Lefschetz theory for exterior algebras and fermionic diagonal coinvariants

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 $\wedge (V\oplus V^{\ast })$ as well as its quotient $D{R}_W:=\wedge (V\oplus V^{\ast })/⟨\wedge (V\oplus V^{\ast }{)}_{+}^W⟩$ by the ideal generated by its homogeneous $W$-invariants with vanishing constant term. We describe the bigraded isomorphism type of $D{R}_W$; 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 $D{R}_W$ to the (type A) Catalan and Narayana numbers and describe a standard monomial basis of $D{R}_W$ using a variant of Motzkin paths. Our methods are type-uniform and involve a Lefschetz-like theory which applies to the exterior algebra $\wedge (V\oplus V^{\ast })$.

16 pages