Generalizing Tanisaki's Ideal Via Ideals of Truncated Symmetric Functions

Authors

Aba Mbirika, University of Wisconsin - Eau Claire
Julianna Tymoczko, Smith CollegeFollow

Document Type

Article

Publication Date

2-2013

Publication Title

Journal of Algebraic Combinatorics

Abstract

Abstract. We define a family of ideals I_h in the polynomial ring Z[_{x1, . . . , xn}] 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 {_{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 I_h, including that if h > h′ in the natural partial order on Dyck paths then Ih ⊂ Ih′ , 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.

Keywords

Symmetric functions, Tanisaki ideal, Springer variety, Hessenberg variety, Gröbner basis

Volume

37

Issue

1

First Page

167

Last Page

199

DOI

dx.doi.org/10.1007/s10801-012-0372-2

ISSN

1572-9192

Comments

Peer reviewed accepted manuscript. Language included at the request of the publisher: The final publication is available at Springer via http://dx.doi.org/10.1007/s10801-012-0372-2

Recommended Citation

Mbirika, Aba and Tymoczko, Julianna, "Generalizing Tanisaki's Ideal Via Ideals of Truncated Symmetric Functions" (2013). Mathematics Sciences: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/11