scholarly journals Combinatorial Formulas for Macdonald and Hall-Littlewood Polynomials

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Cristian Lenart
Keyword(s):  
Weyl Group ◽  
Type A ◽  
Macdonald Polynomials ◽  
Combinatorial Formula ◽  
Young Diagrams ◽  
Affine Weyl Group ◽  
Littlewood Polynomials ◽  
Combinatorial Formulas ◽  
International Audience ◽  
Type C

International audience A breakthrough in the theory of (type $A$) Macdonald polynomials is due to Haglund, Haiman and Loehr, who exhibited a combinatorial formula for these polynomials in terms of fillings of Young diagrams. Recently, Ram and Yip gave a formula for the Macdonald polynomials of arbitrary type in terms of the corresponding affine Weyl group. In this paper, we show that a Haglund-Haiman-Loehr type formula follows naturally from the more general Ram-Yip formula, via compression. Then we extend this approach to the Hall-Littlewood polynomials of type $C$, which are specializations of the corresponding Macdonald polynomials at $q=0$. We note that no analog of the Haglund-Haiman-Loehr formula exists beyond type $A$, so our work is a first step towards finding such a formula.

scholarly journals A $t$-generalization for Schubert Representatives of the Affine Grassmannian

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Avinash J. Dalal ◽  
Jennifer Morse
Keyword(s):  
Weyl Group ◽  
Transition Matrix ◽  
Symmetric Functions ◽  
Type A ◽  
Affine Grassmannian ◽  
Combinatorial Objects ◽  
Affine Weyl Group ◽  
Littlewood Polynomials ◽  
International Audience ◽  
Extra Parameter

International audience We introduce two families of symmetric functions with an extra parameter $t$ that specialize to Schubert representatives for cohomology and homology of the affine Grassmannian when $t=1$. The families are defined by a statistic on combinatorial objects associated to the type-$A$ affine Weyl group and their transition matrix with Hall-Littlewood polynomials is $t$-positive. We conjecture that one family is the set of $k$-atoms. Nous présentons deux familles de fonctions symétriques dépendant d'un paramètre $t$ et dont les spécialisations à $t=1$ correspondent aux classes de Schubert dans la cohomologie et l'homologie des variétés Grassmanniennes affines. Les familles sont définies par des statistiques sur certains objets combinatoires associés au groupe de Weyl affine de type $A$ et leurs matrices de transition dans la base des polynômes de Hall-Littlewood sont $t$-positives. Nous conjecturons qu'une de ces familles correspond aux $k$-atomes.

scholarly journals The ABC's of affine Grassmannians and Hall-Littlewood polynomials

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Avinash J. Dalal ◽  
Jennifer Morse
Keyword(s):  
Schur Functions ◽  
Type A ◽  
Combinatorial Formula ◽  
Transition Matrices ◽  
Nous Obtenons ◽  
Affine Grassmannians ◽  
Littlewood Polynomials ◽  
International Audience ◽  
Schubert Classes ◽  
Affine Type

International audience We give a new description of the Pieri rule for $k$-Schur functions using the Bruhat order on the affine type-$A$ Weyl group. In doing so, we prove a new combinatorial formula for representatives of the Schubert classes for the cohomology of affine Grassmannians. We show how new combinatorics involved in our formulas gives the Kostka-Foulkes polynomials and discuss how this can be applied to study the transition matrices between Hall-Littlewood and $k$-Schur functions. Nous présentons une nouvelle description, issue de l'ordre de Bruhat du groupe de Weyl affine de type $A$, de la règle de Pieri pour les fonctions $k$-Schur. Ce faisant, nous obtenons une nouvelle formule combinatoire pour les représentants des classes de Schubert de la cohomologie des Grassmannienne affines. Nous décrivons aussi comment notre approche permet d'obtenir les polynômes de Kostka-Foulkes et comment elle peut être appliquée à l’étude des matrices de transition entre les polynômes de Hall-Littlewood et les fonctions $k$-Schur.

scholarly journals Flag Gromov-Witten invariants via crystals

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Jennifer Morse ◽  
Anne Schilling
Keyword(s):  
Weyl Group ◽  
Type A ◽  
Schubert Calculus ◽  
Schur Function ◽  
Flag Varieties ◽  
Highest Weight ◽  
Affine Weyl Group ◽  
Nous Montrons ◽  
International Audience ◽  
Complete Flag

