Cyclotomic Solomon Algebras

TL;DR

This paper generalizes Solomon descent algebras to complex reflection groups of type G(r,1,n), describing their structure, representations, and a Hopf algebra isomorphism, including a q-analogue deformation.

Contribution

It introduces cyclotomic Solomon algebras for complex reflection groups, providing explicit structure constants, irreducible representations, and a Hopf algebra connection.

Findings

Structure constants are polynomials in r

Explicit irreducible representations are determined

The algebra admits a q-analogue deformation

Abstract

This paper introduces an analogue of the Solomon descent algebra for the complex reflection groups of type $G(r,1,n)$. As with the Solomon descent algebra, our algebra has a basis given by sums of `distinguished' coset representatives for certain `reflection subgroups'. We explicitly describe the structure constants with respect to this basis and show that they are polynomials in $r$. This allows us to define a deformation, or $q$-analogue, of these algebras which depends on a parameter $q$. We determine the irreducible representations of all of these algebras and give a basis for their radicals. Finally, we show that the direct sum of cyclotomic Solomon algebras is canonically isomorphic to a concatenation Hopf algebra.