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

On conjugacy growth of linear groups

We investigate the conjugacy growth of finitely generated linear groups. We show that finitely generated non-virtually-solvable subgroups of GL_d have uniform exponential conjugacy growth and in fact that the number of distinct polynomials arising as characteristic polynomials of the elements of the ball of radius n for the word metric has exponential growth rate bounded away from 0 in terms of the dimension d only.Comment: 21 page

Entanglement of four qubit systems: a geometric atlas with polynomial compass I (the finite world)

We investigate the geometry of the four qubit systems by means of algebraic geometry and invariant theory, which allows us to interpret certain entangled states as algebraic varieties. More precisely we describe the nullcone, i.e., the set of states annihilated by all invariant polynomials, and also the so called third secant variety, which can be interpreted as the generalization of GHZ-states for more than three qubits. All our geometric descriptions go along with algorithms which allow us to identify any given state in the nullcone or in the third secant variety as a point of one of the 47 varieties described in the paper. These 47 varieties correspond to 47 non-equivalent entanglement patterns, which reduce to 15 different classes if we allow permutations of the qubits.Comment: 48 pages, 7 tables, 13 figures, references and remarks added (v2

The variety of reductions for a reductive symmetric pair

We define and study the variety of reductions for a reductive symmetric pair (G,theta), which is the natural compactification of the set of the Cartan subspaces of the symmetric pair. These varieties generalize the varieties of reductions for the Severi varieties studied by Iliev and Manivel, which are Fano varieties. We develop a theoretical basis to the study these varieties of reductions, and relate the geometry of these variety to some problems in representation theory. A very useful result is the rigidity of semi-simple elements in deformations of algebraic subalgebras of Lie algebras. We apply this theory to the study of other varieties of reductions in a companion paper, which yields two new Fano varieties.Comment: 23 page

On the centralizer of the sum of commuting nilpotent elements

Let X and Y be commuting nilpotent K-endomorphisms of a vector space V, where K is a field of characteristic p >= 0. If F=K(t) is the field of rational functions on the projective line, consider the K(t)-endomorphism A=X+tY of V. If p=0, or if the (p-1)-st power of A is 0, we show here that X and Y are tangent to the unipotent radical of the centralizer of A in GL(V). For all geometric points (a:b) of a suitable open subset of the projective line, it follows that X and Y are tangent to the unipotent radical of the centralizer of aX+bY. This answers a question of J. Pevtsova.Comment: 12 pages. To appear in the Friedlander birthday volume of J. Pure and Applied Algebr