On representations of complex reflection groups G(m,1,n)

TL;DR

This paper develops an inductive approach to understanding the representations of complex reflection groups G(m,1,n), introducing new algebraic tools and spectra analysis to advance the theoretical framework.

Contribution

It introduces a novel inductive method and constructs representations of G(m,1,n) using a new associative algebra linked to standard m-tableaux.

Findings

Derived Jucys-Murphy elements from cyclotomic Hecke algebra

Analyzed the spectrum using degenerate cyclotomic affine Hecke algebra

Constructed representations via a new associative algebra

Abstract

An inductive approach to the representation theory of the chain of the complex reflection groups G(m,1,n) is presented. We obtain the Jucys-Murphy elements of G(m,1,n) from the Jucys--Murphy elements of the cyclotomic Hecke algebra, and study their common spectrum using representations of a degenerate cyclotomic affine Hecke algebra. Representations of G(m,1,n) are constructed with the help of a new associative algebra whose underlying vector space is the tensor product of the group ring of G(m,1,n) with a free associative algebra generated by the standard m-tableaux.