These articles are available here in pdf or postscript format, and most of them are also on the ArXiv.
The amplituhedron A(n,k,m) is the image in the Grassmannian Gr(k,k+m) of the totally nonnegative part of Gr(k,n), under a (map induced by a) linear map which is totally positive. It was introduced by Arkani-Hamed and Trnka in 2013 in order to give a geometric basis for the computation of scattering amplitudes in N=4 supersymmetric Yang-Mills theory. While the case m=4 is most relevant to physics, the amplituhedron is an interesting mathematical object for any m. In this paper we study it in the case m=1. We start by taking an orthogonal point of view and define a related "B-amplituhedron" B(n,k,m), which we show is isomorphic to A(n,k,m). We use this reformulation to describe the amplituhedron in terms of sign variation. We then show that A(n,k,1) can be identified with the complex of bounded faces of a cyclic hyperplane arrangement, and describe how its cells fit together. We deduce that A(n,k,1) is homeomorphic to a ball.
This is a preliminary draft of Chapters 1-3 of our forthcoming textbook with Andrei Zelevinsky (1953 -- 2013). We expect to make additional chapters available in the not so distant future. Comments and suggestions are welcome.
Kenyon and Pemantle (2014) gave a formula for the entries of a square matrix in terms of connected principal and almost-principal minors. Each entry is an explicit Laurent polynomial whose terms are the weights of domino tilings of a half Aztec diamond. They conjectured an analogue of this parametrization for symmetric matrices, where the Laurent monomials are indexed by Catalan paths. In this paper we prove the Kenyon-Pemantle conjecture, and apply this to a statistics problem pioneered by Joe (2006). Correlation matrices are represented by an explicit bijection from the cube to the elliptope.
In an earlier work, Corteel and I introduced staircase tableaux, and used them to give combinatorial formulas for steady state probabilities of the ASEP and also for Askey-Wilson moments. It is well-known that Askey-Wilson polynomials can be viewed as the one-variable case of Koornwinder polynomials (also known as Macdonald polynomials of type BC). In this article we introduce rhombic staircase tableaux, and, building on our previous paper, we use them to give combinatorial formulas for steady state probabilities of the two-species ASEP and also for homogeneous Koornwinder moments. (Homogeneous Koornwinder moments are integrals of homogeneous symmetric polynomials with respect to the Koornwinder measure.) Note that rhombic staircase tableaux simultaneusly generalize staircase tableaux and also the rhombic alternative tableaux of Mandelshtam-Viennot.
In this article we use the cluster structure on the Grassmannian and the combinatorics of plabic graphs to exhibit a new aspect of mirror symmetry for Grassmannians in terms of polytopes. From a given plabic graph G we have two coordinate systems: we have a positive chart for our A-model Grassmannian, and we have a cluster chart for our B-model (Landau-Ginzburg model) Grassmannian. On the A-model side, we use the positive chart to associate a corresponding Newton-Okounkov (A-model) polytope. On the B-model side, we use the cluster chart to express the superpotential as a Laurent polynomial, and by tropicalizing this expression, we obtain a B-model polytope. Our main result is that these two polytopes coincide.
Introduced in the late 1960's, the asymmetric exclusion process (ASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice with open boundaries. It has been known for awhile that there is a tight connection between the partition function of the ASEP and moments of Askey-Wilson polynomials, a family of orthogonal polynomials which are at the top of the hierarchy of classical orthogonal polynomials in one variable. On the other hand, Askey-Wilson polynomials can be viewed as a specialization of the multivariate Macdonald-Koornwinder polynomials (also known as Koornwinder polynomials), which in turn give rise to the Macdonald polynomials associated to any classical root system via a limit or specialization. In light of the fact that Koornwinder polynomials generalize the Askey-Wilson polynomials, it is natural to ask whether one can find a particle model whose partition function is related to Koornwinder polynomials. In this article we answer this question affirmatively, by showing that Koornwinder moments at q=t are closely connected to the partition function for the two-species exclusion process.
