Representations of $0$-Yokonuma-Hecke algebras

TL;DR

This paper classifies simple modules of the 0-Yokonuma-Hecke algebra using two approaches and relates it to the 0-Ariki-Koike-Shoji algebra, also providing classification for the nil Yokonuma-Hecke algebra.

Contribution

It introduces two methods for classifying simple modules of 0-Yokonuma-Hecke algebras and connects these classifications to known results for related algebras.

Findings

Classified simple modules of 0-Yokonuma-Hecke algebra.

Established isomorphism with 0-Ariki-Koike-Shoji algebra for module classification.

Provided classification of simple modules for nil Yokonuma-Hecke algebra.

Abstract

We give two different approaches to classifying the simple modules of $0$-Yokonuma-Hecke algebras $Y_{r,n}(0)$ over an algebraically closed field of characteristic $p$ such that $p$ does not divide $r.$ Using the isomorphism between the $0$-Yokonuma-Hecke algebra and $0$-Ariki-Koike-Shoji algebra, we in fact give another way to obtain the simple modules of the latter, which was previously studied by Hivert, Novelli and Thibon (Adv. Math. $\bf{205}$ (2006) 504-548). In the appendix, we give the classification of simple modules of the nil Yokonuma-Hecke algebra.