# Invariant theory for coincidental complex reflection groups

@article{Reiner2020InvariantTF, title={Invariant theory for coincidental complex reflection groups}, author={Victor Reiner and Anne V. Shepler and Eric N. Sommers}, journal={Mathematische Zeitschrift}, year={2020}, volume={298}, pages={787-820} }

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 he 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…

## 4 Citations

Eulerian representations for real reflection groups

- MathematicsJournal of the London Mathematical Society
- 2022

The Eulerian idempotents, first introduced for the symmetric group and later extended to all reflection groups, generate a family of representations called the Eulerian representations that decompose…

Harmonic differential forms for pseudo-reflection groups I. Semi-invariants

- MathematicsJ. Comb. Theory, Ser. A
- 2021

Alternating super-polynomials and super-coinvariants of finite reflection groups

- Mathematics
- 2019

Motivated by a recent conjecture of Zabrocki, Wallach described the alternants in the super-coinvariant algebra of the symmetric group in one set of commuting and one set of anti-commuting variables…

$q$-Kreweras numbers for coincidental Coxeter groups attached to limit symbols.

- Mathematics
- 2020

For a coincidental Coxeter group, i.e. of type $A_{n-1}, BC_n, H_3,$ or $I_2(m)$, we define the corresponding $q$-Kreweras numbers attached to limit symbols in the sense of Shoji. The construction of…

## References

SHOWING 1-10 OF 43 REFERENCES

Generalized Exponents and Forms

- Mathematics
- 2005

We consider generalized exponents of a finite reflection group acting on a real or complex vector space V. These integers are the degrees in which an irreducible representation of the group occurs in…

Invariant derivations and differential forms for reflection groups

- MathematicsProceedings of the London Mathematical Society
- 2019

Classical invariant theory of a complex reflection group W beautifully describes the W ‐invariant polynomials, the W ‐invariant differential forms, and the relative invariants of any W…

Generalized cluster complexes and Coxeter combinatorics

- Mathematics
- 2005

We introduce and study a family of simplicial complexes associated to an arbitrary finite root system and a nonnegative integer parameter m. For m=1, our construction specializes to the (simplicial)…

Invariants of finite groups and their applications to combinatorics

- Mathematics
- 1979

1 CONTENTS 1. Introduction 2. Molien's theorem 3. Cohen-Macaulay rings 4. Groups generated by pseudo-reflections 5. Three applications 6. Syzygies 7. The canonical module 8. Gorenstein rings 9.…

Y-systems and generalized associahedra

- Mathematics
- 2003

The goals of this paper are two-fold. First, we prove, for an arbitrary finite root system D, the periodicity conjecture of Al. B. Zamolodchikov [24] that concerns Y-systems, a particular class of…

Hopf Algebras in Combinatorics

- Mathematics
- 2014

These notes -- originating from a one-semester class by their second author at the University of Minnesota -- survey some of the most important Hopf algebras appearing in combinatorics. After…

Foulkes characters for complex reflection groups

- Mathematics
- 2015

Foulkes discovered a marvelous set of characters for the symmetric group by summing Specht modules of certain ribbon shapes according to height. These characters have many remarkable properties and…