The classical permutohedron Perm has many beautiful properties. For example, the paths from e to w_0 along the edges of Perm are in bijection with the reduced decompositions of w_0. Moreover, the two-dimensional faces of the permutohedron correspond to braid and commuting moves, which by the Tits Lemma, connect any two reduced expressions of w_0. In this note we introduce some ``bridge polytopes" Br(k,n) which provide a positive Grassmannian analogue of the permutohedron. In this setting, BCFW bridge decompositions of reduced plabic graphs play the role of reduced decompositions. We show that paths along the edges of Br(k,n) encode BCFW bridge decompositions of the longest element pi(k,n) in the circular Bruhat order. We also show that two-dimensional faces of Br(k,n) correspond to certain local moves for plabic graphs, which by a result of Postnikov, connect any two reduced plabic graphs associated to pi(k,n). A useful tool in our proofs is the fact that our polytopes are isomorphic to certain Bruhat interval polytopes.
Let u and v be permutations on n letters, with u <= v in Bruhat order. A Bruhat interval polytope Q_{u,v} is the convex hull of all permutation vectors z = (z(1), z(2),...,z(n)) with u <= z <= v. Bruhat interval polytopes were studied recently by Kodama and the second author, in the context of the Toda lattice and the moment map on the flag variety. In this paper we study combinatorial aspects of Bruhat interval polytopes. For example, we give an inequality description and a dimension formula for Bruhat interval polytopes, and prove that every face of a Bruhat interval polytope is a Bruhat interval polytope. A key tool in the proof of the latter statement is a generalization of the well-known lifting property for Coxeter groups. This leads to a generalization of the standard recurrence for R-polynomials. Finally, we define and study a more general class of polytopes called Bruhat interval polytopes for G/P.
This paper proves a version of mirror symmetry expressing the (small) Dubrovin connection for even-dimensional quadrics in terms of a mirror-dual Landau-Ginzburg model on the complement of an anticanonical divisor in a dual quadric. The cluster algebra structure on the coordinate ring of the mirror plays a key role in the proof. We then go into greater depth for all quadrics, even and odd, treating them as a series starting with Q_{3} and Q_{4}=Gr_{2}(4). This leads to a combinatorial model for the Laurent polynomial superpotential in terms of a quiver. We use this quiver description to compute explicitly a particular flat section of the Dubrovin connection, and recover the constant term of Givental's J-function by a variety of methods.
We prove da Silva's 1987 conjecture that any positively oriented matroid is a positroid; that is, it can be realized by a set of vectors in a real vector space. It follows from this result and the main result of my paper ``Shelling totally nonnegative flag varieties" that the positive matroid Grassmannian (or positive MacPhersonian) is homeomorphic to a closed ball.
We investigate the role that non-crossing partitions play in the study of positroids, a class of matroids introduced by Postnikov. We prove that every positroid can be constructed uniquely by choosing a non-crossing partition on the ground set, and then freely placing the structure of a connected positroid on each of the blocks of the partition. This structural result yields several combinatorial facts about positroids. We show that the face poset of a positroid polytope embeds in a poset of weighted non-crossing partitions. We enumerate connected positroids, and show how they arise naturally in free probability. Finally, we prove that the probability that a positroid on [n] is connected equals 1/e^2 asymptotically.
We study combinatorial aspects of the solution to the full Kostant-Toda (f-KT) hierarchy, when the initial data is given by an arbitrary point on the totally non-negative (tnn) flag variety. The f-KT flows on the tnn flag variety are complete, and their asymptotics are completely determined by Rietsch's cell decomposition of the tnn flag variety. We define the f-KT flow on the weight space via the moment map, and show that the closure of each f-KT flow forms an interesting convex polytope generalizing the permutohedron which we call a Bruhat interval polytope. Bruhat interval polytopes are generalized permutohedra, in the sense of Postnikov, and their edges correspond to cover relations in the Bruhat order.
