The Springer variety is the set of flags stabilized by a nilpotent operator. In 1976, T.A. Springer observed that this variety's cohomology ring carries a symmetric group action, and he offered a deep geometric construction of this action. Sixteen years later, Garsia and Procesi made Springer's work more transparent and accessible by presenting the cohomology ring as a graded quotient of a polynomial ring. They combinatorially describe an explicit basis for this quotient. The goal of this paper is to generalize their work. Our main result deepens their analysis of Springer varieties and extends it to a family of varieties called Hessenberg varieties, a two-parameter generalization of Springer varieties. Little is known about their cohomology. For the class of regular nilpotent Hessenberg varieties, we conjecture a quotient presentation for the cohomology ring and exhibit an explicit basis. Tantalizing new evidence supports our conjecture for a subclass of regular nilpotent varieties called Peterson varieties.

For a fixed ∈ ℕ = {1, 2, 3, …} we say that a point ( , ) in the integer lattice ℤ × ℤ is -visible from the origin if it lies on the graph of a power function ( ) = with ∈ ℚ and no other integer lattice point lies on this curve (i.e., line of sight) between (0, 0) and ( , ). We prove that the proportion of -visible integer lattice points is given by 1∕ ( + 1), where ( ) denotes the Riemann zeta function. We also show that even though the proportion of -visible lattice points approaches 1 as approaches infinity, there exist arbitrarily large rectangular arrays of -invisible lattice points for any fixed . This work specialized to = 1 recovers original results from the classical lattice point visibility setting where the lines of sight are given by linear functions with rational slope through the origin.

Abstract. We define a family of ideals I h in the polynomial ring Z[x 1 , . . . , 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 , . . . , 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 ′ in the natural partial order on Dyck paths then I h ⊂ I h ′ , 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.

Abstract. We present a formula for the values of the sign representations of the complex reflection groups G(r, p, n) in terms of its image under a generalized Robinson-Schensted algorithm.

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