## Cyclotomic Solomon algebras

### Andrew Mathas and Rosa Orellana

#### 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.

Keywords:
Complex reflection groups, Solomon descent algebra.

AMS Subject Classification:
Primary 16W30; secondary 20C05, 05E15.