In this expository paper we give a gentle introduction to cluster algebras. We also explain how cluster algebras naturally appear in Teichmuller theory, and how they were used to reformulate and prove the Zamolodchikov periodicity conjecture in mathematical physics.
Deodhar introduced his decomposition of partial flag varieties as a tool for understanding Kazhdan-Lusztig polynomials. The Deodhar decomposition of the Grassmannian is also useful in the context of soliton solutions to the KP equation. Deodhar components of the Grassmannian are in bijection with certain tableaux called Go-diagrams, and each component is isomorphic to (K *)^{a} × K^{b} for some non-negative integers a and b. Our main result is an explicit parameterization of each Deodhar component in the Grassmannian in terms of networks. More specifically, from a Go-diagram we construct a weighted network and its weight matrix, whose entries enumerate directed paths in the network. By letting the weights in the network vary over K or K* as appropriate, one gets a parameterization of the corresponding Deodhar component. We also give a (minimal) characterization of each Deodhar component in terms of Plucker coordinates. Note that in his study of the totally non-negative part of the Grassmannian, Postnikov constructed parameterizations of positroid cells that used planar networks associated to Le-diagrams. Our construction generalizes his.
This paper is an exposition of our results from ``The Deodhar decomposition of the Grassmannian and the regularity of KP solitons," and ``KP solitons and total positivity on the Grassmannian."
Given a point A in the real Grassmannian, one may construct a soliton solution u_A(x,y,t) to the KP equation. The contour plot of such a solution provides a tropical approximation to the solution when the variables x, y, and t are considered on a large scale and t is fixed. In this paper we use several decompositions of the Grassmannian in order to understand the contour plots of the corresponding soliton solutions. First we use the positroid stratification of the real Grassmannian in order to characterize the unbounded line-solitons in the contour plots at y>>0 and y<<0. Next we use Deodhar's decomposition of the Grassmannian (a refinement of the positroid stratification) to study contour plots at t<<0. More specifically, we index the components of the Deodhar decomposition of the Grassmannian by certain tableaux which we call Go-diagrams, and then use these Go-diagrams to characterize the contour plots of solitons solutions when t<<0. Finally we use these results to show that a soliton solution u_A(x,y,t) is regular for all times t if and only if A comes from the totally non-negative part of the Grassmannian.
We study an asymmetric exclusion process (ASEP) on a semi-infinite lattice with an open left boundary, and an ASEP on a finite lattice with open left and right boundaries -- and we demonstrate a surprising relationship between their stationary measures. We show that the finite correlation functions involving the leftmost L sites on the semi-infinite ASEP can be obtained as a nonphysical specialization of the stationary distribution of the finite ASEP on a lattice of L sites. Namely, if the output and input rates of particles at the right boundary of the finite ASEP are β and δ, respectively, and we set δ = -β, then this specialization corresponds to sending the right boundary of the lattice to infinity. Combining this observation with work of the second author and Corteel, we obtain a combinatorial formula for the finite correlation functions of the semi-infinite ASEP.
Because of the conjectural connection between cluster algebras and dual canonical bases, it is natural to ask whether one may construct a "good" (vector-space) basis of each cluster algebra. In this paper we construct two bases for each cluster algebra coming from a triangulated surface without punctures. We work in the context of a coefficient system coming from a full-rank exchange matrix, for example, principal coefficients. The elements of our bases have positive Laurent expansions with respect to every cluster.
This paper concerns cluster algebras with principal coefficients A(S,M) associated to bordered surfaces (S,M). Given any arc or loop in the surface -- with or without self-intersections -- we associate an element of A(S,M), using products of elements of PSL_2(R). We give a direct proof that our matrix formulas for arcs and loops agree with the combinatorial formulas for arcs and loops in terms of matchings, which were given in [MSW, MSW2]. Finally, we use our matrix formulas to prove skein relations for the cluster algebra elements associated to arcs and loops. This generalizes prior work of Fock and Goncharov, who worked in the coefficient-free case. The results of this paper will be used in [MSW2] in order to construct vector-space bases for A(S,M).
