Generalizing Tanisakis ideal via ideals of truncated symmetric functions
We define a family of ideals I h in the polynomial ring ?[x 1,,x n ] that are parameterized by Hessenberg functionsh (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 defineI h , we use polynomials in a proper subset of the variables {x 1,,x n } 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 ofI h , including that if h>h? in the natural partial order on Dyck paths, then I h ?I h?, and explicitly construct a Grbner basis forI 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.
tanisaki ideal symmetric functions grbner basis springer variety hessenberg variety