OGS canonical forms and exchange laws for the I and for the A-type Coxeter groups

TL;DR

This paper generalizes the OGS canonical form for certain Coxeter groups, especially type A, and introduces exchange laws, elementary factorizations, and algorithms to connect permutation properties with Coxeter length.

Contribution

It extends the OGS canonical form and exchange laws from dihedral groups to A-type Coxeter groups, providing new formulas and algorithms for permutation analysis.

Findings

New explicit formula for Coxeter length of permutations

Introduction of standard OGS elementary factorization

Algorithms for OGS canonical form and inverse descent set

Abstract

We consider a generalization of the fundamental theorem of finitely generated abelian groups for some non-abelian groups, which is called OGS. First, we consider the dihedral group, which is a non-abelian extension of an abelian group by an involution. Then, we focus on a special case, where the abelian group is cyclic, which is the two-generated Coxeter group I{2}(m). We mention interesting connections between the reduced Coxeter presentation and a particular OGS canonical presentation, which we call the standard OGS canonical presentation. These connections motivate us to offer a generalization of the standard OGS to the A-type Coxeter group, which can be considered as the dual family to the I-type Coxeter groups. The n-1 generated A-type Coxeter groups can be considered as the symmetric group S{n} for an arbitrary n. We mention the standard and the dual-standard OGS of S{n}, where, The standard OGS canonical form of S{n} has a special interest in combinatorics, since in 2001, R. M. Adin, and Y. Roichman has proved that sum of the exponents in the canonical form is coincide with the major-index of the permutation, which is equi-disributed with the Coxeter length. In this paper we extend the results of Adin and Roichman very significantly, where we show interesting properties of the exchange laws, we define standard OGS elementary factorization, which connects between the standard OGS and the descent set of a permutation. Then, by using the standard OGS elementary factorization, we find a new explicit formula for the Coxeter length of an element of S{n}, and we give a new algorithm for the standard OGS canonical form and the descent set of the inverse element of an arbitrary element of S{n}.