Generalizing Tanisaki\u27s Ideal Via Ideals of Truncated Symmetric Functions

Abstract. We define a family of ideals Ih in the polynomial ring Z[x1, . . . , xn] that are parametrized by Hessenberg functions h (equivalently Dyck paths or ample partitions). The ideals Ih generalize algebraically a family of ideals called the Tanisaki ideal, which is used in a geometric construction of permutation representations called Springer theory. To define Ih, we use polynomials in a proper subset of the variables {x1, . . . , xn} that are symmetric under the corresponding permutation subgroup. We call these polynomials truncated symmetric functions and show combinatorial identities relating different kinds of truncated symmetric polynomials. We then prove several key properties of Ih, including that if h \u3e h′ in the natural partial order on Dyck paths then Ih ⊂ Ih′ , and explicitly construct a Gröbner basis for Ih. We use a second family of ideals Jh for which some of the claims are easier to see, and prove that Ih = Jh. The ideals Jh arise in work of Ding, Develin-Martin-Reiner, and Gasharov-Reiner on a family of Schubert varieties called partition varieties. Using earlier work of the first author, the current manuscript proves that the ideals Ih = Jh generalize the Tanisaki ideals both algebraically and geometrically, from Springer varieties to a family of nilpotent Hessenberg varieties