Séminaire Lotharingien de Combinatoire, 84B.90 (2020), 12 pp.

Sean Griffin

Ordered Set Partitions, Tanisaki Ideals, and Rank Varieties

Abstract. We introduce a family of ideals In,λ,s in Q[x1,...,xn] for λ a partition of k <= n and an integer s >= ℓ(λ). This family contains both the Tanisaki ideals Iλ and the ideals In,k of Haglund-Rhoades-Shimozono as special cases. We study the corresponding quotient rings Rn,λ,s as symmetric group modules. When n=k and s is arbitrary, we recover the Garsia-Procesi modules, and when λ=(1k) and s=k, we recover the generalized coinvariant algebras of Haglund-Rhoades-Shimozono.

We give a monomial basis for Rn,λ,s in terms of (n,λmbda,s)-staircases, unifying the monomial bases studied by Garsia-Procesi and Haglund-Rhoades-Shimozono. We realize the Sn-module structure of Rn,λ,s in terms of an action on (n,λ,s)-ordered set partitions. We find a formula for the Hilbert series of Rn,λ,s in terms of inversion and diagonal inversion statistics on (n,λ,s)-ordered set partitions. Furthermore, we give an expansion of the graded Frobenius characteristic of our rings in terms of Gessel's fundamental basis and in terms of dual Hall-Littlewood symmetric functions.

We connect our work with Eisenbud-Saltman rank varieties using results of Weyman. As an application of our results on Rn,λ,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.

Received: November 20, 2019. Accepted: February 20, 2020. Final version: April 30, 2020.

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