Grobner-Shirshov Basis for affine Weyl Group $\widetilde{A_n}$

TL;DR

This paper applies algebraic algorithms to derive a basis for the affine Weyl group A_n, enabling classification of all reduced words within this mathematical structure.

Contribution

It introduces a Grobner-Shirshov basis for A_n using Buchberger-Shirshov algorithm, providing a systematic method for understanding its reduced words.

Findings

Derived the reduced Grobner-Shirshov basis for A_n

Classified all reduced words of the affine Weyl group A_n

Established a new algebraic framework for affine Weyl groups

Abstract

Using Buchberger-Shirshov Algorithm and Composition-Diamond lemma we obtain the reduced Grobner-Shirshov bases of $\widetilde{A_n}$ and classify all reduced words of the affine Weyl group $\widetilde{A_n}$.