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Subdivision and Runner Removal Theorems

Tao Qin

Abstract

We develop a combinatorial framework for the subdivision map -- introduced by Maksimau, Mathas and Tubbenhauer -- between the KLR(W) algebras of type $A^{(1)}_{e-1}$ and type $A^{(1)}_{e}$, which provides a partial categorification of the runner removal theorems.

Subdivision and Runner Removal Theorems

Abstract

We develop a combinatorial framework for the subdivision map -- introduced by Maksimau, Mathas and Tubbenhauer -- between the KLR(W) algebras of type and type , which provides a partial categorification of the runner removal theorems.
Paper Structure (39 sections, 55 theorems, 222 equations, 2 figures)

This paper contains 39 sections, 55 theorems, 222 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Lie Data and Combinatorics
  3. Cartan data
  4. Young diagrams and Abaci
  5. Multipartitions, charges and residues
  6. Tableaux
  7. Dominance order and Bruhat order on tableaux
  8. Degree of tableaux
  9. Garnir tableaux
  10. KLR Algebras and Universal Specht Module
  11. KLR algebras
  12. Module category
  13. Universal Specht module
  14. Permutation modules
  15. Homogeneous Garnir relations
  16. ...and 24 more sections

Key Result

Lemma 1

Let $S,T$ be row-standard ${\boldsymbol\nu}$-tableaux. Then

Figures (2)

  • Figure 1: Expansion of $G_3((6),\,(5,1,1))$.
  • Figure 2: Expansion of $G_4((8),\,(7,1,1))$.

Theorems & Definitions (144)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Lemma 1
  • proof
  • Example 5
  • Definition 1
  • Definition 2
  • Example 6
  • ...and 134 more