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Combinatorics of irreducible characters for Lie superalgebra $\frak{gl}(m,n)$

Alexander Sergeev

Abstract

In this paper we give a new formula for characters of finite dimensional irreducible $\frak{gl}(m,n)$ modules. We use two main ingredients: Su-Zhang formula and Brion's theorem.

Combinatorics of irreducible characters for Lie superalgebra $\frak{gl}(m,n)$

Abstract

In this paper we give a new formula for characters of finite dimensional irreducible modules. We use two main ingredients: Su-Zhang formula and Brion's theorem.
Paper Structure (5 sections, 23 theorems, 103 equations)

This paper contains 5 sections, 23 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Su-Zhang formula
  4. Graphs and polyhedra
  5. Character formula

Key Result

Theorem 1.1

Let $L(\chi)$ be a finite dimensional irreducible module over Lie superalgebra $\frak{gl}(m,n)$ with the highest weight $\chi$. Then the following equality holds true where $\Gamma_{f}$ is the graph which is explicitly constructed from the weight diagram $f$ of $\chi$, $S_{\chi}=\{\alpha_1<\dots<\alpha_r\}$ is a maximal orthogonal set of atypical roots and $\theta(\Gamma_f,t_1,\dots,t_r)$ is a La

Theorems & Definitions (62)

  • Theorem 1.1
  • Theorem 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7
  • Definition 2.8
  • Lemma 2.9
  • ...and 52 more