An isomorphism theorem for degenerate cyclotomic Yokonuma-Hecke algebras and applications

TL;DR

This paper establishes an explicit isomorphism between degenerate cyclotomic Yokonuma-Hecke algebras and matrix algebras over tensor products of cyclotomic Hecke algebras, leading to new insights into their structure and representation theory.

Contribution

It provides a novel algebra isomorphism that simplifies the understanding of Yokonuma-Hecke algebras and enables new proofs and criteria for their properties.

Findings

Derived a new proof of the modular representation theory

Established a semisimplicity criterion

Proved that the algebra is symmetric and determined Schur elements

Abstract

Inspired by the work [PA], we establish an explicit algebra isomorphism between the degenerate cyclotomic Yokonuma-Hecke algebra $Y_{r,n}^{d}(q)$ and a direct sum of matrix algebras over tensor products of degenerate cyclotomic Hecke algebras of type $A$. We then develop several applications of this result, including a new proof of the modular representation theory of $Y_{r,n}^{d}(q)$, a semisimplicity criterion for it and cellularity of it. Moreover, we prove that $Y_{r,n}^{d}(q)$ is a symmetric algebra and determine the associated Schur elements by using the isomorphism theorem for it.