Lefschetz theory for exterior algebras and fermionic diagonal coinvariants

TL;DR

This paper develops a Lefschetz-like theory for exterior algebras associated with complex reflection groups, describing the structure and invariants of fermionic diagonal coinvariants, with explicit results for symmetric groups.

Contribution

It introduces a type-uniform Lefschetz theory for exterior algebras and characterizes the bigraded structure of fermionic diagonal coinvariants for complex reflection groups.

Findings

Describes the bigraded isomorphism type of $DR_W$

Relates Hilbert series to Catalan and Narayana numbers

Provides a standard monomial basis using Motzkin paths

Abstract

Let $W$ be an irreducible complex reflection group acting on its reflection representation $V$. We consider the doubly graded action of $W$ on the exterior algebra $\wedge (V \oplus V^*)$ as well as its quotient $DR_W := \wedge (V \oplus V^*)/ \langle \wedge (V \oplus V^*)^{W}_+ \rangle$ by the ideal generated by its homogeneous $W$-invariants with vanishing constant term. We describe the bigraded isomorphism type of $DR_W$; when $W = \mathfrak{S}_n$ is the symmetric group, the answer is a difference of Kronecker products of hook-shaped $\mathfrak{S}_n$-modules. We relate the Hilbert series of $DR_W$ to the (type A) Catalan and Narayana numbers and describe a standard monomial basis of $DR_W$ using a variant of Motzkin paths. Our methods are type-uniform and involve a Lefschetz-like theory which applies to the exterior algebra $\wedge (V \oplus V^*)$.