scholarly journals Combinatorial formulas for ⅃-coordinates in a totally nonnegative Grassmannian, extended abstract, extended abstract

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Kelli Talaska
Keyword(s):  
Inverse Boundary ◽  
Plücker Coordinates ◽  
Boundary Measurement ◽  
Totally Positive ◽  
Combinatorial Formulas ◽  
International Audience

International audience Postnikov constructed a decomposition of a totally nonnegative Grassmannian $(Gr _{kn})_≥0$ into positroid cells. We provide combinatorial formulas that allow one to decide which cell a given point in $(Gr _{kn})_≥0$ belongs to and to determine affine coordinates of the point within this cell. This simplifies Postnikov's description of the inverse boundary measurement map and generalizes formulas for the top cell given by Speyer and Williams. In addition, we identify a particular subset of Plücker coordinates as a totally positive base for the set of non-vanishing Plücker coordinates for a given positroid cell.

scholarly journals Arrangements of equal minors in the positive Grassmannian

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Miriam Farber ◽  
Alexander Postnikov
Keyword(s):  
Cluster Algebra ◽  
The Other ◽  
Eulerian Number ◽  
Other Hand ◽  
Totally Positive ◽  
Positive Matrices ◽  
International Audience ◽  
Separated Sets ◽  
Weakly Separated Sets ◽  
Totally Positive Matrices

International audience We discuss arrangements of equal minors in totally positive matrices. More precisely, we would like to investigate the structure of possible equalities and inequalities between the minors. We show that arrangements of equals minors of largest value are in bijection with <i>sorted sets</i>, which earlier appeared in the context of <i>alcoved polytopes</i> and Gröbner bases. Maximal arrangements of this form correspond to simplices of the alcoved triangulation of the hypersimplex; and the number of such arrangements equals the <i>Eulerian number</i>. On the other hand, we conjecture and prove in many cases that arrangements of equal minors of smallest value are exactly the <i>weakly separated sets</i>. Weakly separated sets, originally introduced by Leclerc and Zelevinsky, are closely related to the \textitpositive Grassmannian and the associated <i>cluster algebra</i>.

scholarly journals Combinatorial formulas for double parabolic R-polynomials

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Justin Lambright ◽  
Mark Skandera
Keyword(s):  
Polynomial Ring ◽  
Hecke Algebra ◽  
Combinatorial Interpretations ◽  
Combinatorial Formulas ◽  
International Audience ◽  
Quantum Polynomial

International audience The well-known R-polynomials in ℤ[q], which appear in Hecke algebra computations, are closely related to certain modified R-polynomials in ℕ[q] whose coefficients have simple combinatorial interpretations. We generalize this second family of polynomials, providing combinatorial interpretations for expressions arising in a much broader class of computations. In particular, we extend results of Brenti, Deodhar, and Dyer to new settings which include parabolic Hecke algebra modules and the quantum polynomial ring. Les bien connues polynômes-R en ℤ[q], qui apparaissent dans les calcules d'algébre de Hecke, sont relationés à certaines polynômes-R modifiés en ℕ[q], dont les coefficients ont simples interprétations combinatoires. Nous généralisons cette deuxième famille de polynômes, fournissant des interprétations combinatoires pour les expressions qui se posent dans une catégorie beaucoup plus vaste de calculs. En particulier, nous étendons des résultats de Brenti, Deodhar, et Dyer à des situations nouvelles, qui comprennent modules paraboliques pour l'algébre de Hecke, et l'anneau des polynômes quantiques.

scholarly journals Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Cristian Lenart
Keyword(s):  
Weyl Group ◽  
Type A ◽  
Macdonald Polynomials ◽  
Combinatorial Formula ◽  
Young Diagrams ◽  
Affine Weyl Group ◽  
Littlewood Polynomials ◽  
Combinatorial Formulas ◽  
International Audience ◽  
Type C

