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Hyperoctahedral Schur Category and Hyperoctahedral Web Category

Razzi Masroor

Abstract

We extend the Schur algebra and the polynomial web category of the symmetric group to the hyperoctahedral group. In particular, we define the hyperoctahedral web category diagrammatically by generators and relations, and prove that it is equivalent to the hyperoctahedral Schur category.

Hyperoctahedral Schur Category and Hyperoctahedral Web Category

Abstract

We extend the Schur algebra and the polynomial web category of the symmetric group to the hyperoctahedral group. In particular, we define the hyperoctahedral web category diagrammatically by generators and relations, and prove that it is equivalent to the hyperoctahedral Schur category.
Paper Structure (17 sections, 15 theorems, 136 equations)

This paper contains 17 sections, 15 theorems, 136 equations.

Table of Contents

  1. Introduction
  2. Relation to previous work
  3. Organization of the paper
  4. Acknowledgements
  5. Preliminary Definitions, Theorems, and Examples for the Schur Algebra
  6. Symmetric and Hyperoctahedral Groups
  7. Schur Algebra
  8. Web Diagrams and Chicken Foot Diagrams
  9. Definitions for the Hyperoctahedral Group
  10. Basic Definitions for the Hyperoctahedral Schur Algebra and Category
  11. Definition of Hyperoctahedral Web Category
  12. Equivalence between $H\mathcal{W}eb$ and $H\mathcal{S}chur$
  13. Verifying Defining Relations
  14. Fullness of Phi
  15. Reduction of Diagrams
  16. ...and 2 more sections

Key Result

Lemma 2.22

The action of $H_n$ on $H_n/H_\lambda$ is isomorphic to its action on $I_\lambda$.

Theorems & Definitions (77)

  • Remark 1.1
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Remark 2.9
  • ...and 67 more