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Spin Multipartitions

Ola Amara-Omari, Mary Schaps

TL;DR

The paper advances spin representation theory by formulating a spin multipartition framework for the twisted affine algebra $A^{(2)}_{2n}$ and proving that the level-one spin Fock spaces $\mathcal{F}_i$ are modules over the quantum enveloping algebra $U_q(\mathfrak{g})$. It introduces an $h$-restricted spin multipartition algorithm that, via Kashiwara crystal theory and the Hopf structure of $U_q(\mathfrak{g})$, aims to string level-one partitions into genuine multipartitions, extending Ariki–Mathas style constructions to the spin setting. The Serre-relations analysis for $\Lambda=\Lambda_i$ verifies the $U_q(\mathfrak{g})$-module structure on the corresponding Fock spaces, including the intricate cases with $a_{jk}=-1$ and $-2$, thereby solidifying the connection between spin blocks, crystal combinatorics, and quantum groups. Overall, the work lays a principled pathway to label spin representations via spin multipartitions and to realize them as modules of a quantum affine algebra, with potential implications for categorification and modular representation theory of Hecke algebras.``

Abstract

We conjecture an algorithm to construct spin multipartitions and prove that all the level one Fock spaces using our combinatorics are modules over the quantum enveloping algebra.

Spin Multipartitions

TL;DR

The paper advances spin representation theory by formulating a spin multipartition framework for the twisted affine algebra and proving that the level-one spin Fock spaces are modules over the quantum enveloping algebra . It introduces an -restricted spin multipartition algorithm that, via Kashiwara crystal theory and the Hopf structure of , aims to string level-one partitions into genuine multipartitions, extending Ariki–Mathas style constructions to the spin setting. The Serre-relations analysis for verifies the -module structure on the corresponding Fock spaces, including the intricate cases with and , thereby solidifying the connection between spin blocks, crystal combinatorics, and quantum groups. Overall, the work lays a principled pathway to label spin representations via spin multipartitions and to realize them as modules of a quantum affine algebra, with potential implications for categorification and modular representation theory of Hecke algebras.``

Abstract

We conjecture an algorithm to construct spin multipartitions and prove that all the level one Fock spaces using our combinatorics are modules over the quantum enveloping algebra.
Paper Structure (5 sections, 8 theorems, 25 equations)

This paper contains 5 sections, 8 theorems, 25 equations.

Table of Contents

  1. INTRODUCTION
  2. DEFINITIONS AND NOTATION
  3. DEFECTS IN THE REDUCED CRYSTAL GRAPH
  4. THE SPIN MULTIPARTITION ALGORITHM
  5. THE SERRE RELATIONS FOR $\Lambda=\Lambda_i$

Key Result

Theorem 1

Let $\mathfrak{g}$ be the affine Lie algebra of type $A^{(2)}_{2n}$ for a positive integer $n$. The level one Fock spaces $\mathcal{F}_i$ given by $i$-corner partitions are modules for the quantum enveloping algebra $U_q(\mathfrak{g})$.

Theorems & Definitions (25)

  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 3.1
  • Lemma 3.1
  • proof
  • Corollary 3.1.1
  • proof
  • Definition 4.1
  • ...and 15 more