scholarly journals Cohomology classes of rank varieties and a counterexample to a conjecture of Liu

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Brendan Pawlowski
Keyword(s):  
Cohomology Class ◽  
Symmetric Functions ◽  
Specht Module ◽  
Nous Montrons ◽  
Group A ◽  
International Audience ◽  
Schubert Classes ◽  
Stanley Symmetric Functions ◽  
Rank Variety ◽  
Discrete Grid

International audience To each finite subset of a discrete grid $\mathbb{N}×\mathbb{N}$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. We give a counterexample to this conjecture.However, we show that for the diagram variety of a permutation diagram, Liu's conjectured cohomology class $\sigma$ is at least an upper bound on the actual class $\tau$, in the sense that $\sigma - \tau$ is a nonnegative linear combination of Schubert classes. To do this, we consider a degeneration of Coskun's rank varieties which contains the appropriate diagram variety as a component. Rank varieties are instances of Knutson-Lam-Speyer's positroid varieties, whose cohomology classes are represented by affine Stanley symmetric functions. We show that the cohomology class of a rank variety is in fact represented by an ordinary Stanley symmetric function. A chaque sous-ensemble fini de $\mathbb{N}×\mathbb{N}$ (un diagramme), on peut associer une sous-variété d’une grassmannienne complexe et une représentation d’un groupe symétrique (un module de Specht). Liu a conjecturé que la classe de cohomologie de la variété d’un diagramme est représentée par la caractéristique de Frobenius du module de Specht correspondant. Nous donnons un contre-exemple à cette conjecture.Cependant, nous montrons que dans le cas de la variété du diagramme de permutation, la classe de cohomologie conjecturée par Liu est au moins un majorant de la classe juste $\tau$ , c’est-à-dire que $\sigma - \tau$ est une combinaison linéaire non-négative des classes de Schubert. Pour ce faire, nous considérons une dégénérescence des variétés de rang de Coskun qui contient la variété appropriée d’un diagramme comme une composante irréductible. Les variétés de rang sont des exemples de variétés de positroïde, dont les classes de cohomologie sont représentées par des fonctions symétriques de Stanley affines. En effet, nous montrons que la classe de cohomologie d’une variété de rang est représentée par une fonction symétrique de Stanley ordinaire.

scholarly journals Cohomology Classes of Interval Positroid Varieties and a Conjecture of Liu

10.37236/6960 ◽  
2018 ◽  
Vol 25 (4) ◽  
Author(s):  
Brendan Pawlowski
Keyword(s):  
Symmetric Group ◽  
Finite Subset ◽  
Upper Bound ◽  
Cohomology Class ◽  
Symmetric Functions ◽  
A Priori ◽  
Specht Module ◽  
Group A ◽  
Schubert Classes ◽  
Stanley Symmetric Functions

To each finite subset of $\mathbb{Z}^2$ (a diagram), one can associate a subvariety of a complex Grassmannian (a diagram variety), and a representation of a symmetric group (a Specht module). Liu has conjectured that the cohomology class of a diagram variety is represented by the Frobenius characteristic of the corresponding Specht module. We give a counterexample to this conjecture.However, we show that for the diagram variety of a permutation diagram, Liu's conjectured cohomology class $\sigma$ is at least an upper bound on the actual class $\tau$, in the sense that $\sigma - \tau$ is a nonnegative linear combination of Schubert classes. To do this, we exhibit the appropriate diagram variety as a component in a degeneration of one of Knutson's interval positroid varieties (up to Grassmann duality). A priori, the cohomology classes of these interval positroid varieties are represented by affine Stanley symmetric functions. We give a different formula for these classes as ordinary Stanley symmetric functions, one with the advantage of being Schur-positive and compatible with inclusions between Grassmannians.

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 Invariance properties for coefficients of symmetric functions

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Emmanuel Briand ◽  
Rosa Orellana ◽  
Mercedes Rosas
Keyword(s):  
Symmetric Functions ◽  
Invariance Properties ◽  
Nous Montrons ◽  
International Audience ◽  
Kronecker Coefficients

International audience We show that several of the main structural constants for symmetric functions (Littlewood-Richardsoncoefficients, Kronecker coefficients, plethysm coefficients, and the Kostka–Foulkes polynomials) share invarianceproperties related to the operations of taking complements with respect to rectangles and adding rectangles. Nous montrons que plusieurs des principales constantes de structure de la théorie des fonctions symétriques(les coefficients de Littlewood–Richardson, les coefficients de Kronecker, les coefficients du pléthysme, et les polynômesde Kostka–Foulkes) ont en commun des symétries décrites par des opérations de complémentation dans des rectangleset d’ajout de rectangles pour les partitions qui les étiquettent.

scholarly journals Supercharacters, symmetric functions in noncommuting variables (extended abstract)

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Marcelo Aguiar ◽  
Carlos André ◽  
Carolina Benedetti ◽  
Nantel Bergeron ◽  
Zhi Chen ◽  
...  
Keyword(s):  
Hopf Algebra ◽  
Fourier Analysis ◽  
Finite Field ◽  
Hopf Algebras ◽  
Symmetric Functions ◽  
Combinatorial Hopf Algebras ◽  
Nous Montrons ◽  
International Audience

International audience We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras. Nous montrons que deux structures en apparence bien différentes peuvent être identifiées: les super-caractères, qui sont un outil commode pour faire de l'analyse de Fourier sur le groupe des matrices unipotentes triangulaires supérieures à coefficients dans un corps fini, et l'anneau des fonctions symétriques en variables non-commutatives. Ces deux structures sont des algèbres de Hopf isomorphes. Cette identification permet de traduire dans une structure les dévelopements conçus pour l'autre, et suggère de nombreux exemples dans le domaine nouveau des algèbres de Hopf combinatoires.

