Ordered set partitions, Garsia-Procesi modules, and rank varieties

We introduce a family of ideals In,λ,sI_{n,\lambda , s} in Q[x1,…,xn]\mathbb {Q}[x_1,\dots , x_n] for λ\lambda a partition of k≤nk\leq n and an integer s≥ℓ(λ)s \geq \ell (\lambda ). This family contains both the Tanisaki ideals IλI_\lambda and the ideals In,kI_{n,k} of Haglund-Rhoades-Shimozono as special cases. We study the corresponding quotient rings Rn,λ,sR_{n,\lambda , s} as symmetric group modules. When n=kn=k and ss is arbitrary, we recover the Garsia-Procesi modules, and when λ=(1k)\lambda =(1^k) and s=ks=k, we recover the generalized coinvariant algebras of Haglund-Rhoades-Shimozono. We give a monomial basis for Rn,λ,sR_{n,\lambda , s} in terms of (n,λ,s)(n,\lambda , s)-staircases, unifying the monomial bases studied by Garsia-Procesi and Haglund-Rhoades-Shimozono. We realize the SnS_n-module structure of Rn,λ,sR_{n,\lambda , s} in terms of an action on (n,λ,s)(n,\lambda , s)-ordered set partitions. We find a formula for the Hilbert series of Rn,λ,sR_{n,\lambda , s} in terms of inversion and diagonal inversion statistics on a set of fillings in bijection with (n,λ,s)(n,\lambda , s)-ordered set partitions. Furthermore, we prove an expansion of the graded Frobenius characteristic of our rings into Gessel’s fundamental quasisymmetric basis. We connect our work with Eisenbud-Saltman rank varieties using results of Weyman. As an application of our results on Rn,λ,sR_{n,\lambda , s}, we give a monomial basis, Hilbert series formula, and graded Frobenius characteristic formula for the coordinate ring of the scheme-theoretic intersection of a rank variety with diagonal matrices.