Generalization of the basis theorem for the D-type Coxeter groups

TL;DR

This paper extends the concept of ordered generating systems (OGS) from symmetric groups to the D-type Coxeter groups, providing new descriptions and connections to Coxeter length.

Contribution

It generalizes the standard OGS from symmetric groups to Dn Coxeter groups, including exchange laws and length connections.

Findings

Established exchange laws for the generalized OGS of Dn

Connected OGS to Coxeter length of elements in Dn

Extended the basis theorem to D-type Coxeter groups

Abstract

The OGS for non-abelian groups is an interesting generalization of the basis of finite abelian groups. The definition of OGS states that every element of a group has a unique presentation as a product of some powers of specific generators of the group, in a specific given order. In case of the symmetric groups Sn there is a paper of R. Shwartz, which demonstrates a strong connection between the OGS and the standard Coxeter presentation of the symmetric group, which is called the standard OGS of Sn. In this paper we generalize the standard OGS of Sn to the finite classical Coxeter group Dn. We describe the exchange laws for the generalized standard OGS of Dn, and we connect it to the Coxeter length of elements of Dn.