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 Operations on partially ordered sets and rational identities of type A

2013 ◽  
Vol Vol. 15 no. 2 (Combinatorics) ◽  
Author(s):  
Adrien Boussicault
Keyword(s):  
Partially Ordered Sets ◽  
Rational Functions ◽  
Hasse Diagram ◽  
Linear Extensions ◽  
Ordered Sets ◽  
Closed Expression ◽  
Schubert Polynomials ◽  
Special Cases ◽  
The Family ◽  
International Audience

Combinatorics International audience We consider the family of rational functions ψw= ∏( xwi - xwi+1 )-1 indexed by words with no repetition. We study the combinatorics of the sums ΨP of the functions ψw when w describes the linear extensions of a given poset P. In particular, we point out the connexions between some transformations on posets and elementary operations on the fraction ΨP. We prove that the denominator of ΨP has a closed expression in terms of the Hasse diagram of P, and we compute its numerator in some special cases. We show that the computation of ΨP can be reduced to the case of bipartite posets. Finally, we compute the numerators associated to some special bipartite graphs as Schubert polynomials.

scholarly journals Adjoint functors and tree duality

2009 ◽  
Vol Vol. 11 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Jan Foniok ◽  
Claude Tardif
Keyword(s):  
Adjoint Functors ◽  
Relational Structures ◽  
International Audience ◽  
Complete Set ◽  
Near Unanimity

Graphs and Algorithms International audience A family T of digraphs is a complete set of obstructions for a digraph H if for an arbitrary digraph G the existence of a homomorphism from G to H is equivalent to the non-existence of a homomorphism from any member of T to G. A digraph H is said to have tree duality if there exists a complete set of obstructions T consisting of orientations of trees. We show that if H has tree duality, then its arc graph delta H also has tree duality, and we derive a family of tree obstructions for delta H from the obstructions for H. Furthermore we generalise our result to right adjoint functors on categories of relational structures. We show that these functors always preserve tree duality, as well as polynomial CSPs and the existence of near-unanimity functions.

scholarly journals Bruhat order, rationally smooth Schubert varieties, and hyperplane arrangements

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Suho Oh ◽  
Hwanchul Yoo
Keyword(s):  
Hyperplane Arrangement ◽  
Schubert Variety ◽  
Bruhat Order ◽  
Hyperplane Arrangements ◽  
Schubert Varieties ◽  
Flag Manifolds ◽  
Poincaré Polynomial ◽  
Generalized Flag Manifolds ◽  
Nous Montrons ◽  
International Audience

International audience We link Schubert varieties in the generalized flag manifolds with hyperplane arrangements. For an element of a Weyl group, we construct a certain graphical hyperplane arrangement. We show that the generating function for regions of this arrangement coincides with the Poincaré polynomial of the corresponding Schubert variety if and only if the Schubert variety is rationally smooth. Nous relions des variétés de Schubert dans le variété flag généralisée avec des arrangements des hyperplans. Pour un élément dún groupe de Weyl, nous construisons un certain arrangement graphique des hyperplans. Nous montrons que la fonction génératrice pour les régions de cet arrangement coincide avec le polynome de Poincaré de la variété de Schubert correspondante si et seulement si la variété de Schubert est rationnellement lisse.

scholarly journals A Murgnahan-Nakayama rule for Schubert polynomials

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Andrew Morrison
Keyword(s):  
Quantum Cohomology ◽  
Chern Character ◽  
Intersection Theory ◽  
Flag Manifolds ◽  
Nous Obtenons ◽  
Schubert Polynomial ◽  
Power Sum ◽  
Schubert Polynomials ◽  
Small Quantum ◽  
International Audience

International audience We expose a rule for multiplying a general Schubert polynomial with a power sum polynomial in $k$ variables. A signed sum over cyclic permutations replaces the signed sum over rim hooks in the classical Murgnahan-Nakayama rule. In the intersection theory of flag manifolds this computes all intersections of Schubert cycles with tautological classes coming from the Chern character. We also discuss extensions of this rule to small quantum cohomology. Nous écrivons une formule pour multiplier les polynômes de Schubert avec les sommes de Newton. Une somme signée de permutations cycliques remplace la somme signée de rubans dans la formule classique de Murgnahan-Nakayama. Nous obtenons donc des relations dans l’anneau de Chow de la variété de drapeaux. Nous discutons également des extensions de cette formule en cohomologie quantique.

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 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.

Remarks on Plücker embedding and totally antisymmetric gauge fields

2020 ◽  
Vol 35 (22) ◽  
pp. 2050184
Author(s):  
J. A. Nieto ◽  
P. A. Nieto-Marín ◽  
E. A. León ◽  
E. García-Manzanárez
Keyword(s):  
Electromagnetic Field ◽  
Field Strength ◽  
Degrees Of Freedom ◽  
Gauge Fields ◽  
Plücker Coordinates ◽  
Electromagnetic Field Strength ◽  
String Structure ◽  
Plücker Relations ◽  
Plücker Embedding ◽  
The Way

We make a number of comments about the way the Plücker embedding, which can be derived via the Grassmann-Plücker relations, can be associated to totally antisymmetric gauge fields. As a first step we discuss the case of the electromagnetic field strength, showing that the Plücker map implies both the true degrees of freedom of the electromagnetic field and the 1-brane (string) structure. The procedure is generalized in order to prove that the true degrees of freedom of a totally antisymmetric field and the p-brane structure are, in part, consequence of the Plücker coordinates.

scholarly journals Dynamic Threshold Strategy for Universal Best Choice Problem

2010 ◽  
Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽  
Author(s):  
Jakub Kozik
Keyword(s):  
Partially Ordered Sets ◽  
Dynamic Threshold ◽  
Ordered Sets ◽  
Threshold Strategy ◽  
Choice Problem ◽  
Best Choice Problem ◽  
Partial Analysis ◽  
Probability Of Success ◽  
New Strategy ◽  
International Audience

International audience We propose a new strategy for universal best choice problem for partially ordered sets. We present its partial analysis which is sufficient to prove that the probability of success with this strategy is asymptotically strictly greater than 1/4, which is the value of the best universal strategy known so far.

scholarly journals On Plücker coordinates of a perfectly oriented planar network

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Kelli Talaska
Keyword(s):  
Positive Integer ◽  
Generating Functions ◽  
Weight Functions ◽  
Combinatorial Formula ◽  
Rational Map ◽  
Positive Weight ◽  
Integer Coefficients ◽  
Plücker Coordinates ◽  
Planar Network ◽  
International Audience

International audience Let $G$ be a perfectly oriented planar graph. Postnikov's boundary measurement construction provides a rational map from the set of positive weight functions on the edges of $G$ onto the appropriate totally nonnegative Grassmann cell. We establish an explicit combinatorial formula for Postnikov's map by expressing each Plücker coordinate of the image as a ratio of two polynomials in the edge weights, with positive integer coefficients. These polynomials are weight generating functions for certain subsets of edges in $G$.

Sign in / Sign up

Export Citation Format

Share Document