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Homomorphisms into Specht modules labelled by hooks in quantum characteristic two

Berta Hudak

Abstract

Let $R_n$ denote the KLR algebra of type $A^{(1)}_{e-1}$. Using the presentation of Specht modules given by Kleschev-Mathas-Ram, Loubert completely determined $\hom_{R_n}(S^μ,S^λ)$ where $μ$ is an arbitrary partition, $λ$ is a hook and $e\neq2$. In this paper, we investigate the same problem when $e=2$. First we give a complete description of the action of the generators on the basis elements of $S^λ$. We use this result to identify a large family of partitions $μ$ such that there exists at least one non-zero homomorphism from $S^μ$ to $S^λ$, explicitly describe these maps and give their grading. Finally, we generalise James's result for the trivial module.

Homomorphisms into Specht modules labelled by hooks in quantum characteristic two

Abstract

Let denote the KLR algebra of type . Using the presentation of Specht modules given by Kleschev-Mathas-Ram, Loubert completely determined where is an arbitrary partition, is a hook and . In this paper, we investigate the same problem when . First we give a complete description of the action of the generators on the basis elements of . We use this result to identify a large family of partitions such that there exists at least one non-zero homomorphism from to , explicitly describe these maps and give their grading. Finally, we generalise James's result for the trivial module.
Paper Structure (18 sections, 32 theorems, 157 equations)

This paper contains 18 sections, 32 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Lie theoretical notation
  4. Partitions and tableaux
  5. Bruhat order
  6. KLR algebras
  7. Universal Specht modules
  8. Specht modules indexed by hook partitions
  9. Action of $\psi_k$ with $k$ even
  10. Three consecutive elements in $\operatorname{Leg}(\mathtt{T})$ or $\operatorname{Arm}(\mathtt{T})$
  11. Two consecutive pairs in $\operatorname{Leg}(\mathtt{T})$ or $\operatorname{Arm}(\mathtt{T})$
  12. At least three consecutive pairs in $\operatorname{Leg}(\mathtt{T})$ or $\operatorname{Arm}(\mathtt{T})$
  13. Homomorphisms into S-lambda
  14. A novel way to label hook partitions
  15. Homomorphisms from hook partitions
  16. ...and 3 more sections

Key Result

Theorem 2.2

kmr. Let $\mu \in {\mathcal{P}}_n$. Then the universal Specht module $S^\mu$ for $R^{\Lambda}_n$ has homogeneous $\mathbb{F}$-basis

Theorems & Definitions (100)

  • Definition 2.1
  • Theorem 2.2
  • Definition 3.1
  • Definition 3.2
  • Remark
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Remark
  • Lemma 3.5
  • ...and 90 more