The geometry and combinatorics of Springer fibers

This survey paper describes Springer fibers, which are used in one of the earliest examples of a geometric representation. We will compare and contrast them with Schubert varieties, another family of subvarieties of the flag variety that play an important role in representation theory and combinatorics, but whose geometry is in many respects simpler. The end of the paper describes a way that Springer fibers and Schubert varieties are related, as well as open questions.Comment: 18 page

Permutation actions on equivariant cohomology

This survey paper describes two geometric representations of the permutation group using the tools of toric topology. These actions are extremely useful for computational problems in Schubert calculus. The (torus) equivariant cohomology of the flag variety is constructed using the combinatorial description of Goresky-Kottwitz-MacPherson, discussed in detail. Two permutation representations on equivariant and ordinary cohomology are identified in terms of irreducible representations of the permutation group. We show how to use the permutation actions to construct divided difference operators and to give formulas for some localizations of certain equivariant classes. This paper includes several new results, in particular a new proof of the Chevalley-Monk formula and a proof that one of the natural permutation representations on the equivariant cohomology of the flag variety is the regular representation. Many examples, exercises, and open questions are provided.Comment: 24 page

Billey's formula in combinatorics, geometry, and topology

In this expository paper we describe a powerful combinatorial formula and its implications in geometry, topology, and algebra. This formula first appeared in the appendix of a book by Andersen, Jantzen, and Soergel. Sara Billey discovered it independently five years later, and it played a prominent role in her work to evaluate certain polynomials closely related to Schubert polynomials. Billey's formula relates many pieces of Schubert calculus: the geometry of Schubert varieties, the action of the torus on the flag variety, combinatorial data about permutations, the cohomology of the flag variety and of the Schubert varieties, and the combinatorics of root systems (generalizing inversions of a permutation). Combinatorially, Billey's formula describes an invariant of pairs of elements of a Weyl group. On its face, this formula is a combination of roots built from subwords of a fixed word. As we will see, it has deeper geometric and topological meaning as well: (1) It tells us about the tangent spaces at each permutation flag in each Schubert variety. (2) It tells us about singular points in Schubert varieties. (3) It tells us about the values of Kostant polynomials. Billey's formula also reflects an aspect of GKM theory, which is a way of describing the torus-equivariant cohomology of a variety just from information about the torus-fixed points in the variety. This paper will also describe some applications of Billey's formula, including concrete combinatorial descriptions of Billey's formula in special cases, and ways to bootstrap Billey's formula to describe the equivariant cohomology of subvarieties of the flag variety to which GKM theory does not apply.Comment: 14 pages, presented at the International Summer School and Workshop on Schubert Calculus in Osaka, Japan, 201

Representation stability of the cohomology of Springer varieties and some combinatorial consequences

A sequence of $S_n$-representations $\{V_n\}$ is said to be uniformly representation stable if the decomposition of $V_n = \bigoplus_{\mu} c_{\mu,n} V(\mu)_n$ into irreducible representations is independent of $n$ for each $\mu$---that is, the multiplicities $c_{\mu,n}$ are eventually independent of $n$ for each $\mu$. Church-Ellenberg-Farb proved that the cohomology of flag varieties (the so-called diagonal coinvariant algebra) is uniformly representation stable. We generalize their result from flag varieties to all Springer fibers. More precisely, we show that for any increasing subsequence of Young diagrams, the corresponding sequence of Springer representations form a graded co-FI-module of finite type (in the sense of Church-Ellenberg-Farb). We also explore some combinatorial consequences of this stability.Comment: 21 pages. Version 2: Improved exposition incorporating suggestions from the referees. The title has been changed slightly. Added Remark 1, and revisions made to the statements and proofs of Proposition 1, Theorem 3, and Corollary