Combinatorial description of the cohomology of the affine flag variety

Affine Flag Variety ◽

International audience We construct the affine version of the Fomin-Kirillov algebra, called the affine FK algebra, to investigatethe combinatorics of affine Schubert calculus for typeA. We introduce Murnaghan-Nakayama elements and Dunklelements in the affine FK algebra. We show that they are commutative as Bruhat operators, and the commutativealgebra generated by these operators is isomorphic to the cohomology of the affine flag variety. As a byproduct, weobtain Murnaghan-Nakayama rules both for the affine Schubert polynomials and affine Stanley symmetric functions. This enable us to expressk-Schur functions in terms of power sum symmetric functions. We also provide the defi-nition of the affine Schubert polynomials, polynomial representatives of the Schubert basis in the cohomology of theaffine flag variety.

The Murnaghan―Nakayama rule for k-Schur functions

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2894 ◽

2011 ◽

Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽

Author(s):

Jason Bandlow ◽

Anne Schilling ◽

Mike Zabrocki

Keyword(s):

Explicit Formula ◽

Symmetric Function ◽

Symmetric Functions ◽

Schur Functions ◽

Schur Function ◽

Noncommutative Symmetric Functions ◽

Power Sum ◽

International Audience

International audience We prove a Murnaghan–Nakayama rule for k-Schur functions of Lapointe and Morse. That is, we give an explicit formula for the expansion of the product of a power sum symmetric function and a k-Schur function in terms of k-Schur functions. This is proved using the noncommutative k-Schur functions in terms of the nilCoxeter algebra introduced by Lam and the affine analogue of noncommutative symmetric functions of Fomin and Greene. Nous prouvons une règle de Murnaghan-Nakayama pour les fonctions de k-Schur de Lapointe et Morse, c'est-à-dire que nous donnons une formule explicite pour le développement du produit d'une fonction symétrique "somme de puissances'' et d'une fonction de k-Schur en termes de fonctions k-Schur. Ceci est prouvé en utilisant les fonctions non commutatives k-Schur en termes d'algèbre nilCoxeter introduite par Lam et l'analogue affine des fonctions symétriques non commutatives de Fomin et Greene.

Schur $P$-positivity and Involution Stanley Symmetric Functions

International Mathematics Research Notices ◽

10.1093/imrn/rnx274 ◽

2017 ◽

Vol 2019 (17) ◽

pp. 5389-5440 ◽

Cited By ~ 5

Author(s):

Zachary Hamaker ◽

Eric Marberg ◽

Brendan Pawlowski

Keyword(s):

Power Series ◽

Generating Functions ◽

Symmetric Function ◽

Symmetric Functions ◽

Schur Functions ◽

Pattern Avoidance ◽

Dominance Order ◽

Schubert Polynomials ◽

Skew Schur Functions

Abstract The involution Stanley symmetric functions$\hat{F}_y$ are the stable limits of the analogs of Schubert polynomials for the orbits of the orthogonal group in the flag variety. These symmetric functions are also generating functions for involution words and are indexed by the involutions in the symmetric group. By construction, each $\hat{F}_y$ is a sum of Stanley symmetric functions and therefore Schur positive. We prove the stronger fact that these power series are Schur $P$-positive. We give an algorithm to efficiently compute the decomposition of $\hat{F}_y$ into Schur $P$-summands and prove that this decomposition is triangular with respect to the dominance order on partitions. As an application, we derive pattern avoidance conditions which characterize the involution Stanley symmetric functions which are equal to Schur $P$-functions. We deduce as a corollary that the involution Stanley symmetric function of the reverse permutation is a Schur $P$-function indexed by a shifted staircase shape. These results lead to alternate proofs of theorems of Ardila–Serrano and DeWitt on skew Schur functions which are Schur $P$-functions. We also prove new Pfaffian formulas for certain related involution Schubert polynomials.

Pieri rule for the affine flag variety

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2517 ◽

2015 ◽

Vol DMTCS Proceedings, 27th... (Proceedings) ◽

Author(s):

Seung Jin Lee

Keyword(s):

Equivariant Cohomology ◽

Flag Variety ◽

Nous Montrons ◽

International Audience ◽

Affine Flag Variety ◽

International audience We prove the affine Pieri rule for the cohomology of the affine flag variety conjectured by Lam, Lapointe, Morse and Shimozono. We study the cap operator on the affine nilHecke ring that is motivated by Kostant and Kumar’s work on the equivariant cohomology of the affine flag variety. We show that the cap operators for Pieri elements are the same as Pieri operators defined by Berg, Saliola and Serrano. This establishes the affine Pieri rule. Nous prouvons la règle affine Pieri pour la cohomologie de la variété affine de drapeau conjecturé par Lam, Lapointe, Morse et Shimozono. Nous étudions l’opérateur de bouchon sur l’affine nilHecke anneau qui est motivé par le travail de Kostant et Kumar sur la cohomologie équivariante de la variété affine de drapeau. Nous montrons que les opérateurs de capitalisation pour les éléments Pieri sont les mêmes que les opérateurs Pieri définies par Berg, Saliola et Serrano. Ceci établit la règle affine Pieri.

