scholarly journals The Prism tableau model for Schubert polynomials

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Anna Weigandt ◽  
Alexander Yong
Keyword(s):  
Young Tableau ◽  
Schubert Varieties ◽  
Symmetric Polynomials ◽  
Schubert Polynomials ◽  
International Audience

International audience The Schubert polynomials lift the Schur basis of symmetric polynomials into a basis for Z[x1; x2; : : :]. We suggest the prism tableau model for these polynomials. A novel aspect of this alternative to earlier results is that it directly invokes semistandard tableaux; it does so as part of a colored tableau amalgam. In the Grassmannian case, a prism tableau with colors ignored is a semistandard Young tableau. Our arguments are developed from the Gr¨obner geometry of matrix Schubert varieties.

scholarly journals Which Schubert varieties are local complete intersections?

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Henning Úlfarsson ◽  
Alexander Woo
Keyword(s):  
Nous Avons ◽  
Pattern Avoidance ◽  
Schubert Varieties ◽  
Complete Intersections ◽  
Minimal Set ◽  
Schubert Polynomials ◽  
Cutting Out ◽  
International Audience ◽  
New Formula ◽  
Local Complete Intersections

International audience We characterize by pattern avoidance the Schubert varieties for $\mathrm{GL}_n$ which are local complete intersections (lci). For those Schubert varieties which are local complete intersections, we give an explicit minimal set of equations cutting out their neighbourhoods at the identity. Although the statement of our characterization only requires ordinary pattern avoidance, showing that the Schubert varieties not satisfying our conditions are not lci appears to require working with more general notions of pattern avoidance. The Schubert varieties defined by inclusions, originally introduced by Gasharov and Reiner, turn out to be an important subclass, and we further develop some of their combinatorics. One application is a new formula for certain specializations of Schubert polynomials. Nous caractérisons par l’évitement des motifs les variétés de Schubert qui sont localement des intersections complètes. Pour les variétés de Schubert qui sont localement des intersections complètes, nous donnons des ensembles explicites des polynômes qui définissent leurs voisinages à l’identité. Bien que notre caractérisation n'utilise que les motifs ordinaire, nous avons besoin des notions plus générales des motifs dans notre démonstration. Les variétés de Schubert définies par des inclusions, introduites par Gasharov et Reiner, sont une sous-classe importante, et nous développons davantage leurs combinatoire. Une application est une nouvelle formule pour une spécialisation des polynômes de Schubert.

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 Kraśkiewicz-Pragacz modules and some positivity properties of Schubert polynomials

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Masaki Watanabe
Keyword(s):  
Schur Function ◽  
Schubert Polynomials ◽  
International Audience

International audience We use the modules introduced by Kraśkiewicz and Pragacz (1987, 2004) to show some positivity propertiesof Schubert polynomials. We give a new proof to the classical fact that the product of two Schubert polynomialsis Schubert-positive, and also show a new result that the plethystic composition of a Schur function with a Schubertpolynomial is Schubert-positive. The present submission is an extended abstract on these results and the full versionof this work will be published elsewhere. Nous employons les modules introduits par Kraśkiewicz et Pragacz (1987, 2004) et démontrons certainespropriétés de positivité des polynômes de Schubert: nous donnons une nouvelle preuve pour le fait classique quele produit de deux polynômes de Schubert est Schubert-positif; nous démontrons aussi un nouveau résultat que lacomposition plethystique d’une fonction de Schur avec un polynôme de Schubert est Schubert-positif. Cet article estun sommaire de ces résultats, et une version pleine de ce travail sera publée ailleurs.

scholarly journals A categorification of the chromatic symmetric polynomial

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Radmila Sazdanović ◽  
Martha Yip
Keyword(s):  
Symmetric Function ◽  
Symmetric Polynomial ◽  
Chromatic Polynomial ◽  
Khovanov Homology ◽  
Symmetric Polynomials ◽  
Nous Obtenons ◽  
Combinatorial Properties ◽  
International Audience ◽  
Chromatic Symmetric Function ◽  
Frobenius Series

