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We study Dressians of matroids using the initial matroids of Dress and Wenzel. These correspond to cells in regular matroid subdivisions of matroid polytopes. An efficient algorithm for computing Dressians is presented, and its implementation is applied to a range of interesting matroids. We give counterexamples to a few plausible statements about matroid subdivisions.

The Dressian and the tropical Grassmannian parameterize abstract and realizable tropical linear spaces; but in general, the Dressian is much larger than the tropical Grassmannian. There are natural positive notions of both of these spaces – the positive Dressian, and the positive tropical Grassmannian (which we introduced roughly fifteen years ago in [J. Algebraic Combin. 22 (2005), pp. 189–210]) – so it is natural to ask how these two positive spaces compare. In this paper we show that the positive Dressian equals the positive tropical Grassmannian. Using the connection between the positive Dressian and regular positroidal subdivisions of the hypersimplex, we use our result to give a new “tropical” proof of da Silva’s 1987 conjecture (first proved in 2017 by Ardila-Rincón-Williams) that all positively oriented matroids are realizable. We also show that the finest regular positroidal subdivisions of the hypersimplex consist of series-parallel matroid polytopes, and achieve equality in Speyer’s f f -vector theorem. Finally we give an example of a positroidal subdivision of the hypersimplex which is not regular, and make a connection to the theory of tropical hyperplane arrangements.

Let FI denote the category whose objects are the sets $[n] = \{1,\ldots, n\}$, and whose morphisms are injections. We study functors from the category FI into the category of sets. We write $\mathfrak{S}_n$ for the symmetric group on $[n]$. Our first main result is that, if the functor $[n] \mapsto X_n$ is "finitely generated" there there is a finite sequence of integers $m_i$ and a finite sequence of subgroups $H_i$ of $\mathfrak{S}_{m_i}$ such that, for $n$ sufficiently large, $X_n \cong \bigsqcup_i \mathfrak{S}_n/(H_i \times \mathfrak{S}_{n-m_i})$ as a set with $\mathfrak{S}_n$ action. Our second main result is that, if $[n] \mapsto X_n$ and $[n] \mapsto Y_n$ are two such finitely generated functors and $R_n \subset X_n \times Y_n$ is an FI-invariant family of relations, then the $(0,1)$ matrices encoding the relation $R_n$, when written in an appropriate basis, vary polynomially with $n$. In particular, if $R_n$ is an FI-invariant family of relations from $X_n$ to itself, then the eigenvalues of this matrix are algebraic functions of $n$. As an application of this theorem we provide a proof of a result about eigenvalues of adjacency matrices claimed by the first and last author. This result recovers, for instance, that the adjacency matrices of the Kneser graphs have eigenvalues which are algebraic functions of $n$, while also expanding this result to a larger family of graphs.

Given the complement of a hyperplane arrangement, let [math] be the closure of the graph of the map inverting each of its defining linear forms. The characteristic polynomial manifests itself in the Hilbert series of [math] in two different-seeming ways, one due to Terao and the other to Huh and Katz. We define an extension of the no broken circuit complex of a matroid and use it to give a direct Gröbner basis argument that the polynomials extracted from the Hilbert series in these two ways agree.

The purpose of this document is to connect two maps related to certain graphs embedded in the disc. The first is Postnikov's boundary measurement map, which combines partition functions of matchings in the graph into a map from an algebraic torus to an open positroid variety in a Grassmannian. The second is a rational map from the open positroid variety to an algebraic torus, given by certain Plucker coordinates which are expected to be a cluster in a cluster structure. This paper clarifies the relationship between these two maps, which has been ambiguous since they were introduced by Postnikov in 2001. The missing ingredient supplied by this paper is a twist automorphism of the open positroid variety, which takes the target of the boundary measurement map to the domain of the (conjectural) cluster. Among other applications, this provides an inverse to the boundary measurement map, as well as Laurent formulas for twists of Plucker coordinates.

