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 ∧(V ⊕ V * ) as well as its quotientby the ideal generated by its homogeneous Winvariants with vanishing constant term. We describe the bigraded isomorphism type of DRW ; when W = Sn is the symmetric group, the answer is a difference of Kronecker products of hookshaped Sn-modules. We relate the Hilbert series of DRW to the (type A) Catalan and Narayana numbers and describe a standard monomial basis of DRW using a variant of Motzkin paths. Our methods are type-uniform and involve a Lefschetz-like theory which applies to the exterior algebra ∧(V ⊕ V * ).Key words and phrases. coinvariant algebra, fermion, exterior algebra, Lefschetz element. 1 And, in fact, to a wider class of groups G; see Remark 4.5. 2 A commuting variable xi is called bosonic; the power x 2 i corresponds to two indistinguishable bosons in State i.

The Bosonic-Fermionic Diagonal Coinvariant Modules Conjecture

“…G(q, t; z; x) := n−1 k=0 ν⊢k m ν (z) Θ eν ∇(e n−k )(q, t; x). Table 1 summarizes the overall situation 8 . Conjecture 1 essentially states all entries may…”

Increasing Subsequences and Kronecker Coefficients

“…and denote by ∧{Θ n , Ξ n } Sn + ⊆ ∧{Θ n , Ξ n } the two-sided ideal generated by S n -invariants with vanishing constant term. The fermionic diagonal coinvariant ring is defined in [4] by (15) F DR n := ∧{Θ n , Ξ n }/ ∧{Θ n , Ξ n } Sn + . This is a doubly graded S n -module and an anticommutative version of the diagonal coinvariant ring [3].…”

“…Theorem 5. (Kim-R. [4]) The (i, j)-graded piece (F DR n ) i,j is zero unless i + j < n. When i + j < n we have (16) Frob((F DR n ) i,j ) = s (n−i,1 i ) * s (n−j,1 j ) − s (n−i+1,1 i−1 ) * s (n−j+1,1 j−1 ) , where we interpret s (n−i+1,1 i−1 ) * s (n−j+1,1 j−1 ) = 0 if i = 0 or j = 0. In particular, for 0 ≤ k ≤ n − 1 if we set V n,k := (F DR n ) k,n−k−1 then V n,k has Frobenius image given by Equation (11).…”