International audience The Stanley chromatic polynomial of a graph $G$ is a symmetric function generalization of the chromatic polynomial, and has interesting combinatorial properties. We apply the ideas of Khovanov homology to construct a homology $H$<sub>*</sub>($G$) of graded $S_n$-modules, whose graded Frobenius series $Frob_G(q,t)$ reduces to the chromatic symmetric function at $q=t=1$. We also obtain analogues of several familiar properties of the chromatic symmetric polynomials in terms of homology. Le polynôme chromatique symétrique d’un graphe $G$ est une généralisation par une fonction symétrique du polynôme chromatique, et possède des propriétés combinatoires intéressantes. Nous appliquons les techniques de l’homologie de Khovanov pour construire une homologie $H$<sub>*</sub>($G$) de modules gradués $S_n$, dont la série bigraduée de Frobeniusse $Frob_G(q,t)$ réduit au polynôme chromatique symétrique à $q=t=1$. Nous obtenons également des analogies pour plusieurs propriétés connues des polynômes chromatiques en termes d’homologie.

scholarly journals Invariant and coinvariant spaces for the algebra of symmetric polynomials in non-commuting variables

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
François Bergeron ◽  
Aaron Lauve
Keyword(s):  
Tensor Product ◽  
Symmetric Group ◽  
Hopf Algebras ◽  
Symmetric Polynomials ◽  
Reflection Groups ◽  
Invariant Polynomials ◽  
Product Decomposition ◽  
Combinatorial Hopf Algebras ◽  
Produit Tensoriel ◽  
International Audience

International audience We analyze the structure of the algebra $\mathbb{K}\langle \mathbf{x}\rangle^{\mathfrak{S}_n}$ of symmetric polynomials in non-commuting variables in so far as it relates to $\mathbb{K}[\mathbf{x}]^{\mathfrak{S}_n}$, its commutative counterpart. Using the "place-action'' of the symmetric group, we are able to realize the latter as the invariant polynomials inside the former. We discover a tensor product decomposition of $\mathbb{K}\langle \mathbf{x}\rangle^{\mathfrak{S}_n}$ analogous to the classical theorems of Chevalley, Shephard-Todd on finite reflection groups. In the case $|\mathbf{x}|= \infty$, our techniques simplify to a form readily generalized to many other familiar pairs of combinatorial Hopf algebras. Nous analysons la structure de l'algèbre $\mathbb{K}\langle \mathbf{x}\rangle^{\mathfrak{S}_n}$ des polynômes symétriques en des variables non-commutatives pour obtenir des analogues des résultats classiques concernant la structure de l'anneau $\mathbb{K}[\mathbf{x}]^{\mathfrak{S}_n}$ des polynômes symétriques en des variables commutatives. Plus précisément, au moyen de "l'action par positions'', on réalise $\mathbb{K}[\mathbf{x}]^{\mathfrak{S}_n}$ comme sous-module de $\mathbb{K}\langle \mathbf{x}\rangle^{\mathfrak{S}_n}$. On découvre alors une nouvelle décomposition de $\mathbb{K}\langle \mathbf{x}\rangle^{\mathfrak{S}_n}$ comme produit tensoriel, obtenant ainsi un analogue des théorèmes classiques de Chevalley et Shephard-Todd. Dans le cas $|\mathbf{x}|= \infty$, nos techniques se simplifient en une forme aisément généralisables à beaucoup d'autres paires d'algèbres de Hopf familières.

scholarly journals Total positivity for the Lagrangian Grassmannian

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Rachel Karpman
Keyword(s):  
Symplectic Form ◽  
Total Positivity ◽  
Combinatorial Structure ◽  
Schubert Varieties ◽  
Flag Varieties ◽  
Young Diagrams ◽  
Lagrangian Grassmannian ◽  
Planar Network ◽  
International Audience ◽  
Isotropic Subspaces

International audience The positroid decomposition of the Grassmannian refines the well-known Schubert decomposition, and has a rich combinatorial structure. There are a number of interesting combinatorial posets which index positroid varieties,just as Young diagrams index Schubert varieties. In addition, Postnikov’s boundary measurement map gives a family of parametrizations for each positroid variety. The domain of each parametrization is the space of edge weights of a weighted planar network. The positroid stratification of the Grassmannian provides an elementary example of Lusztig’s theory of total non negativity for partial flag varieties, and has remarkable applications to particle physics.We generalize the combinatorics of positroid varieties to the Lagrangian Grassmannian, the moduli space of maximal isotropic subspaces with respect to a symplectic form

