Alexander Postnikov

Scattering Amplitudes and the Positive Grassmannian

We establish a direct connection between scattering amplitudes in planar four-dimensional theories and a remarkable mathematical structure known as the positive Grassmannian. The central physical idea is to focus on on-shell diagrams as objects of fundamental importance to scattering amplitudes. We show that the all-loop integrand in N=4 SYM is naturally represented in this way. On-shell diagrams in this theory are intimately tied to a variety of mathematical objects, ranging from a new graphical representation of permutations to a beautiful stratification of the Grassmannian G(k,n) which generalizes the notion of a simplex in projective space. All physically important operations involving on-shell diagrams map to canonical operations on permutations; in particular, BCFW deformations correspond to adjacent transpositions. Each cell of the positive Grassmannian is naturally endowed with positive coordinates and an invariant measure which determines the on-shell function associated with the diagram. This understanding allows us to classify and compute all on-shell diagrams, and give a geometric understanding for all the non-trivial relations among them. Yangian invariance of scattering amplitudes is transparently represented by diffeomorphisms of G(k,n) which preserve the positive structure. Scattering amplitudes in (1+1)-dimensional integrable systems and the ABJM theory in (2+1) dimensions can both be understood as special cases of these ideas. On-shell diagrams in theories with less (or no) supersymmetry are associated with exactly the same structures in the Grassmannian, but with a measure deformed by a factor encoding ultraviolet singularities. The Grassmannian representation of on-shell processes also gives a new understanding of the all-loop integrand for scattering amplitudes, presenting all integrands in a novel dLog form which directly reflects the underlying positive structure.

Quantum Schubert polynomials

where In is the ideal generated by symmetric polynomials in x1,... ,xn without constant term. Another, geometric, description of the cohomology ring of the flag manifold is based on the decomposition of Fln into Schubert cells. These are even-dimensional cells indexed by the elements w of the symmetric group Sn. The corresponding cohomology classes oa, called Schubert classes, form an additive basis in H* (Fln 2) . To relate the two descriptions, one would like to determine which elements of 2[xl, ... , Xn]/In correspond to the Schubert classes under the isomorphism (1.1). This was first done in [2] (see also [8]) for a general case of an arbitrary complex semisimple Lie group. Later, Lascoux and Schiitzenberger [22] came up with a combinatorial version of this theory (for the type A) by introducing remarkable polynomial representatives of the Schubert classes oa called Schubert polynomials and denoted Gw. Recently, motivated by ideas that came from the string theory [31, 30], mathematicians defined, for any Kahler algebraic manifold X, the (small) quantum cohomology ring QH* (X, 2), which is a certain deformation of the classical cohomology ring (see, e.g., [28, 19, 14] and references therein). The additive structure of QH* (X , 2) is essentially the same as that of ordinary cohomology. In particular, QH* (Fln , Z) is canonically isomorphic, as an abelian group, to the tensor product H* (Fln , 2) (0 Z[ql,..., qn-1], where the qi are formal variables (deformation parameters). The multiplicative structure of the quantum cohomology is however

Trees, parking functions, syzygies, and deformations of monomial ideals

For a graph G, we construct two algebras whose dimensions are both equal to the number of spanning trees of G. One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of the polynomial ring modulo certain powers of linear forms. We describe the set of monomials that forms a linear basis in each of these two algebras. The basis elements correspond to G-parking functions that naturally came up in the abelian sandpile model. These ideals are instances of the general class of monotone monomial ideals and their deformations. We show that the Hilbert series of a monotone monomial ideal is always bounded by the Hilbert series of its deformation. Then we define an even more general class of monomial ideals associated with posets and construct free resolutions for these ideals. In some cases these resolutions coincide with Scarf resolutions. We prove several formulas for Hilbert series of monotone monomial ideals and investigate when they are equal to Hilbert series of deformations. In the appendix we discuss the abelian sandpile model.