Soliton solutions of the KP equation have been studied since 1970, when Kadomtsev and Petviashvili proposed a two-dimensional nonlinear dispersive wave equation now known as the KP equation. It is well-known that one can use the Wronskian method to construct a soliton solution to the KP equation from each point of the real Grassmannian. The regular soliton solutions that one obtains in this way come from points of the totally nonnegative part of the Grassmannian. In this paper we exhibit a surprising connection between the theory of total positivity for the Grassmannian, and the structure of regular soliton solutions to the KP equation. This gives new insights into the structure of KP solitons as well as new interpretations of the combinatorial objects indexing positroid cells. In particular, we use this framework to: give an explicit construction of certain soliton contour graphs; demonstrate an intriguing connection between soliton graphs and cluster algebras; and solve the inverse problem for soliton solutions coming from the totally positive Grassmannian.
Regular KP soliton solutions provide a good model for shallow water waves. Coincidentally, my sister Eleanor also studies waves.
This is an announcement of the results of the paper above.
We give explicit formulas for Askey-Wilson moments and the (enhanced) partition function of the ASEP. At various specializations of the parameters, the partition function factors. We also explore combinatorial properties of staircase tableaux, elucidating connections to trees, matchings, permutations, etc. We conclude with a number of open problems.
In this paper we outline a ``Matrix Ansatz" approach to some problems of combinatorial enumeration. The idea is that many interesting quantities can be expressed in terms of products of matrices, where the matrices obey certain relations. We illustrate this approach with applications to moments of orthogonal polynomials, permutations, signed permutations, and tableaux.
We define a multivariate Markov chain on the symmetric group with remarkable enumerative properties. We conjecture that the components of its stationary distribution can be written as positive combinations of Schubert polynomials.
Introduced in the late 1960's, the asymmetric exclusion process (ASEP) is a model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites. In its most general form, particles may enter and exit at the left with probabilities α and γ, and they may exit and enter at the right with probabilities β and δ. In the bulk, the probability of hopping left is q times that of hopping right. The first result of this paper is a combinatorial formula for the stationary distribution of the ASEP with all parameters general, in terms of some new staircase tableaux. This generalizes our previous work for the ASEP with parameters γ=δ=0. Combining our first result with results of Uchiyama-Sasamoto-Wadati, we derive our second result: a combinatorial formula for the moments of Askey-Wilson polynomials. Since the early 1980's there has been a great deal of work giving combinatorial formulas for moments of various other classical orthogonal polynomials. However, this is the first such formula for the Askey-Wilson polynomials, which are at the top of the hierarchy of classical orthogonal polynomials.
This is an announcement of the results of the paper above.
The class of cluster algebras coming from triangulated surfaces was systematically studied by Fomin-Shapiro-Thurston, and it was later shown by Felikson-Shapiro-Tumarkin that this class is very large: it includes all but finitely many (= eleven) of the skew-symmetric cluster algebras of finite mutation type. In this paper we give combinatorial formulas for the Laurent expansion of any cluster variable in any cluster algebra coming from a triangulated surface (with or without punctures), with respect to an arbitrary seed. Moreover, we work in the generality of principal coefficients. An immediate corollary of our formulas is a proof of the positivity conjecture of Fomin and Zelevinsky for cluster algebras from surfaces, in geometric type.