scholarly journals The topology of restricted partition posets

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Richard Ehrenborg ◽  
JiYoon Jung
Keyword(s):  
Symmetric Group ◽  
Homotopy Type ◽  
Integer Partitions ◽  
Order Complex ◽  
Specht Module ◽  
Set Partitions ◽  
Nous Montrons ◽  
Restricted Partition ◽  
International Audience ◽  
Border Strip

International audience For each composition $\vec{c}$ we show that the order complex of the poset of pointed set partitions $Π ^• _{\vec{c}}$ is a wedge of $β\vec{c}$ spheres of the same dimensions, where $β\vec{c}$ is the number of permutations with descent composition ^$\vec{c}$. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module $S^B$ where $B$ is a border strip associated to the composition $\vec{c}$. We also study the filter of pointed set partitions generated by a knapsack integer partitions and show the analogous results on homotopy type and action on the top homology. Pour chaque composition $\vec{c}$ nous montrons que le complexe simplicial des chaînes de l'ensemble ordonné $Π ^• _{\vec{c}}$ des partitions pointées d'un ensemble est un bouquet de $β\vec{c}$ sphères de même dimension, où $β\vec{c}$ est le nombre de permutations ayant la composition de descentes $\vec{c}$. De plus, l'action du groupe symétrique sur le groupe d'homologie de degré maximum est isomorphe au module de Specht $S^B$ où $B$ est la bande frontalière associée à la composition $\vec{c}$. Nous étudions aussi le filtre des partitions pointées d'un ensemble, engendré par des partitions d'entiers de type "sac à dos'' et nous démontrons des résultats analogues pour le type d'homotopie et pour l'action sur le groupe d'homologie de degré maximum.

scholarly journals Product of Stanley symmetric functions

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 ◽  
Stanley Symmetric Functions ◽  
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.

scholarly journals Staircase Macdonald polynomials and the $q$-Discriminant

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Adrien Boussicault ◽  
Jean-Gabriel Luque
Keyword(s):  
Symmetric Functions ◽  
Macdonald Polynomials ◽  
Macdonald Polynomial ◽  
Monomial Basis ◽  
Nous Montrons ◽  
International Audience ◽  
Nous Examinons

International audience We prove that a $q$-deformation $\mathfrak{D}_k(\mathbb{X};q)$ of the powers of the discriminant is equal, up to a normalization, to a specialization of a Macdonald polynomial indexed by a staircase partition. We investigate the expansion of $\mathfrak{D}_k(\mathbb{X};q)$ on different bases of symmetric functions. In particular, we show that its expansion on the monomial basis can be explicitly described in terms of standard tableaux and we generalize a result of King-Toumazet-Wybourne about the expansion of the $q$-discriminant on the Schur basis. Nous montrons qu’une $q$-déformation $\mathfrak{D}_k(\mathbb{X};q)$ des puissances du discriminant est égale, à un coefficient de normalisation près, à un polynôme de Macdonald indexé par une partition escalier pour une certaine spécialisation des paramètres. Nous examinons les développements de $\mathfrak{D}_k(\mathbb{X};q)$ dans différentes bases de fonctions symétriques. En particulier, nous montrons que son écriture dans la base des fonctions monomiales peut être explicitement décrite en terme de tableaux standard et nous généralisons un résultat de King-Toumazet-Wybourne sur le développement du $q$-discriminant dans la base de Schur.

scholarly journals Combinatorial description of the cohomology of the affine flag variety

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Seung Jin Lee
Keyword(s):  
Symmetric Functions ◽  
Schur Functions ◽  
Flag Variety ◽  
Schubert Calculus ◽  
Power Sum ◽  
Schubert Polynomials ◽  
International Audience ◽  
Stanley Symmetric Functions ◽  
Affine Flag Variety ◽  
Affine Flag

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.

scholarly journals Balanced labellings of affine permutations

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Hwanchul Yoo ◽  
Taedong Yun
Keyword(s):  
Symmetric Functions ◽  
Simple Algorithm ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Reduced Decompositions ◽  
Finite Case ◽  
International Audience ◽  
Stanley Symmetric Functions ◽  
Affine Permutations ◽  
Necessary And Sufficient

International audience We study the $\textit{diagrams}$ of affine permutations and their $\textit{balanced}$ labellings. As in the finite case, which was investigated by Fomin, Greene, Reiner, and Shimozono, the balanced labellings give a natural encoding of reduced decompositions of affine permutations. In fact, we show that the sum of weight monomials of the $\textit{column strict}$ balanced labellings is the affine Stanley symmetric function defined by Lam and we give a simple algorithm to recover reduced words from balanced labellings. Applying this theory, we give a necessary and sufficient condition for a diagram to be an affine permutation diagram. Finally, we conjecture that if two affine permutations are $\textit{diagram equivalent}$ then their affine Stanley symmetric functions coincide.

Stanley Symmetric Functions and Peterson Algebras

2014 ◽  
pp. 133-168 ◽  
Author(s):  
Thomas Lam ◽  
Luc Lapointe ◽  
Jennifer Morse ◽  
Anne Schilling ◽  
Mark Shimozono ◽  
...  
Keyword(s):  
Symmetric Functions ◽  
Stanley Symmetric Functions
Sign in / Sign up

Export Citation Format

Share Document