International audience We apply ideas from crystal theory to affine Schubert calculus and flag Gromov-Witten invariants. By defining operators on certain decompositions of elements in the type-$A$ affine Weyl group, we produce a crystal reflecting the internal structure of Specht modules associated to permutation diagrams. We show how this crystal framework can be applied to study the product of a Schur function with a $k$-Schur function. Consequently, we prove that a subclass of 3-point Gromov-Witten invariants of complete flag varieties for $\mathbb{C}^n$ enumerate the highest weight elements under these operators. Nous appliquons des idées provenant de la théorie des bases cristallines au calcul de Schubert affine et aux invariants de drapeaux de Gromov–Witten. Nous définissons des opérateurs sur certaines décompositions d’éléments de groupes de Weyl affines en type $A$ afin de construire une base cristalline encodant la structure interne des modules de Specht associés aux diagrammes de permutations. Nous montrons comment la structure de cristal permet d’étudier le produit d’une fonction de Schur avec une $k$-fonction de Schur. En conséquence, nous prouvons que la sous-classe des invariants de 3-points de Gromov–Witten d’une variété complète de drapeaux complets pour $\mathbb{C}^n$ énumère les éléments de poids maximaux pour ces opérateurs.

scholarly journals Rational smoothness and affine Schubert varieties of type A

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Sara Billey ◽  
Andrew Crites
Keyword(s):  
Weyl Group ◽  
Schubert Variety ◽  
Type A ◽  
Algebraic Combinatorics ◽  
Schubert Varieties ◽  
Topological Properties ◽  
Affine Weyl Group ◽  
International Audience ◽  
Affine Permutations

International audience The study of Schubert varieties in G/B has led to numerous advances in algebraic combinatorics and algebraic geometry. These varieties are indexed by elements of the corresponding Weyl group, an affine Weyl group, or one of their parabolic quotients. Often times, the goal is to determine which of the algebraic and topological properties of the Schubert variety can be described in terms of the combinatorics of its corresponding Weyl group element. A celebrated example of this occurs when G/B is of type A, due to Lakshmibai and Sandhya. They showed that the smooth Schubert varieties are precisely those indexed by permutations that avoid the patterns 3412 and 4231. Our main result is a characterization of the rationally smooth Schubert varieties corresponding to affine permutations in terms of the patterns 4231 and 3412 and the twisted spiral permutations. L'étude des variétés de Schubert dans G/B a mené à plusieurs avancées en combinatoire algébrique. Ces variétés sont indexées soit par l'élément du groupe de Weyl correspondant, soit par un groupe de Weyl affine, soit par un de leurs quotients paraboliques. Souvent, le but est de déterminer quelles propriétés algébriques et topologiques des variétés de Schubert peuvent être décrites en termes des propriétés combinatoires des éléments du groupe de Weyl correspondant. Un exemple bien connu, dû à Lakshmibai et Sandhya, concerne le cas où G/B est de type A. Ils ont montré que les variétés de Schubert lisses sont exactement celles qui sont indexées par les permutations qui évitent les motifs 3412 et 4231. Notre résultat principal est une caractérisation des variétés de Schubert lisses et rationnelles qui correspondent à des permutations affines pour les motifs 4231 et 3412 et les permutations spirales tordues.

scholarly journals A combinatorial approach to Macdonald q, t-symmetry via the Carlitz bijection

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Maria Monks Gillespie
Keyword(s):  
Young Diagram ◽  
Macdonald Polynomials ◽  
Point Of View ◽  
Combinatorial Proof ◽  
Symmetry Relation ◽  
Combinatorial Approach ◽  
Combinatorial Formula ◽  
Monomial Basis ◽  
Littlewood Polynomials ◽  
International Audience

International audience We investigate the combinatorics of the symmetry relation H μ(x; q, t) = H μ∗ (x; t, q) on the transformed Macdonald polynomials, from the point of view of the combinatorial formula of Haglund, Haiman, and Loehr in terms of the inv and maj statistics on Young diagram fillings. By generalizing the Carlitz bijection on permutations, we provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials (q = 0) for the coefficients of the square-free monomials in the variables x. Our work in this case relates the Macdonald inv and maj statistics to the monomial basis of the modules Rμ studied by Garsia and Procesi. We also provide a new proof for the full Macdonald relation in the case when μ is a hook shape.

scholarly journals A Combinatorial Approach to the $q,t$-Symmetry Relation in Macdonald Polynomials

10.37236/5350 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Maria Monks Gillespie
Keyword(s):  
Macdonald Polynomials ◽  
Combinatorial Proof ◽  
Symmetry Relation ◽  
Combinatorial Approach ◽  
Combinatorial Formula ◽  
Recursive Structure ◽  
Littlewood Polynomials

