The reflection category of Flake and Maassen
  11. Universal properties
  12. A presentation for the permutation category
  13. A presentation for the reflection category
  14. An equivalence of symmetric monoidal categories
  15. Relation to the representations of $H_n$
  16. ...and 6 more sections

Key Result

Theorem 1.1

There is a symmetric monoidal equivalence such that the diagram \begin{tikzcd} \underline{\text{Rep}}(H_{n}) \arrow{r}{G} \arrow[d,"\Omega"] & \text{Rep}(H_{n})\arrow{d}{=} \ \\ \text{Par}(\mathbb{Z}_{2},2n)^{Kar} \arrow{r}{H} & \text{Rep}(H_{n}) \end{tikzcd}commutes for all $n\in \mathop{\mathrm{\mathbb{N}}}\nolimits$. Here $G$ and $H$ den

Theorems & Definitions (70)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Remark 2.8
  • ...and 60 more