International audience A breakthrough in the theory of (type $A$) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of fillings of Young diagrams. Recently, Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of the corresponding affine Weyl group. In this paper, we show that a Haglund-Haiman-Loehr type formula follows naturally from the more general Ram-Yip formula, via compression. Then we extend this approach to the Hall-Littlewood polynomials of type $C$, which are specializations of the corresponding Macdonald polynomials at $q=0$. We note that no analog of the Haglund-Haiman-Loehr formula exists beyond type $A$, so our work is a first step towards finding such a formula.

scholarly journals Expected values of statistics on permutation tableaux

2007 ◽  
Vol DMTCS Proceedings vol. AH,... (Proceedings) ◽  
Author(s):  
Sylvie Corteel ◽  
Pawel Hitczenko
Keyword(s):  
Simple Proof ◽  
Statistical Physics ◽  
Unified Approach ◽  
Research Activity ◽  
Considerable Research ◽  
Totally Positive ◽  
International Audience ◽  
Expected Values

International audience Permutation tableaux are new objects that were introduced by Postnikov in the context of enumeration of the totally positive Grassmannian cells. They are known to be in bijection with permutations and recently, they have been connected to PASEP model used in statistical physics. Properties of permutation tableaux became a focus of a considerable research activity. In this paper we study properties of basic statistics defined on permutation tableaux. We present a simple and unified approach based on probabilistic techniques and use it to compute the expected values of basic statistics defined on permutation tableaux. We also provide a non―bijective and very simple proof that there are n! permutation tableaux of length n.

scholarly journals Sign variation, the Grassmannian, and total positivity

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Steven N. Karp
Keyword(s):  
Total Positivity ◽  
Oriented Matroids ◽  
Plücker Coordinates ◽  
The Past ◽  
Sign Patterns ◽  
International Audience

International audience The <i>totally nonnegative Grassmannian</i> is the set of $k$-dimensional subspaces $V$ of &#8477;<sup>$n$</sup> whose nonzero Plücker coordinates (i.e. $k &times; k$ minors of a $k &times; n$ matrix whose rows span $V$) all have the same sign. Total positivity has been much studied in the past two decades from an algebraic, combinatorial, and topological perspective, but first arose in the theory of oscillations in analysis. It was in the latter context that Gantmakher and Krein (1950) and Schoenberg and Whitney (1951) independently showed that a subspace $V$ is totally nonnegative iff every vector in $V$, when viewed as a sequence of $n$ numbers and ignoring any zeros, changes sign fewer than $k$ times. We generalize this result, showing that the vectors in $V$ change sign fewer than $l$ times iff certain sequences of the Plücker coordinates of some <i>generic perturbation</i> of $V$ change sign fewer than $l &minus; k &plus; 1$ times. We give an algorithm which constructs such a generic perturbation. Also, we determine the <i>positroid cell</i> of each totally nonnegative $V$ from sign patterns of vectors in $V$. These results generalize to oriented matroids. La <i>grassmannienne totalement non négative</i> est l’ensemble des sous-espaces $V$ de &#8477;<sup>$n$</sup> de dimension $k$ dont coordonnées plückeriennes non nulles (mineurs de l’ordre $k$ d’une matrice $k &times; n$ dont les lignes engendrent $V$) ont toutes le même signe. La positivité totale a beaucoup été étudiée durant les deux dernières décennies d’une perspective algébrique, combinatoire, et topologique, mais a pris naissance dans la théorie analytique des oscillations. C’est dans ce contexte que Gantmakher et Krein (1950) et Schoenberg et Whitney (1951) ont indépendamment démontré qu’un sous-espace $V$ est totalement non négatif ssi chaque vecteur dans $V$, lorsque considéré comme une séquence de $n$ nombres et dont on ignore les zéros, change de signe moins de $k$ fois. Nous généralisons ce résultat, démontrant que les vecteurs dans $V$ changent de signe moins de $l$ fois ssi certaines séquences des coordonnées plückeriennes d’une <i>perturbation générique</i> de $V$ changent de signe moins de $l &minus; k &plus; 1$ fois. Un algorithme construisant une telle perturbation générique est obtenu. De plus, nous déterminons la <i>cellule positroïde</i> de chaque $V$ totalement non négatif à partir des données de signe des vecteurs dans $V$. Ces résultats sont valides pour les matroïdes orientés.