In a seminal 1994 paper, Lusztig extended the theory of total positivity by introducing the totally non-negative part (G/P)_{\geq 0} of an arbitrary (generalized, partial) flag variety G/P. He referred to this space as a "remarkable polyhedral subspace", and conjectured a decomposition into cells, which was subsequently proven by the first author. Subsequently the second author made the concrete conjecture that this cell decomposed space is the next best thing to a polyhedron, by conjecturing it to be a regular CW complex that is homeomorphic to a closed ball (see the paper "Shelling ..." below). In this article we use discrete Morse theory to prove this conjecture up to homotopy-equivalence. Explicitly, we prove that the boundaries of the cells are homotopic to spheres, and the closures of cells are contractible. The latter part generalizes a result of Lusztig's that (G/P)_{\geq 0} -- the closure of the top-dimensional cell -- is contractible. Concerning our result on the boundaries of cells, even the special case that the boundary of the top-dimensional cell (G/P)_{> 0} is homotopic to a sphere, is new for all G/P other than projective space.
We introduce a new family of noncommutative analogs of the Hall-Littlewood symmetric functions. Our construction relies upon Tevlin's bases and simple q-deformations of the classical combinatorial Hopf algebras. We connect our new Hall-Littlewood functions to permutation tableaux, which gives an exact formula for the q-enumeration of permutation tableaux of a fixed shape. By a result in the paper ``Permutation tableaux and permutation patterns" below, this is also an exact formula for the number of permutations with a fixed set of weak excedances, enumerated according to crossings. And by the main result of the paper ``Tableaux Combinatorics for the asymmetric exclusion process" below, this gives an explicit formula for the steady state probability of each state in the partially asymmetric exclusion process.
The totally nonnegative part of a partial flag variety G/P has been shown by Rietsch to be a union of semi-algebraic cells. In this note we provide glueing maps for each of the cells to prove that the totally nonnegative part of G/P is a CW complex. This generalizes a previous result found in collaboration with Postnikov and Speyer for Grassmannians. We again use a technique of associating an auxiliary toric variety to each parameterization of a cell; but this time we need to use the canonical basis to prove that the parameterizations are given by positive polynomials.
In this paper we explore the combinatorics of the non-negative part of a cominuscule Grassmannian (G/P)+. For each such Grassmannian we define Le-diagrams -- certain fillings of generalized Young diagrams which are in bijection with the cells of (G/P)+. In the classical cases, we describe Le-diagrams explicitly in terms of pattern avoidance. We also define a game on diagrams, by which one can reduce an arbitrary diagram to a Le-diagram. We give enumerative results and relate our Le-diagrams to other combinatorial objects. Surprisingly, the totally non-negative cells in the open Schubert cell of the even and odd orthogonal Grassmannians are (essentially) in bijection with preference functions and atomic preference functions respectively.
In this paper we use toric geometry to investigate the topology of the totally non-negative part of the Grassmannian, a cell complex whose cells can be parameterized in terms of the combinatorics of plane-bipartite graphs. To each cell we associate a related toric variety, whose moment polytope is related to a matroid polytope, and whose combinatorial structure is similar to a Birkhoff polytope and can be completely described in terms of plane-bipartite graphs. We use our technology to prove that the cell decomposition of the non-negative part of the Grassmannian is a CW complex and that the Euler characteristic of the closure of each cell is 1.
The aim of this paper is to calculate face numbers of simple generalized permutohedra and study their f, h, and gamma-vectors. Generalized permutohedra include many famous families of polytopes, including permutohedra, assocahedra, graph-associahedra, and graphical zonotopes. We give several explicit formulas involving descent statistics, and calculate generating functions. In particular, we give a combinatorial interpretation for gamma-vectors of a wide class of simple generalized permutohedra (the chordal nestohedra), proving Gal's conjecture on the nonnegativity of gamma-vectors in this case.
In this paper we give two short proofs of a conjecture of Richard Stanley concerning the equidistribution of derangements and alternating permutations with the maximal number of fixed points.
In this paper we strengthen the connection between permutation tableaux and the PASEP found in our previous paper "Tableaux combinatorics ..." by showing that the PASEP can be "lifted" to a Markov chain on permutation tableaux of a fixed semiperimeter. Because of the bijection between permutation tableaux and permutations, this can also be thought of as a Markov chain on permutations in S_n.