Considered as commutative algebras, cluster algebras can be very unpleasant objects. However, the first author introduced a condition known as "local acyclicity" which implies that cluster algebras behave reasonably. One of the earliest and most fundamental examples of a cluster algebra is the homogenous coordinate ring of the Grassmannian. We show that the Grassmannian is locally acyclic. Morally, we are in fact showing the stronger result that all positroid varieties are locally acyclic. However, it has not been shown that all positroid varieties have cluster structure, so what we actually prove is that certain cluster varieties associated to Postnikov's alternating strand diagrams are locally acylic. Moreover, we actually establish a slightly stronger property than local acyclicity, which we term the Louise property, that is designed to facilitate proofs involving the Mayer-Vietores sequence., Comment: 14 pages, 8 figures, minor edits from previous version, added references to recent work of LeClerc

We prove a Fundamental Theorem of Finite Semidistributive Lattices (FTFSDL), modelled on Birkhoff's Fundamental Theorem of Finite Distributive Lattices. Our FTFSDL is of the form "A poset L is a finite semidistributive lattice if and only if there exists a set Sha with some additional structure, such that L is isomorphic to the admissible subsets of Sha ordered by inclusion; in this case, Sha and its additional structure are uniquely determined by L." The additional structure on Sha is a combinatorial abstraction of the notion of torsion pairs from representation theory and has geometric meaning in the case of posets of regions of hyperplane arrangements. We show how the FTFSDL clarifies many constructions in lattice theory, such as canonical join representations and passing to quotients, and how the semidistributive property interacts with other major classes of lattices. Many of our results also apply to infinite lattices., Comment: 44 pages

We study the action of a differential operator on Schubert polynomials. Using this action, we first give a short new proof of an identity of I. Macdonald (1991). We then prove a determinant conjecture of R. Stanley (2017). This conjecture implies the (strong) Sperner property for the weak order on the symmetric group, a property recently established by C. Gaetz and Y. Gao (2018)., 6 pages

Schur modules give the irreducible polynomial representations of the general linear group G L t \mathrm {GL}_t . Viewing the symmetric group S t \mathfrak {S}_t as a subgroup of G L t \mathrm {GL}_t , we may restrict Schur modules to S t \mathfrak {S}_t and decompose the result into a direct sum of Specht modules, the irreducible representations of S t \mathfrak {S}_t . We give an equivariant Möbius inversion formula that we use to invert this expansion in the representation ring for S t \mathfrak {S}_t for t t large. In addition to explicit formulas in terms of plethysms, we show the coefficients that appear alternate in sign by degree. In particular, this allows us to define a new basis of symmetric functions whose structure constants are stable Kronecker coefficients and which expand with signs alternating by degree into the Schur basis.

We develop a general approach to finding combinatorial models for cluster algebras. The approach is to construct a labeled graph called a framework. When a framework is constructed with certain properties, the result is a model incorporating information about exchange matrices, principal coefficients, g-vectors, and g-vector fans. The idea behind frameworks arises from Cambrian combinatorics and sortable elements, and in this paper, we use sortable elements to construct a framework for any cluster algebra with an acyclic initial exchange matrix. This Cambrian framework yields a model of the entire exchange graph when the cluster algebra is of finite type. Outside of finite type, the Cambrian framework models only part of the exchange graph. In a forthcoming paper, we extend the Cambrian construction to produce a complete framework for a cluster algebra whose associated Cartan matrix is of affine type., 50 pages, 2 figures

Leclerc and Zelevinsky described quasicommuting families of quantum minors in terms of a certain combinatorial condition, called weak separation. They conjectured that all inclusion-maximal weakly separated collections of minors have the same cardinality, and that they can be related to each other by a sequence of mutations. Postnikov studied total positivity on the Grassmannian. He described a stratification of the totally non-negative Grassmannian into positroid strata, and constructed theirparameterization using plabic graphs. In this paper, we link the study of weak separation to plabic graphs. We extend the notion of weak separation to positroids. We generalize the conjectures of Leclerc and Zelevinsky, and related ones of Scott, and prove them. We show that the maximal weakly separated collections in a positroid are in bijective correspondence with the plabic graphs. This correspondence allows us to use the combinatorial techniques of positroids and plabic graphs to prove the (generalized) purity and mutation connectedness conjectures., National Science Foundation (U.S.) (CAREER Award DMS-0504629)

