Julianna S. Tymoczko

Exponents for B-stable ideals

Let G be a simple algebraic group over the complex numbers containing a Borel subgroup B. Given a B-stable ideal I in the nilradical of the Lie algebra of B, we define natural numbers m 1 , m 2 ,..., m k which we call ideal exponents. We then propose two conjectures where these exponents arise, proving these conjectures in types A n , B n , C n and some other types. When / = 0, we recover the usual exponents of G by Kostant (1959), and one of our conjectures reduces to a well-known factorization of the Poincare polynomial of the Weyl group. The other conjecture reduces to a well-known result of Arnold-Brieskorn on the factorization of the characteristic polynomial of the corresponding Coxeter hyperplane arrangement.

Linear conditions imposed on flag varieties

We study subvarieties of the flag variety called Hessenberg varieties, defined by certain linear conditions. These subvarieties arise naturally in applications including geometric representation theory, number theory, and numerical analysis. We describe completely the homology of Hessenberg varieties over GLn(C) and show that they have no odd-dimensional homology. We provide an explicit geometric construction which partitions each Hessenberg variety into pieces homeomorphic to affine space. We characterize these affine pieces by fillings of Young tableaux and show that the dimension of the affine piece can be computed by combinatorial rules generalizing the Eulerian numbers. We give an equivalent formulation of this result in terms of roots. We conclude with a section on open questions.

A positive Monk formula in the S1-equivariant cohomology of type A Peterson varieties

Peterson varieties are a special class of Hessenberg varieties that have been extensively studied, for example, by Peterson, Kostant, and Rietsch, in connection with the quantum cohomology of the flag variety. In this manuscript, we develop a generalized Schubert calculus, and in particular a positive Chevalley–Monk formula, for the ordinary and Borel-equivariant cohomology of the Peterson variety Y in type A n−1 , with respect to a natural S 1 -action arising from the standard action of the maximal torus on flag varieties. As far as we know, this is the first example of positive Schubert calculus beyond the realm of Kac–Moody flag varieties G/P. Our main results are as follows. First, we identify a computationally convenient basis of H * S1 (Y), which we call the basis of Peterson Schubert classes. Second, we derive a manifestly positive, integral Chevalley–Monk formula for the product of a cohomology-degree-2 Peterson Schubert class with an arbitrary Peterson Schubert class. Both H * S1 (Y) and H*(Y) are generated in degree 2. Finally, by using our Chevalley–Monk formula we give explicit descriptions (via generators and relations) of both the S 1 -equivariant cohomology ring H * S1 (Y) and the ordinary cohomology ring H*(Y) of the type A n−1 Peterson variety. Our methods are both directly from and inspired by those of the GKM (Goresky–Kottwitz–MacPherson) theory and classical Schubert calculus. We discuss several open questions and directions for future work.

Permutation representations on Schubert varieties

This paper defines and studies permutation representations on the equivariant cohomology of Schubert varieties, as representations both over

Springer Representations on the Khovanov Springer Varieties

{\Bbb C}

Springer fibers and Schubert points

and over

Valid Plane Trees: Combinatorial Models for RNA Secondary Structures with Watson--Crick Base Pairs

{\Bbb C}[t_1, t_2,\ldots,t_n]

Distinguishing Numbers for Graphs Groups

. We show these group actions are the same as an action of simple transpositions studied geometrically by M. Brion, and give topological meaning to the divided difference operators of Berstein-Gelfand-Gelfand, Demazure, Kostant-Kumar, and others. We analyze these representations using the combinatorial approach to equivariant cohomology introduced by Goresky-Kottwitz-MacPherson. We find that each permutation representation on equivariant cohomology produces a representation on ordinary cohomology that is trivial, though the equivariant representation is not.

An introduction to equivariant cohomology and homology, following Goresky, Kottwitz, and MacPherson

Springer varieties are studied because their cohomology carries a natural action of the symmetric group

Paving Hessenberg varieties by affines

S_n