scholarly journals Defining amplituhedra and Grassmann polytopes

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Steven N. Karp
Keyword(s):  
Dimensional Subspace ◽  
Equivalent Condition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Positivity Condition ◽  
Yang Mills ◽  
Cyclic Polytopes ◽  
Plücker Coordinates ◽  
International Audience ◽  
Necessary And Sufficient

International audience The totally nonnegative Grassmannian Gr≥0 k,n is the set of k-dimensional subspaces V of Rn whose nonzero Plucker coordinates all have the same sign. In their study of scattering amplitudes in N = 4 supersym- metric Yang-Mills theory, Arkani-Hamed and Trnka (2013) considered the image (called an amplituhedron) of Gr≥0 k,n under a linear map Z : Rn → Rr, where k ≤ r and the r × r minors of Z are all positive. One reason they required this positivity condition is to ensure that the map Gr≥0 k,n → Grk,r induced by Z is well defined, i.e. it takes everynelement of Gr≥0 k,n to a k-dimensional subspace of Rr. Lam (2015) gave a sufficient condition for the induced map Gr≥0 k,n → Grk,r to be well defined, in which case he called the image a Grassmann polytope. (In the case k = 1, Grassmann polytopes are just polytopes, and amplituhedra are cyclic polytopes.) We give a necessary and sufficient condition for the induced map Gr≥0 k,n → Grk,r to be well defined, in terms of sign variation. Using previous work we presented at FPSAC 2015, we obtain an equivalent condition in terms of the r × r minors of Z (assuming Z has rank r).

scholarly journals A combinatorial analysis of Severi degrees

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Fu Liu
Keyword(s):  
Power Series ◽  
Special Function ◽  
Joint Work ◽  
Toric Surfaces ◽  
Height Functions ◽  
Combinatorial Objects ◽  
Multivariate Function ◽  
Related Family ◽  
Combinatorial Formulas ◽  
International Audience

International audience Based on results by Brugallé and Mikhalkin, Fomin and Mikhalkin give formulas for computing classical Severi degrees Nd,δ using long-edge graphs. In 2012, Block, Colley and Kennedy considered the logarithmic versionof a special function associated to long-edge graphs which appeared in Fomin-Mikhalkin’s formula, and conjecturedit to be linear. They have since proved their conjecture. At the same time, motivated by their conjecture, we considera special multivariate function associated to long-edge graphs that generalizes their function. The main result of thispaper is that the multivariate function we define is always linear.The first application of our linearity result is that by applying it to classical Severi degrees, we recover quadraticity of Qd,δ and a bound δ for the threshold of polynomiality ofNd,δ.Next, in joint work with Osserman, we apply thelinearity result to a special family of toric surfaces and obtain universal polynomial results having connections to the Göttsche-Yau-Zaslow formula. As a result, we provide combinatorial formulas for the two unidentified power series B1(q) and B2(q) appearing in the Göttsche-Yau-Zaslow formula.The proof of our linearity result is completely combinatorial. We defineτ-graphs which generalize long-edge graphs,and a closely related family of combinatorial objects we call (τ,n)-words. By introducing height functions and aconcept of irreducibility, we describe ways to decompose certain families of (τ,n)-words into irreducible words,which leads to the desired results.

scholarly journals Order Filter Model for Minuscule Plücker Relations

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
David C Lax
Keyword(s):  
Flag Manifolds ◽  
Ordered Sets ◽  
Projective Varieties ◽  
Filter Model ◽  
Plücker Coordinates ◽  
International Audience ◽  
Complete Set ◽  
Plücker Relations ◽  
Coordinate Rings ◽  
Lie Algebra Representations