Deformations of Coxeter Hyperplane Arrangements

We investigate several hyperplane arrangements that can be viewed as deformations of Coxeter arrangements. In particular, we prove a conjecture of Linial and Stanley that the number of regions of the arrangement xi?xj=1, 1?i

Affine approach to quantum Schubert calculus

This paper presents a formula for products of Schubert classes in the quantum cohomology ring of the Grassmannian. We introduce a generalization of Schur symmetric polynomials for shapes that are naturally embedded in a torus. Then we show that the coefficients in the expansion of these toric Schur polynomials, in terms of the regular Schur polynomials, are exactly the 3-point Gromov-Witten invariants, which are the structure constants of the quantum cohomology ring. This construction implies three symmetries of the Gromov-Witten invariants of the Grassmannian with respect to the groups S3, (Z/nZ)2, and Z/2Z. The last symmetry is a certain curious duality of the quantum cohomology which inverts the quantum parameter q . Our construction gives a solution to a problem posed by Fulton and Woodward about the characterization of the powers of the quantum parameter q which occur with nonzero coefficients in the quantum product of two Schubert classes. The curious duality switches the smallest such power of q with the highest power. We also discuss the affine nil-Temperley-Lieb algebra that gives a model for the quantum cohomology.

Alcoved Polytopes, I

The aim of this paper is to study alcoved polytopes, which are polytopes arising from affine Coxeter arrangements. This class of convex polytopes includes many classical polytopes, for example, the hypersimplices. We compare two constructions of triangulations of hypersimplices due to Stanley and Sturmfels and explain them in terms of alcoved polytopes. We study triangulations of alcoved polytopes, the adjacency graphs of these triangulations, and give a combinatorial formula for volumes of these polytopes. In particular, we study a class of matroid polytopes, which we call the multi-hypersimplices.

Combinatorics of hypergeometric functions associated with positive roots

In this paper we study the hypergeometric system on unipotent matrices. This system gives a holonomic D-module. We find the number of independent solutions of this system at a generic point. This number is equal to the famous Catalan number. An explicit basis of Γ-series in solution space of this system is constructed in the paper. We also consider restriction of this system to certain strata. We introduce several combinatorial constructions with trees, polyhedra, and triangulations related to this subject.

Mixed Bruhat Operators and Yang-Baxter Equations for Weyl Groups

We introduce and study a family of operators which act in the group algebra of a Weyl group W and provide a multiparameter solution to the quantum Yang-Baxter equations of the corresponding type. These operators are then used to derive new combinatorial properties of W and to obtain new proofs of known results concerning the Bruhat order of W. The paper is organized as follows. Section 2 is devoted to preliminaries on Coxeter groups and associated Yang-Baxter equations. In Theorem 3.1 of Section 3, we describe our solution of these equations. In Section 4, we consider a certain limiting case of our solution, which leads to the quantum Bruhat operators. These operators play an important role in the explicit description of the (small) quantum cohomology ring of G/B. Section 5 contains the proof of Theorem 3.1. Section 6 is devoted to combinatorial applications of our operators. For an arbitrary element u ∈W,we define a graded partial order onW called the tilted Bruhat order; this partial order has unique minimal element u. (The usual Bruhat order corresponds to the special case where u = e, the identity element.) We then prove that tilted Bruhat orders are lexicographically shellable graded posets whose every interval is Eulerian. This generalizes the well-known results of D.-N. Verma, A. Bjorner, M. Wachs, and M. Dyer.

Syzygies of oriented matroids

We construct minimal cellular resolutions of squarefree monomial ideals arising from hyperplane arrangements, matroids, and oriented matroids. These are StanleyReisner ideals of complexes of independent sets and of triangulations of Lawrence matroid polytopes. Our resolution provides a cellular realization of R. Stanley’s formula for their Betti numbers. For unimodular matroids our resolutions are related to hyperplane arrangements on tori, and we recover the resolutions constructed by D. Bayer, S. Popescu, and B. Sturmfels []. We resolve the combinatorial problems posed in [3] by computing Mobius invariants of graphic and cographic arrangements in terms of Hermite polynomials.

Schur positivity and Schur log-concavity

We prove Okounkovs conjecture, a conjecture of Fomin-Fulton-Li-Poon, and a special case of Lascoux-Leclerc-Thibons conjecture on Schur positivity and give several more general statements using a recent result of Rhoades and Skandera. We also give an intriguing log-concavity property of Schur functions.