Internal Zonotopal Algebras and the Monomial Reflection Groups

TL;DR

This paper investigates the internal zonotopal algebra associated with the Gale dual of the reflection arrangement of the complex reflection group G(m,1,n), revealing its structure, representation stability, and connections to known combinatorial objects.

Contribution

It provides a formula for the top degree component as an induced character and explores the algebra's representation-theoretic and combinatorial properties.

Findings

Derived a formula for the top degree component as an induced character.

Established connections to the Whitehouse representation in type A.

Explored analogs of decreasing trees in type B.

Abstract

The group $G(m,1,n)$ consists of $n$-by-$n$ monomial matrices whose entries are $m$th roots of unity. It is generated by $n$ complex reflections acting on $\mathbf{C}^n$. The reflecting hyperplanes give rise to a (hyperplane) arrangement $\mathcal{G} \subset \mathbf{C}^n$. The internal zonotopal algebra of an arrangement is a finite dimensional algebra first studied by Holtz and Ron. Its dimension is the number of bases of the associated matroid with zero internal activity. In this paper we study the structure of the internal zonotopal algebra of the Gale dual of the reflection arrangement of $G(m,1,n)$, as a representation of this group. Our main result is a formula for the top degree component as an induced character from the cyclic group generated by a Coxeter element. We also provide results on representation stability, a connection to the Whitehouse representation in type~A, and an analog of decreasing trees in type~B.