Defining sequences for fundamental root systems and Coxeter graphs for super Weyl groups

Abstract

The super Weyl group of a basic classical Lie superalgebra was introduced and studied in \cite{PS}, which turns out to play an important role for the study of representations of the basic classical Lie superalgebras and algebraic supergroups (see \cite{PS, LS}). These groups turn out to be some quotients of Coxeter groups. It is deserved to specially investigate super Weyl groups via revealing the related Coxeter systems. The purpose of this paper is twofold. One is to describe the Coxeter systems for super Weyl groups of basic classical Lie superalgebras. The other one is to introduce defining sequences which are a kind of new descriptions of fundamental root systems for classical Lie superalgebras of type $A,B,C$ and $D$. Based on defining sequences, we decide the Coxeter groups associated with those super Weyl groups via Coxeter graphs.