Calibrated representations of affine Yokonuma-Hecke algebras

TL;DR

This paper classifies all irreducible calibrated representations of affine Yokonuma-Hecke algebras over complex numbers, providing a comprehensive understanding of their structure and applications, including classifications in degenerate cases and positive characteristic fields.

Contribution

It offers the first complete classification of irreducible calibrated representations of affine Yokonuma-Hecke algebras, extending to degenerate and positive characteristic cases.

Findings

Classified irreducible calibrated representations over or affine Yokonuma-Hecke algebras.

Developed applications of the classification in algebraic structures.

Constructed irreducible representations in degenerate and positive characteristic settings.

Abstract

Inspired by the work [Ra1], we directly give a complete classification of irreducible calibrated representations of affine Yokonuma-Hecke algebras $\widehat{Y}_{r,n}(q)$ over $\mathbb{C},$ which are indexed by $r$-tuples of placed skew shapes. We then develop several applications of this result. In the appendix, inspired by [Ru], we classify and construct irreducible completely splittable representations of degenerate affine Yokonuma-Hecke algebras $D_{r,n}$ and the wreath product $(\mathbb{Z}/r\mathbb{Z})\wr \mathfrak{S}_{n}$ over an algebraically closed field of characteristic $p> 0$ such that $p$ does not divide $r$.