scholarly journals Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Pavel Galashin ◽  
Darij Grinberg ◽  
Gaku Liu
Keyword(s):  
Symmetric Functions ◽  
Schur Functions ◽  
The Other ◽  
Countable Family ◽  
Plane Partitions ◽  
International Audience ◽  
K Theory ◽  
Reverse Plane ◽  
Grothendieck Polynomials

International audience The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the K-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters such that the generalization still defines symmetric functions. We outline two self-contained proofs of this fact, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries 1 and 2.

scholarly journals Refined Dual Stable Grothendieck Polynomials and Generalized Bender-Knuth Involutions

10.37236/5737 ◽  
2016 ◽  
Vol 23 (3) ◽  
Author(s):  
Pavel Galashin ◽  
Darij Grinberg ◽  
Gaku Liu
Keyword(s):  
Symmetric Functions ◽  
Schur Functions ◽  
The Other ◽  
Countable Family ◽  
Plane Partitions ◽  
K Theory ◽  
Reverse Plane ◽  
Grothendieck Polynomials

The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the $K$-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters, and we prove that this generalization still defines symmetric functions. For this fact, we give two self-contained proofs, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries $1$ and $2$.

scholarly journals The expansion of Hall-Littlewood functions in the dual Grothendieck polynomial basis

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Jason Bandlow ◽  
Jennifer Morse
Keyword(s):  
Symmetric Functions ◽  
Polynomial Basis ◽  
Dual Basis ◽  
Nous Obtenons ◽  
Plane Partitions ◽  
Generalize Charge ◽  
Littlewood Polynomials ◽  
International Audience ◽  
Reverse Plane ◽  
Grothendieck Polynomials

International audience A combinatorial expansion of the Hall-Littlewood functions into the Schur basis of symmetric functions was first given by Lascoux and Schützenberger, with their discovery of the charge statistic. A combinatorial expansion of stable Grassmannian Grothendieck polynomials into monomials was first given by Buch, using set-valued tableaux. The dual basis of the stable Grothendieck polynomials was given a combinatorial expansion into monomials by Lam and Pylyavskyy using reverse plane partitions. We generalize charge to set-valued tableaux and use all of these combinatorial ideas to give a nice expansion of Hall-Littlewood polynomials into the dual Grothendieck basis. \par En associant une charge à un tableau, une formule combinatoire donnant le développement des polynômes de Hall-Littlewood en termes des fonctions de Schur a été obtenue par Lascoux et Schützenberger. Une formule combinatoire donnant le développement des polynômes de Grothendieck Grassmanniens stables en termes des fonctions monomiales a quant à elle été obtenue par Buch à l'aide de tableaux à valeurs sur des ensembles. Finalement, une formule faisant intervenir des partitions planaires inverses a été obtenue par Lam et Pylyavskyy pour donner le développement de la base duale aux polynômes de Grothendieck stables en termes de monômes. Nous généralisons le concept de charge aux tableaux à valeurs sur des ensembles et, en nous servant de toutes ces notions combinatoires, nous obtenons une formule élégante donnant le développement des polynômes de Hall-Littlewood en termes de la base de Grothendieck duale.

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 Combinatorial Expansions in $K$-Theoretic Bases

10.37236/2320 ◽  
2012 ◽  
Vol 19 (4) ◽  
Author(s):  
Jason Bandlow ◽  
Jennifer Morse
Keyword(s):  
Generating Functions ◽  
Symmetric Functions ◽  
Schur Functions ◽  
Geometric Information ◽  
Combinatorial Description ◽  
Littlewood Polynomials ◽  
Stanley Symmetric Functions ◽  
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 $\{G_\lambda\}$.

scholarly journals Peterson Isomorphism in K-theory and Relativistic Toda Lattice

2018 ◽  
Vol 2020 (19) ◽  
pp. 6421-6462 ◽  
Author(s):  
Takeshi Ikeda ◽  
Shinsuke Iwao ◽  
Toshiaki Maeno
Keyword(s):  
Symmetric Functions ◽  
Toda Lattice ◽  
Flag Variety ◽  
The Other ◽  
Initial Condition ◽  
Birational Morphism ◽  
Affine Grassmannian ◽  
Other Hand ◽  
K Theory ◽  
Relativistic Toda Lattice

Abstract The K-homology ring of the affine Grassmannian of $SL_{n}(\mathbb{C})$ was studied by Lam, Schilling, and Shimozono. It is realized as a certain concrete Hopf subring of the ring of symmetric functions. On the other hand, for the quantum K-theory of the flag variety $F\,\! l_{n}$, Kirillov and Maeno provided a conjectural presentation based on the results obtained by Givental and Lee. We construct an explicit birational morphism between the spectrums of these two rings. Our method relies on Ruijsenaars’s relativistic Toda lattice with unipotent initial condition. From this result, we obtain a K-theory analogue of the so-called Peterson isomorphism for (co)homology. We provide a conjecture on the detailed relationship between the Schubert bases, and, in particular, we determine the image of Lenart–Maeno’s quantum Grothendieck polynomial associated with a Grassmannian permutation.