Using the combinatorial formula for the transformed Macdonald polynomials of Haglund, Haiman, and Loehr, we investigate the combinatorics of the symmetry relation $\widetilde{H}_\mu(\mathbf{x};q,t)=\widetilde{H}_{\mu^\ast}(\mathbf{x};t,q)$. We provide a purely combinatorial proof of the relation in the case of Hall-Littlewood polynomials ($q=0$) when $\mu$ is a partition with at most three rows, and for the coefficients of the square-free monomials in $\mathbf{x}$ for all shapes $\mu$. We also provide a proof for the full relation in the case when $\mu$ is a hook shape, and for all shapes at the specialization $t=1$. Our work in the Hall-Littlewood case reveals a new recursive structure for the cocharge statistic on words.

scholarly journals New Combinatorial Formulas for Cluster Monomials of Type $A$ Quivers

10.37236/6464 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Kyungyong Lee ◽  
Li Li ◽  
Ba Nguyen
Keyword(s):  
Cluster Algebra ◽  
Theta Functions ◽  
Type A ◽  
Perfect Matchings ◽  
Cluster Algebras ◽  
Combinatorial Formula ◽  
Natural Basis ◽  
Combinatorial Formulas ◽  
New Formula

Lots of research focuses on the combinatorics behind various bases of cluster algebras. This paper studies the natural basis of a type $A$ cluster algebra, which consists of all cluster monomials. We introduce a new kind of combinatorial formula for the cluster monomials in terms of the so-called globally compatible collections. We give bijective proofs of these formulas by comparing with the well-known combinatorial models of the $T$-paths and of the perfect matchings in a snake diagram. For cluster variables of a type $A$ cluster algebra, we give a bijection that relates our new formula with the theta functions constructed by Gross, Hacking, Keel and Kontsevich.

scholarly journals Young Diagrams and Intersection Numbers for Toric Manifolds associated with Weyl Chambers

10.37236/4307 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Hiraku Abe
Keyword(s):  
Weyl Group ◽  
Root System ◽  
Ring Structure ◽  
Classical Type ◽  
Toric Manifold ◽  
Combinatorial Formula ◽  
Young Diagrams ◽  
Intersection Numbers ◽  
Toric Manifolds

We study intersection numbers of invariant divisors in the toric manifold associated with the fan determined by the collection of Weyl chambers for each root system of classical type and of exceptional type $G_2$. We give a combinatorial formula for intersection numbers of certain subvarieties which are naturally indexed by elements of the Weyl group. These numbers describe the ring structure of the cohomology of the toric manifold.

scholarly journals Affine descents and the Steinberg torus

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Kevin Dilks ◽  
T. Kyle Petersen ◽  
John R. Stembridge
Keyword(s):  
Weyl Group ◽  
Generating Functions ◽  
Cell Complex ◽  
Premier Lieu ◽  
Coxeter Complex ◽  
Affine Weyl Group ◽  
Nous Montrons ◽  
International Audience ◽  
Translation Subgroup

International audience Let $W \ltimes L$ be an irreducible affine Weyl group with Coxeter complex $\Sigma$, where $W$ denotes the associated finite Weyl group and $L$ the translation subgroup. The Steinberg torus is the Boolean cell complex obtained by taking the quotient of $\Sigma$ by the lattice $L$. We show that the ordinary and flag $h$-polynomials of the Steinberg torus (with the empty face deleted) are generating functions over $W$ for a descent-like statistic first studied by Cellini. We also show that the ordinary $h$-polynomial has a nonnegative $\gamma$-vector, and hence, symmetric and unimodal coefficients. In the classical cases, we also provide expansions, identities, and generating functions for the $h$-polynomials of Steinberg tori. Nous considérons un groupe de Weyl affine irréductible $W \ltimes L$ avec complexe de Coxeter $\Sigma$, où $W$ désigne le groupe de Weyl fini associé et $L$ le sous-groupe des translations. Le tore de Steinberg est le complexe cellulaire Booléen obtenu comme le quotient de $\Sigma$ par $L$. Nous montrons que les $h$-polynômes, ordinaires et de drapeaux, du tore de Steinberg (sans la face vide) sont des fonctions génératrices sur $W$ pour une statistique de type descente, étudiée en premier lieu par Cellini. Nous montrons également qu'un $h$-polynôme ordinaire possède un $\gamma$-vecteur positif, et par conséquent, a des coefficients symétriques et unimodaux. Dans les cas classiques, nous donnons également des développements, des identités et des fonctions génératrices pour les $h$-polynômes des tores de Steinberg.

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