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Coxeter Graphs for Super Weyl Groups of Exceptional Classical Lie Superalgebras

Yuhui Shen, Zhiyang Tan

Abstract

Super Weyl group plays an important role in the study of representations of basic classical Lie superalgebras. The Coxeter graphs for super Weyl groups of basis classical Lie superalgebras have been given in \cite{CLS}, where the authors also made a proposal on the Coxeter graphs for the super Weyl groups of exceptional classical Lie superalgebras $D(2,1,α)$, $F(4)$ and $G(3)$. In this paper, we present all fundamental systems of the exceptional Lie superalgebras and get the Coxeter graphs for the corresponding super Weyl groups, verifying a proposal in \cite{CLS}.

Coxeter Graphs for Super Weyl Groups of Exceptional Classical Lie Superalgebras

Abstract

Super Weyl group plays an important role in the study of representations of basic classical Lie superalgebras. The Coxeter graphs for super Weyl groups of basis classical Lie superalgebras have been given in \cite{CLS}, where the authors also made a proposal on the Coxeter graphs for the super Weyl groups of exceptional classical Lie superalgebras , and . In this paper, we present all fundamental systems of the exceptional Lie superalgebras and get the Coxeter graphs for the corresponding super Weyl groups, verifying a proposal in \cite{CLS}.

Paper Structure

This paper contains 10 sections, 3 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. The root system and Coxeter graph for $D(2,1,\alpha)$
  3. The root system
  4. The Weyl group and Coxeter graph
  5. The root system and Coxeter graph for $F(4)$
  6. Root system for $F(4)$
  7. Coxeter graph
  8. The root system and Coxeter graph for $G(3)$
  9. Root system of $G(3)$
  10. Coxeter graph

Key Result

Theorem 2.1

The Coxeter graph of $\widehat{\mathscr{C}}$ for $D(2,1,\alpha)$ is

Theorems & Definitions (3)

  • Theorem 2.1
  • Theorem 3.1
  • Theorem 4.1