The Dressian Dr(k,n) parametrizes all tropical linear spaces, and it carries a natural fan structure as a subfan of the secondaryfan of the hypersimplex \Delta(k,n). We explore the combinatorics of the rays of Dr(k,n), that is, the most degenerate tropical planes, for arbitrary k and n. This is related to a new rigidity concept for configurations of n-k points in the tropical (k-1)-torus. Additional conditions are given for k=3. On the way, we compute the entire fan Dr(3,8)., Comment: Minor revision: several improvements, new figures, some statements made more precise

The second author has introduced non-crossing tableaux, objects whose non-nesting analogues are semi-standard Young tableaux. We relate non-crossing tableaux to Gelfand-Tsetlin patterns and develop the non-crossing analogue of standard monomial theory. Leclerc and Zelevinsky's weakly separated sets are special cases of non-crossing tableaux, and we suggest that non-crossing tableaux may help illuminate the theory of weakly separated sets., Comment: 23 pages, 7 figures

Let G ( d , n ) denote the Grassmannian of d-planes in C n and let T be the torus ( C ∗ ) n / diag ( C ∗ ) which acts on G ( d , n ) . Let x be a point of G ( d , n ) and let T x ¯ be the closure of the T-orbit through x. Then the class of the structure sheaf of T x ¯ in the K-theory of G ( d , n ) depends only on which Plucker coordinates of x are nonzero – combinatorial data known as the matroid of x. In this paper, we will define a certain map of additive groups from K ○ ( G ( d , n ) ) to Z [ t ] . Letting g x ( t ) denote the image of ( − 1 ) n − dim T x [ O T x ¯ ] , g x behaves nicely under the standard constructions of matroid theory, such as direct sum, two-sum, duality and series and parallel extensions. We use this invariant to prove bounds on the complexity of Kapranov's Lie complexes [M. Kapranov, Chow quotients of Grassmannians I, Adv. Soviet Math. 16 (2) (1993) 29–110], Hacking, Keel and Tevelev's very stable pairs [P. Hacking, S. Keel, E. Tevelev, Compactification of the moduli space of hyperplane arrangements, J. Algebraic Geom. 15 (2006) 657–680] and the author's tropical linear spaces when they are realizable in characteristic zero [D. Speyer, Tropical linear spaces, SIAM J. Discrete Math. 22 (4) (2008) 1527–1558]. Namely, in characteristic zero, a Lie complex or the underlying ( d − 1 ) -dimensional scheme of a very stable pair can have at most ( n − i − 1 ) ! ( d − i ) ! ( n − d − i ) ! ( i − 1 ) ! strata of dimensions n − i and d − i , respectively. This prove the author's f-vector conjecture, from [D. Speyer, Tropical linear spaces, SIAM J. Discrete Math. 22 (4) (2008) 1527–1558], in the case of a tropical linear space realizable in characteristic 0.

In this paper we use toric geometry to investigate the topology of the totally non-negative part of the Grassmannian, denoted (Gr k,n )?0. This is a cell complex whose cells Δ G can be parameterized in terms of the combinatorics of plane-bipartite graphs G. To each cell Δ G we associate a certain polytope P(G). The polytopes P(G) are analogous to the well-known Birkhoff polytopes, and we describe their face lattices in terms of matchings and unions of matchings of G. We also demonstrate a close connection between the polytopes P(G) and matroid polytopes. We use the data of P(G) to define an associated toric variety X G . We use our technology to prove that the cell decomposition of (Gr k,n )?0 is a CW complex, and furthermore, that the Euler characteristic of the closure of each cell of (Gr k,n )?0 is 1.

