# Invariant theory for coincidental complex reflection groups

## TL;DR

This paper investigates the invariant theory of coincidental complex reflection groups, correcting a previous conjecture and deriving new product formulas for related polynomials and combinatorial identities.

## Contribution

It demonstrates that Molchanov's Hilbert series conjecture is valid for all coincidental complex reflection groups with modifications, and derives new formulas for $q$-Narayana, $q$-Kirkman polynomials, and cluster complexes.

## Key findings

- Molchanov's conjecture is false in general but holds for coincidental groups with modifications.
- Derived simple product formulas for $q$-Narayana and $q$-Kirkman polynomials.
- Established a $q$-analogue of the $h$-vector to $f$-vector transformation for certain cluster complexes.

## Abstract

V.F. Molchanov considered the Hilbert series for the space of invariant skew-symmetric tensors and dual tensors with polynomial coefficients under the action of a real reflection group, and speculated that it had a certain product formula involving the exponents of the group. We show that Molchanov's speculation is false in general but holds for all coincidental complex reflection groups when appropriately modified using exponents and co-exponents. These are the irreducible well-generated (i.e., duality) reflection groups with exponents forming an arithmetic progression and include many real reflection groups and all non-real Shephard groups, e.g., the Shephard-Todd infinite family $G(d,1,n)$. We highlight consequences for the $q$-Narayana and $q$-Kirkman polynomials, giving simple product formulas for both, and give a $q$-analogue of the identity transforming the $h$-vector to the $f$-vector for the coincidental finite type cluster/Cambrian complexes of Fomin--Zelevinsky and Reading.