scholarly journals A symmetric Bloch–Okounkov theorem

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.

Inequalities for Generalized Symmetric Functions

1969 ◽  
Vol 12 (5) ◽  
pp. 615-623 ◽  
Author(s):  
K.V. Menon
Keyword(s):  
Symmetric Function ◽  
Symmetric Functions ◽  
Elementary Symmetric Function ◽  
Generating Series ◽  
Image Position ◽  
Complete Symmetric Function

The generating series for the elementary symmetric function Er, the complete symmetric function Hr, are defined byrespectively.

scholarly journals Towards the Calculation of Jack Polynomials

2021 ◽  
Author(s):  
◽  
Leigh Alan Roberts
Keyword(s):  
Recent Result ◽  
Symmetric Functions ◽  
Mathematical Statistics ◽  
Jack Polynomials ◽  
Symmetric Polynomials ◽  
Geometric Progressions ◽  
Elementary Symmetric Polynomials

<p>Jack polynomials are useful in mathematical statistics, but they are awkward to calculate, and their uses have chiefly been theoretical. In this thesis a determinantal expansion of Jack polynomials in elementary symmetric polynomials is found, complementing a recent result in the literature on expansions as determinants in monomial symmetric functions. These results offer enhanced possibilities for the calculation of these polynomials, and for finding workable approximations to them. The thesis investigates the structure of the determinants concerned, finding which terms can be expected to dominate, and quantifying the sparsity of the matrices involved. Expressions are found for the elementary and monomial symmetric polynomials when the variates involved assume the form of arithmetic and geometric progressions. The latter case in particular is expected to facilitate the construction of algorithms suitable for approximating Jack polynomials.</p>

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

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.

scholarly journals Explicit generating series for connection coefficients

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 $[a,b,1^{n-a-b}]$.

scholarly journals Symmetric functions on spaces $\ell_p(\mathbb{{R}}^n)$ and $\ell_p(\mathbb{{C}}^n)$

2020 ◽  
Vol 12 (1) ◽  
pp. 5-16
Author(s):  
T.V. Vasylyshyn
Keyword(s):  
Linear Combination ◽  
Banach Spaces ◽  
Symmetric Functions ◽  
Symmetric Polynomial ◽  
Complex Numbers ◽  
Symmetric Polynomials ◽  
Algebraic Basis

This work is devoted to study algebras of continuous symmetric, that is, invariant with respect to permutations of coordinates of its argument, polynomials and $*$-polynomials on Banach spaces $\ell_p(\mathbb{R}^n)$ and $\ell_p(\mathbb{C}^n)$ of $p$-power summable sequences of $n$-dimensional vectors of real and complex numbers resp., where $1\leq p < +\infty.$ We construct the subset of the algebra of all continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n)$ such that every continuous symmetric polynomial on the space $\ell_p(\mathbb{R}^n)$ can be uniquely represented as a linear combination of products of elements of this set. In other words, we construct an algebraic basis of the algebra of all continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n).$ Using this result, we construct an algebraic basis of the algebra of all continuous symmetric $*$-polynomials on the space $\ell_p(\mathbb{C}^n).$ Results of the paper can be used for investigations of algebras, generated by continuous symmetric polynomials on the space $\ell_p(\mathbb{R}^n),$ and algebras, generated by continuous symmetric $*$-polynomials on the space $\ell_p(\mathbb{C}^n).$

scholarly journals Bijection Between Oriented Maps and Weighted Non-Oriented Maps

10.37236/6718 ◽  
2017 ◽  
Vol 24 (3) ◽  
Author(s):  
Agnieszka Czyżewska-Jankowska ◽  
Piotr Śniady
Keyword(s):  
Symmetric Functions ◽  
Explicit Formulas ◽  
Remarkable Property ◽  
Power Sum ◽  
Generating Series ◽  
Face Structure

We consider bicolored maps, i.e. graphs which are drawn on surfaces, and construct a bijection between (i) oriented maps with arbitary face structure, and (ii) (weighted) non-oriented maps with exactly one face. Above, each non-oriented map is counted with a multiplicity which is based on the concept of the orientability generating series and the measure of orientability of a map. This bijection has the remarkable property of preserving the underlying bicolored graph. Our bijection shows equivalence between two explicit formulas for the top-degree of Jack characters, i.e. (suitably normalized) coefficients in the expansion of Jack symmetric functions in the basis of power-sum symmetric functions.

scholarly journals Bijective evaluation of the connection coefficients of the double coset algebra

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.

scholarly journals On Determinantal Symmetric Functions

1934 ◽  
Vol 4 (1) ◽  
pp. 47-52
Author(s):  
Zia-ud-Din
Keyword(s):  
Symmetric Functions ◽  
Symmetric Polynomials ◽  
The Difference ◽  
Dual Identities ◽  
Image Position

§ 1. The theory of symmetric polynomials abounds in dual identities and symmetries of various kinds. It has been investigated from the determinantal standpoint largely by means of quotients of alternants, such as,the denominator being the difference product of a, b, c, …, a simple alternant.

scholarly journals Towards the Calculation of Jack Polynomials

2021 ◽  
Author(s):  
◽  
Leigh Alan Roberts
Keyword(s):  
Recent Result ◽  
Symmetric Functions ◽  
Mathematical Statistics ◽  
Jack Polynomials ◽  
Symmetric Polynomials ◽  
Geometric Progressions ◽  
Elementary Symmetric Polynomials

<p>Jack polynomials are useful in mathematical statistics, but they are awkward to calculate, and their uses have chiefly been theoretical. In this thesis a determinantal expansion of Jack polynomials in elementary symmetric polynomials is found, complementing a recent result in the literature on expansions as determinants in monomial symmetric functions. These results offer enhanced possibilities for the calculation of these polynomials, and for finding workable approximations to them. The thesis investigates the structure of the determinants concerned, finding which terms can be expected to dominate, and quantifying the sparsity of the matrices involved. Expressions are found for the elementary and monomial symmetric polynomials when the variates involved assume the form of arithmetic and geometric progressions. The latter case in particular is expected to facilitate the construction of algorithms suitable for approximating Jack polynomials.</p>

scholarly journals Lipschitz symmetric functions on Banach spaces with symmetric bases

2021 ◽  
Vol 13 (3) ◽  
pp. 727-733
Author(s):  
M.V. Martsinkiv ◽  
S.I. Vasylyshyn ◽  
T.V. Vasylyshyn ◽  
A.V. Zagorodnyuk
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Symmetric Functions ◽  
Large Family ◽  
Symmetric Polynomials ◽  
Symmetric Basis ◽  
Several Variables ◽  
Symmetric Bases ◽  
Unbounded Subset

We investigate Lipschitz symmetric functions on a Banach space $X$ with a symmetric basis. We consider power symmetric polynomials on $\ell_1$ and show that they are Lipschitz on the unbounded subset consisting of vectors $x\in \ell_1$ such that $|x_n|\le 1.$ Using functions $\max$ and $\min$ and tropical polynomials of several variables, we constructed a large family of Lipschitz symmetric functions on the Banach space $c_0$ which can be described as a semiring of compositions of tropical polynomials over $c_0$.

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