scholarly journals On Schubert calculus in elliptic cohomology

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Cristian Lenart ◽  
Kirill Zainoulline
Keyword(s):  
Formal Group ◽  
Schubert Calculus ◽  
Elliptic Cohomology ◽  
Reduced Word ◽  
Combinatorial Result ◽  
International Audience ◽  
Schubert Classes ◽  
K Theory ◽  
Nous Utilisons ◽  
Group Law

International audience An important combinatorial result in equivariant cohomology and $K$-theory Schubert calculus is represented by the formulas of Billey and Graham-Willems for the localization of Schubert classes at torus fixed points. These formulas work uniformly in all Lie types, and are based on the concept of a root polynomial. We define formal root polynomials associated with an arbitrary formal group law (and thus a generalized cohomology theory). We usethese polynomials to simplify the approach of Billey and Graham-Willems, as well as to generalize it to connective $K$-theory and elliptic cohomology. Another result is concerned with defining a Schubert basis in elliptic cohomology (i.e., classes independent of a reduced word), using the Kazhdan-Lusztig basis of the corresponding Hecke algebra. Un résultat combinatoire important dans le calcul de Schubert pour la cohomologie et la $K$-théorie équivariante est représenté par les formules de Billey et Graham-Willems pour la localisation des classes de Schubert aux points fixes du tore. Ces formules sont uniformes pour tous les types de Lie, et sont basés sur le concept d’un polynôme de racines. Nous définissons les polynômes formels de racines associées à une loi arbitraire de groupe formel (et donc à une théorie de cohomologie généralisée). Nous utilisons ces polynômes pour simplifier les preuves de Billey et Graham-Willems, et aussi pour généraliser leurs résultats à la $K$-théorie connective et la cohomologie elliptique. Un autre résultat concerne la définition d’une base de Schubert dans cohomologie elliptique (c’est à dire, des classes indépendantes d’un mot réduit), en utilisant la base de Kazhdan-Lusztig de l’algèbre de Hecke correspondant.

scholarly journals Formal Group Laws and Chromatic Symmetric Functions of Hypergraphs

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Jair Taylor
Keyword(s):  
Power Series ◽  
Symmetric Functions ◽  
Formal Group ◽  
Combinatorial Interpretation ◽  
Recursive Structure ◽  
Combinatorial Objects ◽  
Formal Group Law ◽  
International Audience ◽  
Formal Group Laws ◽  
Group Law

International audience If $f(x)$ is an invertible power series we may form the symmetric function $f(f^{-1}(x_1)+f^{-1}(x_2)+...)$ which is called a formal group law. We give a number of examples of power series $f(x)$ that are ordinary generating functions for combinatorial objects with a recursive structure, each of which is associated with a certain hypergraph. In each case, we show that the corresponding formal group law is the sum of the chromatic symmetric functions of these hypergraphs by finding a combinatorial interpretation for $f^{-1}(x)$. We conjecture that the chromatic symmetric functions arising in this way are Schur-positive. Si $f(x)$ est une série entière inversible, nous pouvons former la fonction symétrique $f(f^{-1}(x_1)+f^{-1}(x_2)+...)$ que nous appelons une loi de groupe formel. Nous donnons plusieurs exemples de séries entières $f(x)$ qui sont séries génératrices ordinaires pour des objets combinatoires avec une structure récursive, chacune desquelles est associée à un certain hypergraphe. Dans chaque cas, nous donnons une interprétation combinatoire à $f^{-1}(x)$, ce qui nous permet de montrer que la loi de groupe formel correspondante est la somme des fonctions symétriques chromatiques de ces hypergraphes. Nous conjecturons que les fonctions symétriques chromatiques apparaissant de cette manière sont Schur-positives.

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.

scholarly journals Genomic Tableaux and Combinatorial $K$-Theory

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Oliver Pechenik ◽  
Alexander Yong
Keyword(s):  
Schubert Calculus ◽  
Nous Obtenons ◽  
International Audience ◽  
Structure Constants ◽  
K Theory ◽  
Orthogonal Grassmannians

International audience We introduce genomic tableaux, with applications to Schubert calculus. We report a combinatorial rule for structure coefficients in the torus-equivariant $K$-theory of Grassmannians for the basis of Schubert structure sheaves. This rule is positive in the sense of [Anderson-Griffeth-Miller ’11]. We thereby deduce an earlier conjecture of [Thomas-Yong ’13] for the coefficients. Moreover, our rule specializes to give a new Schubert calculus rule in the (non-equivariant) $K$-theory of Grassmannians. From this perspective, we also obtain a new rule for $K$-theoretic Schubert structure constants of maximal orthogonal Grassmannians, and give conjectural bounds on such constants for Lagrangian Grassmannians. Nous introduisons la notion de tableau génomique, pour l’appliquer au calcul de Schubert. Nous énonçons une règle combinatoire pour les coefficients de structure de la $K$-théorie tore-équivariante des grassmanniennes, dans la base définie par les classes des faisceaux structuraux des variétés de Schubert. Cette règle est positive au sens de [Anderson-Griffeth-Miller ’11]. Nous en déduisons une conjecture de [Thomas-Yong ’13]. De plus, notre règle se spécialise en une règle nouvelle pour le calcul de Schubert dans la $K$-théorie (non équivariante) des grassmanniennes. Nous obtenons également une nouvelle règle pour les coefficients de structure de la $K$-théorie des grassmanniennes orthogonales maximales dans la base de Schubert, et nous conjecturons certaines bornes pour ces coefficients dans le cas des grassmanniennes lagrangiennes.

