Super quasi-symmetric functions via Young diagrams

Jean-Christophe Aval; Valentin Féray; Jean-Christophe Novelli; J.-Y. Thibon

doi:10.46298/dmtcs.2390

Super quasi-symmetric functions via Young diagrams

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2390 ◽

2014 ◽

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

Author(s):

Jean-Christophe Aval ◽

Valentin Féray ◽

Jean-Christophe Novelli ◽

J.-Y. Thibon

Keyword(s):

Symmetric Functions ◽

Explicit Construction ◽

Multiplication Table ◽

Young Diagrams ◽

Generating Series ◽

International Audience

International audience We consider the multivariate generating series $F_P$ of $P-$partitions in infinitely many variables $x_1, x_2, \ldots$ . For some family of ranked posets $P$, it is natural to consider an analog $N_P$ with two infinite alphabets. When we collapse these two alphabets, we trivially recover $F_P$. Our main result is the converse, that is, the explicit construction of a map sending back $F_P$ onto $N_P$. We also give a noncommutative analog of the latter. An application is the construction of a basis of $\mathbf{WQSym}$ with a non-negative multiplication table, which lifts a basis of $\textit{QSym}$ introduced by K. Luoto.

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Cumulants of Jack symmetric functions and b-conjecture (extended abstract)

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.6322 ◽

2020 ◽

Vol DMTCS Proceedings, 28th... ◽

Author(s):

Maciej Dolega ◽

Valentin Féray

Keyword(s):

Nonnegative Integer ◽

Symmetric Functions ◽

Rational Functions ◽

Independent Interest ◽

Jack Polynomials ◽

Factorization Property ◽

Integer Coefficients ◽

Continuous Deformation ◽

Generating Series ◽

International Audience

International audience Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; t, 1 + β) that might be interpreted as a continuous deformation of the rooted hypermap generating series. They made the following conjecture: coefficients of ψ(x, y, z; t, 1+β) are polynomials in β with nonnegative integer coefficients. We prove partially this conjecture, nowadays called b-conjecture, by showing that coefficients of ψ(x, y, z; t, 1 + β) are polynomials in β with rational coefficients. Until now, it was only known that they are rational functions of β. A key step of the proof is a strong factorization property of Jack polynomials when α → 0 that may be of independent interest.

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Explicit generating series for connection coefficients

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3025 ◽

2012 ◽

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

Author(s):

Ekaterina A. Vassilieva

Keyword(s):

Symmetric Functions ◽

Double Coset ◽

Type A ◽

Explicit Computation ◽

Zonal Polynomials ◽

Connection Coefficients ◽

Generating Series ◽

International Audience ◽

Commutative Subalgebras ◽

Orientable Surfaces

International audience This paper is devoted to the explicit computation of generating series for the connection coefficients of two commutative subalgebras of the group algebra of the symmetric group, the class algebra and the double coset algebra. As shown by Hanlon, Stanley and Stembridge (1992), these series gives the spectral distribution of some random matrices that are of interest to statisticians. Morales and Vassilieva (2009, 2011) found explicit formulas for these generating series in terms of monomial symmetric functions by introducing a bijection between partitioned hypermaps on (locally) orientable surfaces and some decorated forests and trees. Thanks to purely algebraic means, we recover the formula for the class algebra and provide a new simpler formula for the double coset algebra. As a salient ingredient, we compute an explicit formulation for zonal polynomials indexed by partitions of type $[a,b,1^{n-a-b}]$. Cet article est dédié au calcul explicite des séries génératrices des constantes de structure de deux sous-algèbres commutatives de l'algèbre de groupe du groupe symétrique, l'algèbre de classes et l'algèbre de double classe latérale. Tel que montrè par Hanlon, Stanley and Stembridge (1992), ces séries déterminent la distribution spectrale de certaines matrices aléatoires importantes en statistique. Morales et Vassilieva (2009, 2011) ont trouvè des formules explicites pour ces séries génératrices en termes des monômes symétriques en introduisant une bijection entre les hypercartes partitionnées sur des surfaces (localement) orientables et certains arbres et forêts décorées. Grâce à des moyens purement algébriques, nous retrouvons la formule pour l'algèbre de classe et déterminons une nouvelle formule plus simple pour l'algèbre de double classe latérale. En tant que point saillant de notre démonstration nous calculons une formulation explicite pour les polynômes zonaux indexés par des partitions de type $[a,b,1^{n-a-b}]$.

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On Kerov polynomials for Jack characters (extended abstract)

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2322 ◽

2013 ◽

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

Author(s):

Valentin Féray ◽

Maciej Dołęga

Keyword(s):

Symmetric Functions ◽

Partial Result ◽

Plancherel Measure ◽

Jack Polynomials ◽

Young Diagrams ◽

Power Sum ◽

International Audience ◽

Random Young Diagrams

International audience We consider a deformation of Kerov character polynomials, linked to Jack symmetric functions. It has been introduced recently by M. Lassalle, who formulated several conjectures on these objects, suggesting some underlying combinatorics. We give a partial result in this direction, showing that some quantities are polynomials in the Jack parameter $\alpha$ with prescribed degree. Our result has several interesting consequences in various directions. Firstly, we give a new proof of the fact that the coefficients of Jack polynomials expanded in the monomial or power-sum basis depend polynomially in $\alpha$. Secondly, we describe asymptotically the shape of random Young diagrams under some deformation of Plancherel measure.