scholarly journals Enumeration of Cylindric Plane Partitions

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Robin Langer
Keyword(s):  
Natural Generalization ◽  
Path Model ◽  
Plane Partition ◽  
Combinatorial Interpretation ◽  
Plane Partitions ◽  
Generating Series ◽  
International Audience ◽  
The Individual ◽  
The Right ◽  
Reverse Plane

International audience Cylindric plane partitions may be thought of as a natural generalization of reverse plane partitions. A generating series for the enumeration of cylindric plane partitions was recently given by Borodin. As in the reverse plane partition case, the right hand side of this identity admits a simple factorization form in terms of the "hook lengths'' of the individual boxes in the underlying shape. The first result of this paper is a new bijective proof of Borodin's identity which makes use of Fomin's growth diagram framework for generalized RSK correspondences. The second result of this paper is a $(q,t)$-analog of Borodin's identity which extends previous work by Okada in the reverse plane partition case. The third result of this paper is an explicit combinatorial interpretation of the Macdonald weight occurring in the $(q,t)$-analog in terms of the non-intersecting lattice path model for cylindric plane partitions. Les partitions planes cylindriques sont une généralisation naturelle des partitions planes renversées. Une série génératrice pour énumération des partitions planes cylindriques a été donnée récemment par Borodin. Comme dans le cas des partitions planes renversées, la partie droite de cette identité peut être factoriser en terme de "longueur d’équerres'' des carrés dans la forme sous-jacente. Le premier résultat de cet article est une nouvelle preuve bijective de l'identité de Borodin qui utilise le cadre de "diagramme de croissance'' de Fomin pour la correspondance de RSK généralisée. Le deuxième résultat de cette article est une $(q,t)$-déformation d'identité de Borodin qui généralise un résultat de Okada dans le cas des partitions planes renversées. Le troisième résultat de cet article est une formule combinatoire explicite pour le poids de Macdonald qui utilise le modèle des chemins non-intersectant pour les partitions planes cylindriques.

scholarly journals The Murnaghan―Nakayama rule for k-Schur functions

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.

scholarly journals Tableaux and plane partitions of truncated shapes (extended abstract)

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Greta Panova
Keyword(s):  
Generating Function ◽  
Schur Functions ◽  
Young Tableaux ◽  
Young Diagrams ◽  
Plane Partitions ◽  
Nous Trouvons ◽  
International Audience

International audience We consider a new kind of straight and shifted plane partitions/Young tableaux — ones whose diagrams are no longer of partition shape, but rather Young diagrams with boxes erased from their upper right ends. We find formulas for the number of standard tableaux in certain cases, namely a shifted staircase without the box in its upper right corner, i.e. truncated by a box, a rectangle truncated by a staircase and a rectangle truncated by a square minus a box. The proofs involve finding the generating function of the corresponding plane partitions using interpretations and formulas for sums of restricted Schur functions and their specializations. The number of standard tableaux is then found as a certain limit of this function. Nous considérons un nouveau type de partitions planes, ou de tableaux de Young, droits ou décalés, obtenus en privant leurs diagrammes de certaines cellules en haut à droite, et dans certains cas nous trouvons des formules d'énumération pour les tableaux standard. Les preuves impliquent le calcul de la fonction génératrice pour les partitions planes correspondantes, en utilisant des interprétations et des formules pour les sommes de fonctions de Schur restreintes et leurs spécialisations. Le nombre de tableaux standard est alors obtenu comme une certaine limite de cette fonction.

scholarly journals A Combinatorial Approach to Multiplicity-Free Richardson Subvarieties of the Grassmannian

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Michelle Snider
Keyword(s):  
Combinatorial Proof ◽  
Combinatorial Approach ◽  
Möbius Inversion ◽  
International Audience ◽  
K Theory ◽  
Multiplicity Free ◽  
Grothendieck Polynomials

International audience We consider Buch's rule for K-theory of the Grassmannian, in the Schur multiplicity-free cases classified by Stembridge. Using a result of Knutson, one sees that Buch's coefficients are related to Möbius inversion. We give a direct combinatorial proof of this by considering the product expansion for Grassmannian Grothendieck polynomials. We end with an extension to the multiplicity-free cases of Thomas and Yong. On examine la règle de Buch pour la K-théorie de la variété grassmannienne dans les cas sans multiplicité de Schur, qui ont étés classifiés par Stembridge. En utilisant un résultat de Knutson, on démontre que les coefficients de Buch sont liés à l'inversion de Möbius. On en fait une preuve directe et combinatoire qui passe par le developpement de produits de polynômes de Grothendieck. Pour conclure, on donne une application de cette théorie aux cas sans multiplicité de Thomas et Yong.

scholarly journals An Involution Principle-Free Bijective Proof of Stanley's Hook-Content Formula

1998 ◽  
Vol Vol. 3 no. 1 ◽  
Author(s):  
Christian Krattenthaler
Keyword(s):  
Generating Function ◽  
Plane Partitions ◽  
Bijective Proof ◽  
International Audience ◽  
Reverse Plane

International audience A bijective proof for Stanley's hook-content formula for the generating function for column-strict reverse plane partitions of a given shape is given that does not involve the involution principle of Garsia and Milne. It is based on the Hillman-Grassl algorithm and Schützenberger's \emphjeu de taquin.

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