scholarly journals Hopf algebras and the logarithm of the S-transform in free probability ― Extended abstract

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Mitja Mastnak ◽  
Alexandru Nica
Keyword(s):  
Hopf Algebras ◽  
Free Probability ◽  
Algebraic Approach ◽  
S Transform ◽  
Multiplicative Convolution ◽  
Free Multiplicative Convolution ◽  
International Audience

International audience This document is an extended abstract of the paper `Hopf algebras and the logarithm of the S-transform in free probability' in which we introduce a Hopf algebraic approach to the study of the operation $\boxtimes$ (free multiplicative convolution) from free probability.

scholarly journals Multivariate Fuss-Narayana Polynomials and their Application to Random Matrices

10.37236/2799 ◽  
2013 ◽  
Vol 20 (2) ◽  
Author(s):  
Romuald Lenczewski ◽  
Rafal Salapata
Keyword(s):  
Random Matrices ◽  
Shape Parameter ◽  
Free Probability ◽  
Homogeneous Polynomials ◽  
Noncrossing Partitions ◽  
Multiplicative Convolution ◽  
Free Multiplicative Convolution

It has been shown recently that the limit moments of $W(n)=B(n)B^{*}(n)$, where $B(n)$ is a product of $p$ independent rectangular random matrices, are certain homogeneous polynomials $P_{k}(d_0,d_1, \ldots , d_{p})$ in the asymptotic dimensions of these matrices. Using the combinatorics of noncrossing partitions, we explicitly determine these polynomials and show that they are closely related to polynomials which can be viewed as {\it multivariate Fuss-Narayana polynomials}. Using this result, we compute the moments of $\varrho_{t_1}\boxtimes \varrho_{t_2}\boxtimes\ldots \boxtimes \varrho_{t_m}$ for any positive $t_1,t_2, \ldots , t_m$, where $\boxtimes$ is the free multiplicative convolution in free probability and $\varrho_{t}$ is the Marchenko-Pastur distribution with shape parameter $t$.

scholarly journals Combinatorial Hopf Algebras of Simplicial Complexes

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Carolina Benedetti ◽  
Joshua Hallam ◽  
John Machacek
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Symmetric Functions ◽  
Simplicial Complexes ◽  
Combinatorial Hopf Algebras ◽  
International Audience ◽  
Donnent Lieu

International audience We consider a Hopf algebra of simplicial complexes and provide a cancellation-free formula for its antipode. We then obtain a family of combinatorial Hopf algebras by defining a family of characters on this Hopf algebra. The characters of these Hopf algebras give rise to symmetric functions that encode information about colorings of simplicial complexes and their $f$-vectors. We also use characters to give a generalization of Stanley’s $(-1)$-color theorem. Nous considérons une algèbre de Hopf de complexes simpliciaux et fournissons une formule sans multiplicité pour son antipode. On obtient ensuite une famille d'algèbres de Hopf combinatoires en définissant une famille de caractères sur cette algèbre de Hopf. Les caractères de ces algèbres de Hopf donnent lieu à des fonctions symétriques qui encode de l’information sur les coloriages du complexe simplicial ainsi que son vecteur-$f$. Nousallons également utiliser des caractères pour donner une généralisation du théorème $(-1)$ de Stanley.

scholarly journals SIF Permutations and Chord-Connected Permutations

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Natasha Blitvić
Keyword(s):  
Probability Theory ◽  
Generating Function ◽  
Probability Distributions ◽  
Free Probability ◽  
Perfect Matchings ◽  
Free Probability Theory ◽  
Diagrammatic Representation ◽  
International Audience ◽  
Binary Strings

