Interval Garside Structures Related to the Affine Artin Groups of Type $\tilde{A}$

TL;DR

This paper constructs new interval Garside structures for affine Artin groups of type tA, moving beyond dual presentations, marking the first such development for these groups.

Contribution

It introduces the first successful construction of interval Garside structures for affine Artin groups unrelated to dual presentations.

Findings

Constructed interval Garside structures for tA using alternative presentations.

Demonstrated that these structures are not derived from dual presentations.

Extended Garside theory to new classes of affine Artin groups.

Abstract

Garside theory emerged from the study of Artin groups and their generalizations. Finite-type Artin groups admit two types of interval Garside structures corresponding to their standard and dual presentations. Concerning affine Artin groups, Digne established interval Garside structures for two families of these groups by using their dual presentations. Recently, McCammond established that none of the remaining dual presentations (except for one additional case) correspond to interval Garside structures. In this paper, shifting attention from dual presentations to other nice presentations for the affine Artin group of type $\tilde{A}$ discovered by Shi and Corran--Lee--Lee, I will construct interval Garside structures related to this group. This construction is the first successful attempt to establish interval Garside structures not related to the dual presentations in the case of affine Artin groups.