Generalizing Tanisaki's ideal via ideals of truncated symmetric functions

We define a family of ideals $I_h$ in the polynomial ring $\mathbb{Z}[x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n]$ that are parametrized by Hessenberg functions $h$ (equivalently Dyck paths or ample partitions). The ideals $I_h$ 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 $I_h$, we use polynomials in a proper subset of the variables $x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_n$ that are symmetric under the corresponding permutation subgroup. We call these polynomials {\em truncated symmetric functions} and show combinatorial identities relating different kinds of truncated symmetric polynomials. We then prove several key properties of $I_h$, including that if $h>h^{\mathrm{\prime }}$ in the natural partial order on Dyck paths then $I_h\subset I_{h^{\mathrm{\prime }}}$, and explicitly construct a Gr"{o}bner basis for $I_h$. We use a second family of ideals $J_h$ for which some of the claims are easier to see, and prove that $I_h=J_h$. The ideals $J_h$ 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 $I_h=J_h$ generalize the Tanisaki ideals both algebraically and geometrically, from Springer varieties to a family of nilpotent Hessenberg varieties.