We prove a number of conjectures due to Dinesh Thakur concerning sums of the form $\sum_P h(P)$ where the sum is over monic irreducible polynomials $P$ in $\mathbb{F}_q[T]$, the function $h$ is a rational function and the sum is considered in the $T^{-1}$-adic topology. As an example of our results, in $\mathbb{F}_2[T]$, the sum $\sum_P \tfrac{1}{P^k - 1}$ always converges to a rational function, and is $0$ for $k=1$.

Fix a variety X X with a transitive (left) action by an algebraic group G G . Let E \mathcal {E} and F \mathcal {F} be coherent sheaves on X X . We prove that for elements g g in a dense open subset of G G , the sheaf T o r i X ( E , g F ) \mathcal {T}\hspace {-.7ex}or^X_i(\mathcal {E}, g \mathcal {F}) vanishes for all i > 0 i > 0 . When E \mathcal {E} and F \mathcal {F} are structure sheaves of smooth subschemes of X X in characteristic zero, this follows from the Kleiman–Bertini theorem; our result has no smoothness hypotheses on the supports of E \mathcal {E} or F \mathcal {F} , or hypotheses on the characteristic of the ground field.

We give a combinatorial model for the exchange graph and g-vector fan associated to any acyclic exchange matrix B of affine type. More specifically, we construct a reflection framework for B in the sense of [N. Reading and D. E. Speyer, "Combinatorial frameworks for cluster algebras"] and establish good properties of this framework. The framework (and in particular the g-vector fan) is constructed by combining a copy of the Cambrian fan for B with an antipodal copy of the Cambrian fan for -B., 41 pages, 14 figures. Version 2: Expository changes in introduction and in background on cluster algebras. Version 3: Minor changes and corrections. (We acknowledge helpful suggestions by anonymous referees.)

Tropical algebraic geometry is the geometry of the tropical semiring (?, min, +). The theory of total positivity is a natural generalization of the study of matrices with all minors positive. In this paper we introduce the totally positive part of the tropicalization of an arbitrary affine variety, an object which has the structure of a polyhedral fan. We then investigate the case of the Grassmannian, denoting the resulting fan Trop+ Grk,n. We show that Trop+ Gr2,n is the Stanley-Pitman fan, which is combinatorially the fan dual to the (type An?3) associahedron, and that Trop+ Gr3,6 and Trop+ Gr3,7 are closely related to the fans dual to the types D4 and E6 associahedra. These results are strikingly reminiscent of the results of Fomin and Zelevinsky, and Scott, who showed that the Grassmannian has a natural cluster algebra structure which is of types An?3, D4, and E6 for Gr2,n, Gr3,6, and Gr3,7. We suggest a general conjecture about the positive part of the tropicalization of a cluster algebra.

In tropical algebraic geometry, the solution sets of polynomial equations are piecewise-linear. We introduce the tropical variety of a polynomial ideal, and we identify it with a polyhedral subcomplex of the Grobner fan. The tropical Grassmannian arises in this manner from the ideal of quadratic Plucker relations. It parametrizes all tropical linear spaces. Lines in tropical projective space are trees, and their tropical Grassmannian G2; n equals the space of phylogenetic trees studied by Billera, Holmes and Vogtmann. Higher Grassmannians oer a natural generalization of the space of trees. Their faces correspond to monomial-free initial ideals of the Plucker ideal. The tropical Grassmannian G3; 6 is a simplicial complex glued from 1035 tetrahedra.

We provide a short proof of a classical result of Kasteleyn, and prove several variants thereof. One of these results has become key in the parametrization of positroid varieties, and thus deserves the short direct proof which we provide.

