Séminaire Lotharingien de Combinatoire, 84B.56 (2020), 12 pp.
Brendon Rhoades and Andrew Timothy Wilson
Vandermondes, Superspace, and Delta Conjecture Modules
Abstract.
Superspace is an algebra Ωn with n commuting generators x1, ..., xn
and n anticommuting generators θ1, ..., θn. We present an extension
δn,k of the Vandermonde
determinant to Ωn which depends on positive
integers k <= n.
We use superspace Vandermondes to build representations of the symmetric
group Sn. In particular, we construct a doubly graded Sn-module
Vn,k whose bigraded Frobenius grFrob(Vn,k;q,t)
conjecturally equals the symmetric function
Δ'ek-1en appearing in the
Haglund-Remmel-Wilson
Delta Conjecture. We prove the specialization
of our conjecture at t=0.
We use a differentiation action of Ωn on itself to build bigraded quotients Wn,k of Ωn
which extend the Delta Conjecture coinvariant rings Rn,k defined by Haglund-Rhoades-Shimozono
and studied geometrically by Pawlowski-Rhoades. Despite the fact that the Hilbert polynomials of the Rn,k
are not palindromic, we show that Wn,k exhibits a superspace version of Poincaré Duality.
Received: November 20, 2019.
Accepted: February 20, 2020.
Final version: April 30, 2020.
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