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

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.