Publications (51)
Poset vectors and generalized permutohedra
Dorian Croitoru, SuHo Oh, Alexander Postnikov
We show that given a poset P and and a subposet Q, the integer points obtained by restricting linear extensions of P to Q can be explained via integer lattice points of a generaliz…
Trees, parking functions, syzygies, and deformations of monomial ideals
Alexander Postnikov, Boris Shapiro
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…
Trianguloids and Triangulations of Root Polytopes
Pavel Galashin, Gleb Nenashev, Alexander Postnikov
Triangulations of a product of two simplices and, more generally, of root polytopes are closely related to Gelfand-Kapranov-Zelevinsky's theory of discriminants, to tropical geomet…
Algebras of curvature forms on homogeneous manifolds
Alexander Postnikov, Boris Shapiro, Mikhail Shapiro
Let C(X) be the algebra generated by the curvature 2-forms of the standard hermitian line bundles over the complex homogeneous manifold X=G/B. We calculate the Hilbert polynomial o…
Purity and separation for oriented matroids
Pavel Galashin, Alexander Postnikov
Leclerc and Zelevinsky, motivated by the study of quasi-commuting quantum flag minors, introduced the notions of strongly separated and weakly separated collections. These notions…
A Gelfand Model for Wreath Products
Ron M. Adin, Alexander Postnikov, Yuval Roichman
A Gelafand model for wreath products is constructed. The proof relies on a combinatorial interpretation of the characters of the model, extending a classical result o…
Branched polymers and hyperplane arrangements
Karola Meszaros, Alexander Postnikov
We generalize the construction of connected branched polymers and the notion of the volume of the space of connected branched polymers studied by Brydges and Imbrie, and Kenyon and…
Alcoved Polytopes I
Thomas Lam, Alexander Postnikov
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,…
Affine approach to quantum Schubert calculus
Alexander Postnikov
This article presents a formula for products of Schubert classes in the quantum cohomology ring of the Grassmannian. We introduce a generalization of Schur symmetric polynomials fo…
Root system chip-firing II: Central-firing
Pavel Galashin, Sam Hopkins, Thomas McConville +1
Jim Propp recently proposed a labeled version of chip-firing on a line and conjectured that this process is confluent from some initial configurations. This was proved by Hopkins-M…
Root system chip-firing I: Interval-firing
Pavel Galashin, Sam Hopkins, Thomas McConville +1
Jim Propp recently introduced a variant of chip-firing on a line where the chips are given distinct integer labels. Hopkins, McConville, and Propp showed that this process is confl…
Smoothness of Schubert varieties via patterns in root systems
Sara Billey, Alexander Postnikov
The aim of this article is to present a smoothness criterion for Schubert varieties in generalized flag manifolds in terms of patterns in root systems. We generalize Lakshmib…
Matching polytopes, toric geometry, and the non-negative part of the Grassmannian
Alexander Postnikov, David Speyer, Lauren Williams
In this paper we use toric geometry to investigate the topology of the totally non-negative part of the Grassmannian (Gr_{kn})_{\geq 0}. This is a cell complex whose cells Delta_G…
Deformations of Coxeter hyperplane arrangements
Alexander Postnikov, Richard P. Stanley
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 numbe…
Proof of a conjecture of Bergeron, Ceballos and Labbé
Darij Grinberg, Alexander Postnikov
The reduced expressions for a given element of a Coxeter group can be regarded as the vertices of a directed graph ; its arcs correspond to the braid m…
Mixed Bruhat operators and Yang-Baxter equations for Weyl groups
Francesco Brenti, Sergey Fomin, Alexander Postnikov
We introduce and study a family of operators which act in the span of a Weyl group and provide a multi-parameter solution to the quantum Yang-Baxter equations of the correspond…
Cosmological Polytopes and the Wavefunction of the Universe
Nima Arkani-Hamed, Paolo Benincasa, Alexander Postnikov
We present a connection between the physics of cosmological time evolution and the mathematics of positive geometries, roughly analogous to similar connections seen in the context…
Combinatorial Gelfand Models
Ron M. Adin, Alexander Postnikov, Yuval Roichman
A combinatorial construction of a Gelafand model for the symmetric group and its Iwahori-Hecke algebra is presented.