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.

scholarly journals DHD-puzzles

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Sabine Beil
Keyword(s):  
Schubert Calculus ◽  
The Other ◽  
Fixed Boundary ◽  
Puzzle Piece ◽  
Other Hand ◽  
International Audience ◽  
K Theory ◽  
The One

International audience In this work triangular puzzles that are composed of unit triangles with labelled edges are considered. To be more precise, the labelled unit triangles that we allow are on the one hand the puzzle pieces that compute Schubert calculus and on the other hand the flipped K-theory puzzle piece. The motivation for studying such puzzles comes from the fact that they correspond to a class of oriented triangular fully packed loop configurations. The main result that is presented is an expression for the number of these puzzles with a fixed boundary in terms of Littlewood- Richardson coefficients.

Stable cohomotopy and cobordism of abelian groups

1981 ◽  
Vol 90 (2) ◽  
pp. 273-278 ◽  
Author(s):  
C. T. Stretch
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Formal Group ◽  
Characteristic Class ◽  
Burnside Ring ◽  
Classifying Space ◽  
Formal Group Law ◽  
Stable Cohomotopy ◽  
Group Law

The object of this paper is to prove that for a finite abelian group G the natural map is injective, where Â(G) is the completion of the Burnside ring of G and σ0(BG) is the stable cohomotopy of the classifying space BG of G. The map â is detected by means of an M U* exponential characteristic class for permutation representations constructed in (11). The result is a generalization of a theorem of Laitinen (4) which treats elementary abelian groups using ordinary cohomology. One interesting feature of the present proof is that it makes explicit use of the universality of the formal group law of M U*. It also involves a computation of M U*(BG) in terms of the formal group law. This may be of independent interest. Since writing the paper the author has discovered that M U*(BG) has previously been calculated by Land-weber(5).

scholarly journals Formal groups and Z -entropies

2016 ◽  
Vol 472 (2195) ◽  
pp. 20160143 ◽  
Author(s):  
Piergiulio Tempesta
Keyword(s):  
Algebraic Topology ◽  
Formal Group ◽  
Point Of View ◽  
Trace Form ◽  
Formal Groups ◽  
Form Class ◽  
New Family ◽  
Mathematical Properties ◽  
Composition Process ◽  
Group Law

We shall prove that the celebrated Rényi entropy is the first example of a new family of infinitely many multi-parametric entropies. We shall call them the Z-entropies . Each of them, under suitable hypotheses, generalizes the celebrated entropies of Boltzmann and Rényi. A crucial aspect is that every Z -entropy is composable (Tempesta 2016 Ann. Phys. 365 , 180–197. ( doi:10.1016/j.aop.2015.08.013 )). This property means that the entropy of a system which is composed of two or more independent systems depends, in all the associated probability space, on the choice of the two systems only. Further properties are also required to describe the composition process in terms of a group law. The composability axiom, introduced as a generalization of the fourth Shannon–Khinchin axiom (postulating additivity), is a highly non-trivial requirement. Indeed, in the trace-form class, the Boltzmann entropy and Tsallis entropy are the only known composable cases. However, in the non-trace form class, the Z -entropies arise as new entropic functions possessing the mathematical properties necessary for information-theoretical applications, in both classical and quantum contexts. From a mathematical point of view, composability is intimately related to formal group theory of algebraic topology. The underlying group-theoretical structure determines crucially the statistical properties of the corresponding entropies.

scholarly journals The elliptic Weyl character formula

2014 ◽  
Vol 150 (7) ◽  
pp. 1196-1234 ◽  
Author(s):  
Nora Ganter
Keyword(s):  
Line Bundle ◽  
Lie Groups ◽  
Theoretical Framework ◽  
Flag Variety ◽  
Schubert Calculus ◽  
Character Formula ◽  
Elliptic Cohomology ◽  
Image Position ◽  
Weyl Character Formula

AbstractWe calculate equivariant elliptic cohomology of the partial flag variety$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}G/H$, where$H\subseteq G$are compact connected Lie groups of equal rank. We identify the${\rm RO}(G)$-graded coefficients${\mathcal{E}} ll_G^*$as powers of Looijenga’s line bundle and prove that transfer along the map$$\begin{equation*} \pi \,{:}\,G/H\longrightarrow {\rm pt} \end{equation*}$$is calculated by the Weyl–Kac character formula. Treating ordinary cohomology,$K$-theory and elliptic cohomology in parallel, this paper organizes the theoretical framework for the elliptic Schubert calculus of [N. Ganter and A. Ram,Elliptic Schubert calculus, in preparation].

scholarly journals Multiplicity-Free Schubert Calculus

2010 ◽  
Vol 53 (1) ◽  
pp. 171-186 ◽  
Author(s):  
Hugh Thomas ◽  
Alexander Yong
Keyword(s):  
Algebraic Geometry ◽  
Schubert Calculus ◽  
Chow Ring ◽  
Free Products ◽  
Linear Basis ◽  
Combinatorial Classification ◽  
Schubert Classes ◽  
Multiplicity Free

AbstractMultiplicity-free algebraic geometry is the study of subvarieties Y ⊆ X with the “smallest invariants” as witnessed by a multiplicity-free Chow ring decomposition of [Y] ∈ A*(X) into a predetermined linear basis.This paper concerns the case of Richardson subvarieties of the Grassmannian in terms of the Schubert basis. We give a nonrecursive combinatorial classification of multiplicity-free Richardson varieties, i.e., we classify multiplicity-free products of Schubert classes. This answers a question of W. Fulton.

Sign in / Sign up

Export Citation Format

Share Document