Generalizing Tanisaki's ideal via ideals of truncated symmetric functions

Mbirika, Aba ; Tymoczko, Julianna

Journal of Algebraic Combinatorics, Tome 37 (2013) no. 1, pp. 167-199.

Résumé

Summary: We define a family of ideals I $_{ h }$ in the polynomial ring $\Bbb $Z[x $_{1},\cdots $,x $_{ n }$] that are parameterized 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},\cdots $,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 of I $_{ h }$, including that if h>h$^{\prime}$ in the natural partial order on Dyck paths, then I $\_{ h }\subset I \_{ h$^prime, and explicitly construct a Gröbner 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.

Détail

Keywords: symmetric functions, tanisaki ideal, Springer variety, Hessenberg variety, Gröbner basis