International audience The Plücker relations which define the Grassmann manifolds as projective varieties are well known. Grass-mann manifolds are examples of minuscule flag manifolds. We study the generalized Plücker relations for minuscule flag manifolds independent of Lie type. To do this we combinatorially model the Plücker coordinates based on Wild-berger’s construction of minuscule Lie algebra representations; it uses the colored partially ordered sets known asminuscule posets. We obtain, uniformly across Lie type, descriptions of the Plücker relations of “extreme weight”. We show that these are “supported” by “double-tailed diamond” sublattices of minuscule lattices. From this, we obtain a complete set of Plücker relations for the exceptional minuscule flag manifolds. These Plücker relations are straightening laws for their coordinate rings.

scholarly journals The twist for positroids

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Greg Muller ◽  
David E. Speyer
Keyword(s):  
Planar Graphs ◽  
Cluster Structure ◽  
The Other ◽  
Twist Map ◽  
Boundary Measurement ◽  
Bruhat Cells ◽  
International Audience ◽  
Initial Seed ◽  
Plabic Graphs

International audience There are two reasonable ways to put a cluster structure on a positroid variety. In one, the initial seed is a set of Plu ̈cker coordinates. In the other, the initial seed consists of certain monomials in the edge weights of a plabic graph. We will describe an automorphism of the positroid variety, the twist, which takes one to the other. For the big positroid cell, this was already done by Marsh and Scott; we generalize their results to all positroid varieties. This also provides an inversion of the boundary measurement map which is more general than Talaska's, in that it works for all reduced plabic graphs rather than just Le-diagrams. This is the analogue for positroid varieties of the twist map of Berenstein, Fomin and Zelevinsky for double Bruhat cells. Our construction involved the combinatorics of dimer configurations on bipartite planar graphs.

scholarly journals An equivariant rim hook rule for quantum cohomology of Grassmannians

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Elizabeth Beazley ◽  
Anna Bertiger ◽  
Kaisa Taipale
Keyword(s):  
Quantum Cohomology ◽  
Schubert Calculus ◽  
Flag Varieties ◽  
Effective Algorithm ◽  
Quantum Parameter ◽  
Complex Torus ◽  
Combinatorial Formulas ◽  
Il Y A ◽  
International Audience ◽  
Schubert Classes

International audience A driving question in (quantum) cohomology of flag varieties is to find non-recursive, positive combinatorial formulas for expressing the quantum product in a particularly nice basis, called the Schubert basis. Bertram, Ciocan-Fontanine and Fulton provide a way to compute quantum products of Schubert classes in the Grassmannian of $k$-planes in complex $n$-space by doing classical multiplication and then applying a combinatorial rimhook rule which yields the quantum parameter. In this paper, we provide a generalization of this rim hook rule to the setting in which there is also an action of the complex torus. Combining this result with Knutson and Tao's puzzle rule provides an effective algorithm for computing the equivariant quantum Littlewood-Richardson coefficients. Interestingly, this rule requires a specialization of torus weights that is tantalizingly similar to maps in affine Schubert calculus. Une question importante dans la cohomologie quantique des variétés de drapeaux est de trouver des formules positives non récursives pour exprimer le produit quantique dans une base particulièrement bonne, appelée la base de Schubert. Bertram, Ciocan-Fontanine et Fulton donnent une façon de calculer les produits quantiques de classes de Schubert dans la Grassmannienne de $k$-plans dans l’espace complexe de dimension $n$ en faisant la multiplication classique et appliquant une règle combinatoire “rimhook” qui donne le paramètre quantique. Dans cet article, nous donnons une généralisation de ce règle rimhook au contexte où il y a aussi une action du tore complexe. Combiné avec la règle “puzzle” de Knutson et Tao, cela donne une algorithme effective pour calculer les coefficients équivariants de Littlewood-Richard. Il est intéressant d'observer que cette règle demande une spécialisation des poids du tore qui est similaire d’une manière tentante aux applications dans le calcul de Schubert affiné.

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