The (partially) asymmetric exclusion process (PASEP) is an important model from statistical mechanics which involves particles hopping on a one-dimensional lattice. It has been cited as a model for traffic flow and protein synthesis. In this paper we use the matrix ansatz to prove a combinatorial interpretation for the steady state probability of being in any configuration of the PASEP. Surprisingly, our formula is in terms of permutation tableaux, certain combinatorial objects indexing cells in the non-negative part of the Grassmannian.
It is conjectured that the non-negative part of a real flag variety (as defined by Lusztig) is homeomorphic to a ball. A stronger conjecture says that its Lusztig-Rietsch cell decomposition is a regular CW complex homeomorphic to a ball. Here we use tools from poset topology to prove the combinatorial analog of this statement: that the poset (partially ordered set) of Lusztig-Rietsch cells is the face poset of a regular CW complex homeomorphic to a ball. This result holds in complete generality -- for any partial flag variety of any type.
Permutation tableaux are a distinguished subset of Postnikov's "Le-diagrams," which index cells in the non-negative part of the Grassmannian. In this paper we show that the bijection from the set of permutation tableaux to permutations translates many natural tableaux statistics into natural permutation statistics. One application is an additional combinatorial interpretation for the q-Eulerian polynomials introduced in my paper "Enumeration of totally positive Grassmann cells": this polynomial enumerates permutations according to descents and occurrences of certain generalized permutation patterns.
My thesis comprises the four papers below. The main difference is an appendix with pictures of some posets of Le-diagrams and decorated permutations.
We consider oriented matroids coming from Coxeter arrangements, and study their Bergman complexes and positive Bergman complexes. We relate these objects to nested set complexes and graph associahedra. Additionally, we prove that for an arbitrary orientable matroid, its Bergman complex is covered in a nice way by the various positive Bergman complexes one gets by considering different orientations.
The Bergman complex can be thought of as a generalization for matroids of the notion of a tropical variety. There is a natural notion of the "totally positive" part of the Bergman complex of an oriented matroid. We relate this object to the Las Vergnas face lattice, thereby proving that it is homeomorphic to a ball.
We introduce the totally positive part of a tropical variety -- an object which has the structure of a polyhedral fan -- and study this object in the case of the Grassmannian. For the Grassmannians G(2,n), G(3,6), G(3,7), and G(3,8), the polyhedral fans in question turn out to be (essentially) the generalized associahedra of types A, D_4, E_6, and E_8, respectively. These results are reminiscent of the fact that the Grassmannian's coordinate ring has a cluster algebra structure which in these cases has types A, D_4, E_6, E_8. We formulate a conjecture generalizing these results.
The nonnegative part of the Grassmannian is the subset of the real Grassmannian where all Plucker coordinates are non-negative (this definition was given by Postnikov; it turns out to agree with Lusztig's definition). We prove an explicit formula for the rank-generating function for its Lusztig-Postnikov-Rietsch cell decomposition. This leads us to introduce a new q-analog of the Eulerian numbers, which enumerates permutations according to weak excedences and "crossings." (Subsequently Corteel showed that this polynomial also has an interpretation in terms of the PASEP, which led to our joint work on the PASEP.)
This was written when I attended the Duluth REU.
This was written when I was a high school student at RSI.
I've had the pleasure of collaborating with: Federico Ardila, Robin Chapman, Sylvie Corteel, Sergey Fomin, Matthieu Josuat-Verges, Steven Karp, Carly Klivans, Yuji Kodama, Thomas Lam, Olya Mandelshtam, Gregg Musiker, Jean-Christophe Novelli, Clelia Pech, Alex Postnikov, Vic Reiner, Konstanze Rietsch, Felipe Rincon, Tomohiro Sasamoto, Ralf Schiffler, David Speyer, Richard Stanley, Dennis Stanton, Einar Steingrimsson, Bernd Sturmfels, Kelli Talaska, Jean-Yves Thibon, Emmanuel Tsukerman, Andrei Zelevinsky.