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 Double Schubert polynomials for the classical Lie groups

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Takeshi Ikeda ◽  
Leonardo Mihalcea ◽  
Hiroshi Naruse
Keyword(s):  
Weyl Group ◽  
Lie Groups ◽  
Equivariant Cohomology ◽  
Natural Extension ◽  
Type B ◽  
Infinite Rank ◽  
Schubert Polynomials ◽  
International Audience ◽  
Schubert Classes ◽  
Double Schubert Polynomials

International audience For each infinite series of the classical Lie groups of type $B$, $C$ or $D$, we introduce a family of polynomials parametrized by the elements of the corresponding Weyl group of infinite rank. These polynomials represent the Schubert classes in the equivariant cohomology of the corresponding flag variety. They satisfy a stability property, and are a natural extension of the (single) Schubert polynomials of Billey and Haiman, which represent non-equivariant Schubert classes. When indexed by maximal Grassmannian elements of the Weyl group, these polynomials are equal to the factorial analogues of Schur $Q$- or $P$-functions defined earlier by Ivanov. Pour chaque série infinie des groupe de Lie classiques de type $B$,$C$ ou $D$, nous présentons une famille de polynômes indexées par de éléments de groupe de Weyl correspondant de rang infini. Ces polynômes représentent des classes de Schubert dans la cohomologie équivariante des variétés de drapeaux. Ils ont une certain propriété de stabilité, et ils étendent naturellement des polynômes Schubert (simples) de Billey et Haiman, que représentent des classes de Schubert dans la cohomologie non-équivariante. Quand ils sont indexées par des éléments Grassmanniennes de groupes de Weyl, ces polynômes sont égaux à des analogues factorielles de fonctions $Q$ et $P$ de Schur, étudiées auparavant par Ivanov.

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 Interval positroid varieties and a deformation of the ring of symmetric functions

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Allen Knutson ◽  
Mathias Lederer
Keyword(s):  
Symmetric Functions ◽  
Schur Functions ◽  
Dimensional Subspace ◽  
Schubert Varieties ◽  
International Audience ◽  
Structure Constants ◽  
Schubert Classes ◽  
K Theory ◽  
Two Parameter ◽  
Parameter Deformation

International audience Define the <b>interval rank</b> $r_[i,j] : Gr_k(\mathbb C^n) →\mathbb{N}$ of a k-plane V as the dimension of the orthogonal projection $π _[i,j](V)$ of V to the $(j-i+1)$-dimensional subspace that uses the coordinates $i,i+1,\ldots,j$. By measuring all these ranks, we define the <b>interval rank stratification</b> of the Grassmannian $Gr_k(\mathbb C^n)$. It is finer than the Schubert and Richardson stratifications, and coarser than the positroid stratification studied by Lusztig, Postnikov, and others, so we call the closures of these strata <b>interval positroid varieties</b>. We connect Vakil's "geometric Littlewood-Richardson rule", in which he computed the homology classes of Richardson varieties (Schubert varieties intersected with opposite Schubert varieties), to Erd&odblac;s-Ko-Rado shifting, and show that all of Vakil's varieties are interval positroid varieties. We build on his work in three ways: (1) we extend it to arbitrary interval positroid varieties, (2) we use it to compute in equivariant K-theory, not just homology, and (3) we simplify Vakil's (2+1)-dimensional "checker games" to 2-dimensional diagrams we call "IP pipe dreams". The ring Symm of symmetric functions and its basis of Schur functions is well-known to be very closely related to the ring $\bigoplus_a,b H_*(Gr_a(\mathbb{C}^{(a+b)})$ and its basis of Schubert classes. We extend the latter ring to equivariant K-theory (with respect to a circle action on each $\mathbb{C}^{(a+b)}$, and compute the structure constants of this two-parameter deformation of Symm using the interval positroid technology above.

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