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Bijective evaluation of the connection coefficients of the double coset algebra

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2944 ◽

2011 ◽

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

Author(s):

Alejandro H. Morales ◽

Ekaterina A. Vassilieva

Keyword(s):

Random Matrix Theory ◽

Random Matrix ◽

Spectral Distribution ◽

Symmetric Functions ◽

Matrix Theory ◽

Double Coset ◽

Hyperoctahedral Group ◽

Connection Coefficients ◽

Generating Series ◽

International Audience

International audience This paper is devoted to the evaluation of the generating series of the connection coefficients of the double cosets of the hyperoctahedral group. Hanlon, Stanley, Stembridge (1992) showed that this series, indexed by a partition $ν$, gives the spectral distribution of some random matrices that are of interest in random matrix theory. We provide an explicit evaluation of this series when $ν =(n)$ in terms of monomial symmetric functions. Our development relies on an interpretation of the connection coefficients in terms of locally orientable hypermaps and a new bijective construction between partitioned locally orientable hypermaps and some permuted forests. Cet article est dédié à l'évaluation des séries génératrices des coefficients de connexion des classes doubles (cosets) du groupe hyperoctaédral. Hanlon, Stanley, Stembridge (1992) ont montré que ces séries indexées par une partition $ν$ donnent la distribution spectrale de certaines matrices aléatoires jouant un rôle important dans la théorie des matrices aléatoires. Nous fournissons une évaluation explicite de ces séries dans le cas $ν =(n)$ en termes de monômes symétriques. Notre développement est fondé sur une interprétation des coefficients de connexion en termes d'hypercartes localement orientables et sur une nouvelle bijection entre les hypercartes localement orientables partitionnées et certaines forêts permutées.

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A symmetric Bloch–Okounkov theorem

Research in the Mathematical Sciences ◽

10.1007/s40687-021-00253-8 ◽

2021 ◽

Vol 8 (2) ◽

Author(s):

Jan-Willem M. van Ittersum

Keyword(s):

Symmetric Functions ◽

Graded Algebra ◽

Symmetric Polynomials ◽

Generating Series

AbstractThe algebra of so-called shifted symmetric functions on partitions has the property that for all elements a certain generating series, called the q-bracket, is a quasimodular form. More generally, if a graded algebra A of functions on partitions has the property that the q-bracket of every element is a quasimodular form of the same weight, we call A a quasimodular algebra. We introduce a new quasimodular algebra $$\mathcal {T}$$ T consisting of symmetric polynomials in the part sizes and multiplicities.

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Combinatorics of k-shapes and Genocchi numbers

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2928 ◽

2011 ◽

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

Author(s):

Florent Hivert ◽

Olivier Mallet

Keyword(s):

Symmetric Functions ◽

New Model ◽

Work In Progress ◽

Genocchi Numbers ◽

Combinatorial Model ◽

International Audience

International audience In this paper we present a work in progress on a conjectural new combinatorial model for the Genocchi numbers. This new model called irreducible k-shapes has a strong algebraic background in the theory of symmetric functions and leads to seemingly new features on the theory of Genocchi numbers. In particular, the natural q-analogue coming from the degree of symmetric functions seems to be unknown so far. Dans cet article, nous présentons un travail en cours sur un nouveau modèle combinatoire conjectural pour les nombres de Genocchi. Ce nouveau modèle est celui des k-formes irréductibles, qui repose sur de solides bases algébriques en lien avec la théorie des fonctions symétriques et qui conduit à des aspects apparemment nouveaux de la théorie des nombres de Genocchi. En particulier, le q-analogue naturel venant du degré des fonctions symétriques semble inconnu jusqu'ici.

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How to get the weak order out of a digraph ?

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2538 ◽

2015 ◽

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

Author(s):

Francois Viard

Keyword(s):

Wreath Product ◽

Symmetric Function ◽

Coxeter Groups ◽

Symmetric Functions ◽

Weak Order ◽

Möbius Function ◽

Mobius Function ◽

Acyclic Digraph ◽

International Audience

International audience We construct a poset from a simple acyclic digraph together with a valuation on its vertices, and we compute the values of its Möbius function. We show that the weak order on Coxeter groups $A$$n-1$, $B$$n$, $Ã$$n$, and the flag weak order on the wreath product ℤ$r$ ≀ $S$$n$ introduced by Adin, Brenti and Roichman (2012), are special instances of our construction. We conclude by briefly explaining how to use our work to define quasi-symmetric functions, with a special emphasis on the $A$$n-1$ case, in which case we obtain the classical Stanley symmetric function. On construit une famille d’ensembles ordonnés à partir d’un graphe orienté, simple et acyclique munit d’une valuation sur ses sommets, puis on calcule les valeurs de leur fonction de Möbius respective. On montre que l’ordre faible sur les groupes de Coxeter $A$$n-1$, $B$$n$, $Ã$$n$, ainsi qu’une variante de l’ordre faible sur les produits en couronne ℤ$r$ ≀ $S$$n$ introduit par Adin, Brenti et Roichman (2012), sont des cas particuliers de cette construction. On conclura en expliquant brièvement comment ce travail peut-être utilisé pour définir des fonction quasi-symétriques, en insistant sur l’exemple de l’ordre faible sur $A$$n-1$ où l’on obtient les séries de Stanley classiques.

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