Every real hyperbolic form in three variables can be realized as the determinant of a linear net of Hermitian matrices containing a positive definite matrix. Such representations are an algebraic certificate for the hyperbolicity of the polynomial and their existence has been proved in several different ways. However, the resulting algorithms for computing determinantal representations are computationally intensive. In this note, we present an algorithm that reduces a large part of the problem to linear algebra and discuss its numerical implementation., Comment: 8 pages, 2 figures

Scattering amplitudes in 4d $$ \mathcal{N}=4 $$ super Yang-Mills theory (SYM) can be described by Grassmannian contour integrals whose form depends on whether the external data is encoded in momentum space, twistor space, or momentum twistor space. After a pedagogical review, we present a new, streamlined proof of the equivalence of the three integral formulations. A similar strategy allows us to derive a new Grassmannian integral for 3d $$ \mathcal{N}=6 $$ ABJM theory amplitudes in momentum twistor space: it is a contour integral in an orthogonal Grassmannian with the novel property that the internal metric depends on the external data. The result can be viewed as a central step towards developing an amplituhedron formulation for ABJM amplitudes. Various properties of Grassmannian integrals are examined, including boundary properties, pole structure, and a homological interpretation of the global residue theorems for $$ \mathcal{N}=4 $$ SYM.

A labeling of a graph is a function f from the vertex set to some subset of the natural numbers. The image of a vertex is called its label. We assign the label | f ( u )− f ( v )| to the edge incident with vertices u and v . In a k-equitable labeling the image of f is the set {0,1,2,…, k −1}. We require both the vertex labels and the edge labels to be as equally distributed as possible, i.e., if v i denotes the number of vertices labeled i and e i denotes the number of edges labeled i , we require | v i − v j |⩽1 and | e i − e j |⩽1 for every i , j in {0,1,2,…, k −1}. Equitable graph labelings were introduced by I. Cahit as a generalization for graceful labeling. We prove that every tree is 3-equitable.

In tropical geometry, given a curve in a toric variety, one defines a corresponding graph embedded in Euclidean space. We study the problem of reversing this process for curves of genus zero and one. Our methods focus on describing curves by parameterizations, not by their defining equations; we give parameterizations by rational functions in the genus-zero case and by nonarchimedean elliptic functions in the genus-one case. For genus-zero curves, those graphs which can be lifted can be characterized in a completely combinatorial manner. For genus-one curves, we show that certain conditions identified by Mikhalkin are sufficient and we also identify a new necessary condition.

Plabic graphs are interesting combinatorial objects used to study the totally nonnegative Grassmannian. Faces of plabic graphs are labeled by $k$-element sets of positive integers, and a collection of such $k$-element sets are the face labels of a plabic graph if that collection forms a maximal weakly separated collection. There are moves that one can apply to plabic graphs, and thus to maximal weakly separated collections, analogous to mutations of seeds in cluster algebras. In this short note, we show that if two maximal weakly separated collections can be mutated from one to another, then one can do so while freezing the face labels they have in common., Comment: 8 pages, 1 figure

We describe a new way to relate an acyclic, skew-symmetrizable cluster algebra to the representation theory of a finite dimensional hereditary algebra. This approach is designed to explain the c-vectors of the cluster algebra. We obtain a necessary and sufficient combinatorial criterion for a collection of vectors to be the c-vectors of some cluster in the cluster algebra associated to a given skew-symmetrizable matrix. Our approach also yields a simple proof of the known result that the c-vectors of an acyclic cluster algebra are sign-coherent, from which Nakanishi and Zelevinsky have showed that it is possible to deduce in an elementary way several important facts about cluster algebras.

To every matroid, we associate a class in the K-theory of the Grassmannian. We study this class using the method of equivariant localization. In particular, we provide a geometric interpretation of the Tutte polynomial. We also extend results of the second author concerning the behavior of such classes under direct sum, series and parallel connection and two-sum; these results were previously only established for realizable matroids, and their earlier proofs were more difficult., À chaque matroïde, nous associons une classe dans la K-théorie de la grassmannienne. Nous étudions cette classe en utilisant la méthode de localisation équivariante. En particulier, nous fournissons une interprétation géométrique du polynôme de Tutte. Nous étendons également les résultats du second auteur concernant le comportement de ces classes pour la somme directe, les connexions série et parallèle et la 2-somme; ces résultats n'ont été déjà établis que pour les matroïdes réalisables, et leurs preuves précédentes étaient plus difficiles.