Product of Stanley symmetric functions

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3064 ◽

2012 ◽

Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽

Author(s):

Nan Li

Keyword(s):

Nous Avons ◽

Symmetric Function ◽

Symmetric Functions ◽

Nous Obtenons ◽

Schubert Polynomial ◽

Schubert Polynomials ◽

International Audience ◽

Natural Expansion ◽

Natural Way

International audience We study the problem of expanding the product of two Stanley symmetric functions $F_w·F_u$ into Stanley symmetric functions in some natural way. Our approach is to consider a Stanley symmetric function as a stabilized Schubert polynomial $F_w=\lim _n→∞\mathfrak{S}_{1^n×w}$, and study the behavior of the expansion of $\mathfrak{S} _{1^n×w}·\mathfrak{S} _{1^n×u}$ into Schubert polynomials, as $n$ increases. We prove that this expansion stabilizes and thus we get a natural expansion for the product of two Stanley symmetric functions. In the case when one permutation is Grassmannian, we have a better understanding of this stability. Nous étudions le problème de développement du produit de deux fonctions symétriques de Stanley $F_w·F_u$ en fonctions symétriques de Stanley de façon naturelle. Notre méthode consiste à considérer une fonction symétrique de Stanley comme un polynôme du Schubert stabilisè $F_w=\lim _n→∞\mathfrak{S}_{1^n×w}$, et à étudier le comportement de développement de $\mathfrak{S} _{1^n×w}·\mathfrak{S} _{1^n×u}$ en polynômes de Schubert lorsque $n$ augmente. Nous prouvons que cette développement se stabilise et donc nous obtenons une développement naturelle pour le produit de deux fonctions symétriques de Stanley. Dans le cas où l'une des permutations est Grassmannienne, nous avons une meilleure compréhension de cette stabilité.

Combinatorial Expansions in $K$-Theoretic Bases

The Electronic Journal of Combinatorics ◽

10.37236/2320 ◽

2012 ◽

Vol 19 (4) ◽

Cited By ~ 2

Author(s):

Jason Bandlow ◽

Jennifer Morse

Keyword(s):

Generating Functions ◽

Symmetric Functions ◽

Schur Functions ◽

Geometric Information ◽

Combinatorial Description ◽

Littlewood Polynomials ◽

K Theory ◽

Combinatorial Representation

We study the class $\mathcal C$ of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape. Included in this class are the Hall-Littlewood polynomials, $k$-Schur functions, and Stanley symmetric functions; functions whose Schur coefficients encode combinatorial, representation theoretic and geometric information. While Schur functions represent the cohomology of the Grassmannian variety of $GL_n$, Grothendieck functions $\{G_\lambda\}$ represent the $K$-theory of the same space. In this paper, we give a combinatorial description of the coefficients when any element of $\mathcal C$ is expanded in the $G$-basis or the basis dual to $\{G_\lambda\}$.

Dual combinatorics of zonal polynomials

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2913 ◽

2011 ◽

Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽

Author(s):

Valentin Féray ◽

Piotr Sniady

Keyword(s):

Irreducible Character ◽

Symmetric Functions ◽

Symmetric Groups ◽

Combinatorial Formula ◽

Zonal Polynomials ◽

Power Sums ◽

Power Sum ◽

Recent Developments ◽

International Audience ◽

Special Case

International audience In this paper we establish a new combinatorial formula for zonal polynomials in terms of power-sums. The proof relies on the sign-reversing involution principle. We deduce from it formulas for zonal characters, which are defined as suitably normalized coefficients in the expansion of zonal polynomials in terms of power-sum symmetric functions. These formulas are analogs of recent developments on irreducible character values of symmetric groups. The existence of such formulas could have been predicted from the work of M. Lassalle who formulated two positivity conjectures for Jack characters, which we prove in the special case of zonal polynomials. Dans cet article, nous établissons une nouvelle formule combinatoire pour les polynômes zonaux en fonction des fonctions puissance. La preuve utilise le principe de l'involution changeant les signes. Nous en déduisons des formules pour les caractères zonaux, qui sont définis comme les coefficients des polynômes zonaux écrits sur la base des fonctions puissance, normalisés de manière appropriée. Ces formules sont des analogues de développements récents sur les caractères du groupe symétrique. L'existence de telles formules aurait pu être prédite à partir des travaux de M. Lassalle, qui a proposé deux conjectures de positivité sur les caractères de Jack, que nous prouvons dans le cas particulier des polynômes zonaux.

Affine flag varieties

Berkeley Lectures on p-adic Geometry ◽

10.23943/princeton/9780691202082.003.0021 ◽

2020 ◽

pp. 191-206

Author(s):

Peter Scholze ◽

Jared Weinstein

Keyword(s):

Group Scheme ◽

Flag Variety ◽

Flag Varieties ◽

Connected Component ◽

Witt Vector ◽

Classical Definition ◽

Definition Of ◽

Affine Flag Variety ◽

This chapter reviews affine flag varieties. It generalizes some of the previous results to the case where G over Zp is a parahoric group scheme. In fact, slightly more generally, it allows the case that the special fiber is not connected, with connected component of the identity G? being a parahoric group scheme. This case comes up naturally in the classical definition of Rapoport-Zink spaces. The chapter first discusses the Witt vector affine flag variety over Fp. This is an increasing union of perfections of quasiprojective varieties along closed immersions. In the case that G° is parahoric, one gets ind-properness.

Extending torsors on the punctured Spec(A inf)

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2021-0077 ◽

2022 ◽

Vol 0 (0) ◽

Author(s):

Johannes Anschütz

Keyword(s):

Flag Variety ◽

Full Generality ◽

Witt Vector ◽

Affine Grassmannian ◽

Group Schemes ◽

Affine Flag Variety ◽

Abstract We prove that torsors under parahoric group schemes on the punctured spectrum of Fontaine’s ring A inf {A_{\mathrm{inf}}} , extend to the whole spectrum. Using descent we can extend a similar result for the ring 𝔖 {\mathfrak{S}} of Kisin and Pappas to full generality. Moreover, we treat similarly the case of equal characteristic. As applications we extend results of Ivanov on exactness of the loop functor and present the construction of a canonical specialization map from the B dR + {B^{+}_{\mathrm{dR}}} -affine Grassmannian to the Witt vector affine flag variety.