Lefschetz theory for exterior algebras and fermionic diagonal coinvariants

2020/03/23 by Kim, Jongwon, Rhoades, Brendon · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2003.10031

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 ⊕ V^*) as well as its quotient DR_W := \wedge (V ⊕ V^*)/ ⟨ \wedge (V ⊕ V^*)^W₊ ⟩ by the ideal generated by its homogeneous W-invariants with vanishing constant term. We describe the bigraded isomorphism type of DR_W; when W = \mathfrakS_n is the symmetric group, the answer is a difference of Kronecker products of hook-shaped \mathfrakS_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 ⊕ V^*).

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