Given a point z in P^1, let F(z) be the osculating flag to the rational normal curve at point z. The study of Schubert problems with respect to such flags F(z_1), F(z_2), ..., F(z_r) has been studied both classically and recently, especially when the points z_i are real. Since the rational normal curve has an action of PGL_2, it is natural to consider the points (z_1, ..., z_r) as living in the moduli space of r distinct point in P^1 -- the famous M_{0,r}. One can then ask to extend the results on Schubert intersections to the compactification \bar{M}_{0,r}. The first part of this paper achieves this goal. We construct a flat, Cohen-Macaulay family over \bar{M}_{0,r}, whose fibers over M_{0,r} are isomorphic to G(d,n) and, given partitions lambda_1, ..., lambda_r, we construct a flat Cohen-Macualay family over \bar{M}_{0,r} whose fiber over (z_1, ..., z_r) in M_{0,r} is the intersection of the Schubert varieties indexed by lambda_i with respect to the osculating flags F(z_i). In the second part of the paper, we investigate the topology of the real points of our family, in the case that sum |lambda_i| = dim G(d,n). We show that our family is a finite covering space of \bar{M}_{0,r}, and give an explicit CW decomposition of this cover whose faces are indexed by objects from the theory of Young tableaux.

While the intersection of the Grassmannian Bruhat decompositions for all coordinate flags is an intractable mess, the intersection of only the cyclic shifts of one Bruhat decomposition turns out to have many of the good properties of the Bruhat and Richardson decompositions. This decomposition coincides with the projection of the Richardson stratification of the flag manifold, studied by Lusztig, Rietsch, Brown-Goodearl-Yakimov and the present authors. However, its cyclic-invariance is hidden in this description. Postnikov gave many cyclic-invariant ways to index the strata, and we give a new one, by a subset of the affine Weyl group we call bounded juggling patterns. We call the strata positroid varieties. Applying results from the authors' previous work, we show that positroid varieties are normal, Cohen-Macaulay, have rational singularities, and are defined as schemes by the vanishing of Plucker coordinates. We prove that their associated cohomology classes are represented by affine Stanley functions. This last fact lets us connect Postnikov's and Buch-Kresch-Tamvakis' approaches to quantum Schubert calculus., Comment: Most of this material appeared in our preprint arXiv:0903.3694 . We generalized many of the results of that paper to all Cartan types and published them separately in arXiv:1008.3939 . This paper contains only those remaining results which are special to Grassmannians. Many of the proofs are also shortened and improved. 44 pages (as compared to 58 in 0903.3694)

While the projections of Schubert varieties in a full generalized flag manifold G/B to a partial flag manifold $G/P$ are again Schubert varieties, the projections of Richardson varieties (intersections of Schubert varieties with opposite Schubert varieties) are not always Richardson varieties. The stratification of G/P by projections of Richardson varieties arises in the theory of total positivity and also from Poisson and noncommutative geometry. In this paper we show that many of the geometric properties of Richardson varieties hold more generally for projected Richardson varieties; they are normal, Cohen-Macaulay, have rational singularities, and are compatibly Frobenius split with respect to the standard splitting. Indeed, we show that the projected Richardson varieties are the only compatibly split subvarieties, providing an example of the recent theorem [Schwede, Kumar-Mehta] that a Frobenius split scheme has only finitely many compatibly split subvarieties. (The G/B case was treated by [Hague], whose proof we simplify somewhat.) One combinatorial analogue of a Richardson variety is the order complex of the corresponding Bruhat interval in W; this complex is known to be an EL-shellable ball [Bjorner-Wachs '82]. We prove that the projection of such a complex into the order complex of the Bruhat order on W/W_P is again a shellable ball. This requires extensive analysis of "P-Bruhat order", a generalization of the k-Bruhat order of [Bergeron-Sottile '98]. In the case that G/P is minuscule (e.g. a Grassmannian), we show that its Grobner degeneration takes each projected Richardson variety to the Stanley-Reisner scheme of its corresponding ball., New appendix, proving Richardon varieties have rational singularities in all characterisitcs. Many other minor edits, suggested by referees. Contains, and greatly improves on, material from arXiv:0903.3694