Faces of Generalized Permutohedra
Alexander Postnikov, Victor Reiner, Lauren Williams
The aim of the paper is to calculate face numbers of simple generalized permutohedra, and study their f-, h- and gamma-vectors. These polytopes include permutohedra, associahedra,…
On-Shell Structures of MHV Amplitudes Beyond the Planar Limit
Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo +2
We initiate an exploration of on-shell functions in SYM beyond the planar limit by providing compact, combinatorial expressions for all leading singularities of MHV…
Total positivity, Grassmannians, and networks
Alexander Postnikov
The aim of this paper is to discuss a relationship between total positivity and planar directed networks. We show that the inverse boundary problem for these networks is naturally…
Littlewood-Richardson Coefficients via Yang-Baxter Equation
Oleg Gleizer, Alexander Postnikov
The purpose of this paper is to present an interpretation for the decomposition of the tensor product of two or more irreducible representations of GL(N) in terms of a system of qu…
Combinatorics and geometry of power ideals
Federico Ardila, Alexander Postnikov
We investigate ideals in a polynomial ring which are generated by powers of linear forms. Such ideals are closely related to the theories of fat point ideals, Cox rings, and box sp…
Honeycombs and Sums of Hermitian Matrices, Revisited
Ankur Moitra, Alexander Postnikov, Dora Woodruff
We give a new proof of the celebrated theorem of Knutson and Tao that the spectra of triples of Hermitian matrices exactly correspond to positions of boundary rays of h…
Higher secondary polytopes and regular plabic graphs
Pavel Galashin, Alexander Postnikov, Lauren Williams
Given a configuration of points in , we introduce the higher secondary polytopes , which have the property that agr…
Symmetries of Gromov-Witten invariants
Alexander Postnikov
The group (Z/nZ)^2 is shown to act on the Gromov-Witten invariants of the complex flag manifold. We also deduce several corollaries of this result.
On Characters of Weyl Groups
Ron M. Adin, Alexander Postnikov, Yuval Roichman
In this note a combinatorial formula related to the symmetric group is generalized to an arbitrary finite Weyl group.
A positive formula for the Ehrhart-like polynomials from root system chip-firing
Sam Hopkins, Alexander Postnikov
In earlier work in collaboration with Pavel Galashin and Thomas McConville we introduced a version of chip-firing for root systems. Our investigation of root system chip-firing led…
Bruhat order, smooth Schubert varieties, and hyperplane arrangements
Suho Oh, Alexander Postnikov, Hwanchul Yoo
The aim of this article is to link Schubert varieties in the flag manifold with hyperplane arrangements. For a permutation, we construct a certain graphical hyperplane arrangement.…
Affine Weyl groups in K-theory and representation theory
Cristian Lenart, Alexander Postnikov
We give an explicit combinatorial Chevalley-type formula for the equivariant K-theory of generalized flag varieties G/P which is a direct generalization of the classical Chevalley…
Schur times Schubert via the Fomin-Kirillov algebra
Karola Meszaros, Greta Panova, Alexander Postnikov
We study multiplication of any Schubert polynomial by a Schur polynomial (the Schubert polynomial of a Grassmannian permutation) and the expansion of this p…
Alcoved Polytopes II
Thomas Lam, Alexander Postnikov
This is the second of two papers where we study polytopes arising from affine Coxeter arrangements. Our results include a formula for their volumes, and also compatible definitions…
Generalized parking functions, descent numbers, and chain polytopes of ribbon posets
Denis Chebikin, Alexander Postnikov
We consider the inversion enumerator I_n(q), which counts labeled trees or, equivalently, parking functions. This polynomial has a natural extension to generalized parking function…
Quantum Bruhat graph and Schubert polynomials
Alexander Postnikov
The quantum Bruhat graph, which is an extension of the graph formed by covering relations in the Bruhat order, is naturally related to the quantum cohomology ring of G/B. We enhanc…
Polypositroids
Thomas Lam, Alexander Postnikov
We initiate the study of a class of polytopes, which we coin polypositroids, defined to be those polytopes that are simultaneously generalized permutohedra (or polymatroids) and al…