International audience A <i>stabilized-interval-free </i> (SIF) permutation on [n], introduced by Callan, is a permutation that does not stabilize any proper interval of [n]. Such permutations are known to be the irreducibles in the decomposition of permutations along non-crossing partitions. That is, if $s_n$ denotes the number of SIF permutations on [n], $S(z)=1+\sum_{n\geq1} s_n z^n$, and $F(z)=1+\sum_{n\geq1} n! z^n$, then $F(z)= S(zF(z))$. This article presents, in turn, a decomposition of SIF permutations along non-crossing partitions. Specifically, by working with a convenient diagrammatic representation, given in terms of perfect matchings on alternating binary strings, we arrive at the \emphchord-connected permutations on [n], counted by $\{c_n\}_{n\geq1}$, whose generating function satisfies $S(z)= C(zS(z))$. The expressions at hand have immediate probabilistic interpretations, via the celebrated <i>moment-cumulant formula </i>of Speicher, in the context of the <i>free probability theory </i>of Voiculescu. The probability distributions that appear are the exponential and the complex Gaussian.

scholarly journals Skew Littlewood―Richardson rules from Hopf algebras

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Thomas Lam ◽  
Aaron Lauve ◽  
Frank Sottile
Keyword(s):  
Hopf Algebras ◽  
Symplectic Group ◽  
Schur Functions ◽  
Affine Grassmannian ◽  
International Audience ◽  
Pieri Rules ◽  
Skew Schur Functions ◽  
Nous Utilisons

International audience We use Hopf algebras to prove a version of the Littlewood―Richardson rule for skew Schur functions, which implies a conjecture of Assaf and McNamara. We also establish skew Littlewood―Richardson rules for Schur $P-$ and $Q-$functions and noncommutative ribbon Schur functions, as well as skew Pieri rules for k-Schur functions, dual k-Schur functions, and for the homology of the affine Grassmannian of the symplectic group. Nous utilisons des algèbres de Hopf pour prouver une version de la règle de Littlewood―Richardson pour les fonctions de Schur gauches, qui implique une conjecture d'Assaf et McNamara. Nous établissons également des règles de Littlewood―Richardson gauches pour les $P-$ et $Q-$fonctions de Schur et les fonctions de Schur rubbans non commutatives, ainsi que des règles de Pieri gauches pour les $k-$fonctions de Schur, les $k-$fonctions de Schur duales, et pour l'homologie de la Grassmannienne affine du groupe symplectique.

scholarly journals Brick polytopes, lattices and Hopf algebras

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Vincent Pilaud
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Transitive Closure ◽  
Weak Order ◽  
Binary Trees ◽  
International Audience ◽  
Congruence Classes

International audience Generalizing the connection between the classes of the sylvester congruence and the binary trees, we show that the classes of the congruence of the weak order on Sn defined as the transitive closure of the rewriting rule UacV1b1 ···VkbkW ≡k UcaV1b1 ···VkbkW, for letters a < b1,...,bk < c and words U,V1,...,Vk,W on [n], are in bijection with acyclic k-triangulations of the (n + 2k)-gon, or equivalently with acyclic pipe dreams for the permutation (1,...,k,n + k,...,k + 1,n + k + 1,...,n + 2k). It enables us to transport the known lattice and Hopf algebra structures from the congruence classes of ≡k to these acyclic pipe dreams, and to describe the product and coproduct of this algebra in terms of pipe dreams. Moreover, it shows that the fan obtained by coarsening the Coxeter fan according to the classes of ≡k is the normal fan of the corresponding brick polytope

scholarly journals Quasisymmetric functions from combinatorial Hopf monoids and Ehrhart Theory

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Jacob White
Keyword(s):  
Commutative Algebra ◽  
Hopf Algebras ◽  
Simplicial Complexes ◽  
Hilbert Functions ◽  
Quasisymmetric Functions ◽  
Ehrhart Theory ◽  
International Audience ◽  
Hopf Monoid

International audience We investigate quasisymmetric functions coming from combinatorial Hopf monoids. We show that these invariants arise naturally in Ehrhart theory, and that some of their specializations are Hilbert functions for relative simplicial complexes. This class of complexes, called forbidden composition complexes, also forms a Hopf monoid, thus demonstrating a link between Hopf algebras, Ehrhart theory, and commutative algebra. We also study various specializations of quasisymmetric functions.

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