Enumerative combinatorics
Rook placements and cellular decomposition of partition varieties
Discrete Mathematics
Descent identities, Hessenberg varieties, and the Weil conjectures
Journal of Combinatorial Theory Series A
On the Gröbner bases of some symmetric systems and their application to coding theory
Journal of Symbolic Computation
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
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We define a family of ideals I h in the polynomial ring 驴[x 1,驴,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,驴,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 hh驴 in the natural partial order on Dyck paths, then I h 驴I h驴, 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.