Root polytopes, Tutte polynomials, and a duality theorem for bipartite graphs
Tamás Kálmán, Alexander Postnikov
Let G be a connected bipartite graph with color classes E and V and root polytope Q. Regarding the hypergraph (V,E) induced by G, we prove that its interior polynomial is equivalen…
Permutohedra, associahedra, and beyond
Alexander Postnikov
The volume and the number of lattice points of the permutohedron P_n are given by certain multivariate polynomials that have remarkable combinatorial properties. We give several di…
Positive Grassmannian and polyhedral subdivisions
Alexander Postnikov
The nonnegative Grassmannian is a cell complex with rich geometric, algebraic, and combinatorial structures. Its study involves interesting combinatorial objects, such as positroid…
A Combinatorial Model for Crystals of Kac-Moody Algebras
Cristian Lenart, Alexander Postnikov
We present a simple combinatorial model for the characters of the irreducible integrable highest weight modules for complex symmetrizable Kac-Moody algebras. This model can be view…
Chains in the Bruhat order
Alexander Postnikov, Richard P. Stanley
We study a family of polynomials whose values express degrees of Schubert varieties in the generalized complex flag manifold G/B. The polynomials are given by weighted sums over sa…
Trapezodial property of the generalized Alexander polynomial
Tamás Kálmán, Karola Mészáros, Alexander Postnikov
Fox's conjecture from 1962, that the absolute values of the coefficients of the Alexander polynomial of an alternating link are trapezoidal, has remained stubbornly open to this da…
Kostka numbers and Fourier duality
Michael Finkelberg, Alexander Postnikov, Vadim Schechtman
We relate the Fourier transform of perverse sheaves smooth along the coordinate hyperplane configuration in a complex vector space to the Deligne-Lusztig duality of unipotent repre…
On the X-rays of permutations
Cecilia Bebeacua, Toufik Mansour, Alexander Postnikov +1
The X-ray of a permutation is defined as the sequence of antidiagonal sums in the associated permutation matrix. X-rays of permutation are interesting in the context of Discrete To…
PP-wave string interactions from perturbative Yang-Mills theory
Neil R. Constable, Daniel Z. Freedman, Matthew Headrick +4
Recently, Berenstein et al. have proposed a duality between a sector of N=4 super-Yang-Mills theory with large R-charge J, and string theory in a pp-wave background. In the limit c…
Scattering Amplitudes and the Positive Grassmannian
Nima Arkani-Hamed, Jacob L. Bourjaily, Freddy Cachazo +3
We establish a direct connection between scattering amplitudes in planar four-dimensional theories and a remarkable mathematical structure known as the positive Grassmannian. The c…
Arrangements of equal minors in the positive Grassmannian
Miriam Farber, Alexander Postnikov
We discuss arrangements of equal minors of totally positive matrices. More precisely, we investigate the structure of equalities and inequalities between the minors. We show that a…
Repeatable patterns and the maximum multiplicity of a generator in a reduced word
Christian Gaetz, Yibo Gao, Pakawut Jiradilok +2
We study the maximum multiplicity of a simple transposition in a reduced word for the longest permutation , a…
Coxeter submodular functions and deformations of Coxeter permutahedra
Federico Ardila, Federico Castillo, Christopher Eur +1
We describe the cone of deformations of a Coxeter permutahedron, or equivalently, the nef cone of the toric variety associated to a Coxeter complex. This family of polytopes contai…
What power of two divides a weighted Catalan number?
Alexander Postnikov, Bruce Sagan
Given a sequence of integers b = (b_0,b_1,b_2,...) one gives a Dyck path P of length 2n the weight wt(P) = b_{h_1} b_{h_2} ... b_{h_n}, where h_i is the height of the ith ascent of…
Hecke Algebra Actions on the Coinvariant Algebra
Ron M. Adin, Alexander Postnikov, Yuval Roichman
Two actions of the Hecke algebra of type A on the corresponding polynomial ring are studied. Both are deformations of the natural action of the symmetric group on polynomials, and…
Schur positivity and Schur log-concavity
Thomas Lam, Alexander Postnikov, Pavlo Pylyavskyy
We prove Okounkov's conjecture, a conjecture of Fomin-Fulton-Li-Poon, and a special case of Lascoux-Leclerc-Thibon's conjecture on Schur positivity and give several more general st…