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

TL;DR

This paper extends the concept of the ordered generating system (OGS) from symmetric groups to the finite classical Coxeter group Bn, detailing exchange laws and their relation to Coxeter length and descent sets.

Contribution

It generalizes the standard OGS from symmetric groups to the Coxeter group Bn, providing new exchange laws and connections to Coxeter length and descent sets.

Findings

Established exchange laws for the generalized OGS of Bn

Connected OGS to Coxeter length and descent set of Bn

Extended the basis theorem to B-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 Bn. We describe the exchange laws for the generalized standard OGS of Bn, and we connect it to the Coxeter length and the descent set of Bn.