We introduce a recurrence which we term the multidimensional cube recurrence, generalizing the octahedron recurrence studied by Propp, Fomin and Zelevinsky, Speyer, and Fock and Goncharov and the three-dimensional cube recurrence studied by Fomin and Zelevinsky, and Carroll and Speyer. The states of this recurrence are indexed by tilings of a polygon with rhombi, and the variables in the recurrence are indexed by vertices of these tilings. We travel from one state of the recurrence to another by performing elementary flips. We show that the values of the recurrence are independent of the order in which we perform the flips; this proof involves nontrivial combinatorial results about rhombus tilings which may be of independent interest. We then show that the multidimensional cube recurrence exhibits the Laurent phenomenon -- any variable is given by a Laurent polynomial in the other variables. We recognize a special case of the multidimensional cube recurrence as giving explicit equations for the isotropic Grassmannians IG(n-1,2n). Finally, we describe a tropical version of the multidimensional cube recurrence and show that, like the tropical octahedron recurrence, it propagates certain linear inequalities., Comment: Final version to appear in Adv. in Math. Improved exposition on Spin groups, other minor changes

Each Coxeter element c of a Coxeter group W defines a subset of W called the c-sortable elements. The choice of a Coxeter element of W is equivalent to the choice of an acyclic orientation of the Coxeter diagram of W. In this paper, we define a more general notion of Omega-sortable elements, where Omega is an arbitrary orientation of the diagram, and show that the key properties of c-sortable elements carry over to the Omega-sortable elements. The proofs of these properties rely on reduction to the acyclic case, but the reductions are nontrivial; in particular, the proofs rely on a subtle combinatorial property of the weak order, as it relates to orientations of the Coxeter diagram. The c-sortable elements are closely tied to the combinatorics of cluster algebras with an acyclic seed; the ultimate motivation behind this paper is to extend this connection beyond the acyclic case., Comment: Final version as published. An error corrected in the previous counterexample, other minor improvements

We study the so-called looping case of [email protected]?s game of numbers, which concerns the (finite) orbits in the reflection representation of affine Weyl groups situated on the boundary of the Tits cone. We give a simple proof that all configurations in the orbit are obtainable from each other by playing the numbers game, and give a strategy for going from one configuration to another. This strategy gives rise to a partition of the finite Weyl group into finitely many graded posets, one for each extending vertex of the associated extended Dynkin diagram. These posets are self-dual and mutually isomorphic, and their Hasse diagrams are dual to the triangulation of the unit hypercube by reflecting hyperplanes. Unlike the weak and Bruhat orders, the top degree is cubic in the number of vertices of the graph. We explicitly compute the rank generating function of the poset.

Let W be an infinite irreducible Coxeter group with (s_1, ..., s_n) the simple generators. We give a simple proof that the word s_1 s_2 ... s_n s_1 s_2 >... s_n ... s_1 s_2 ... s_n is reduced for any number of repetitions of s_1 s_2 >... s_n. This result was proved for simply-laced, crystallographic groups by Kleiner and Pelley using methods from the theory of quiver representations. Our proof only using basic facts about Coxeter groups and the geometry of root systems., 7 pages, no figures

The tropical variety of a $d$-dimensional prime ideal in a polynomial ring with complex coefficients is a pure $d$-dimensional polyhedral fan. This fan is shown to be connected in codimension one. We present algorithmic tools for computing the tropical variety, and we discuss our implementation of these tools in the Gr\"obner fan software \texttt{Gfan}. Every ideal is shown to have a finite tropical basis, and a sharp lower bound is given for the size of a tropical basis for an ideal of linear forms., Comment: 22 pages, 2 figures