scholarly journals Combinatorial interpretation and positivity of Kerov's character polynomials

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Valentin Féray
Keyword(s):  
Symmetric Group ◽  
Young Diagram ◽  
Irreducible Character ◽  
Combinatorial Approach ◽  
Combinatorial Interpretation ◽  
Free Cumulants ◽  
International Audience

International audience Kerov's polynomials give irreducible character values of the symmetric group in term of the free cumulants of the associated Young diagram. Using a combinatorial approach with maps, we prove in this article a positivity result on their coefficients, which extends a conjecture of S. Kerov. Les polynômes de Kerov expriment les valeurs des caractères irréductibles du groupe symétrique en fonction des cumulants libres du diagramme de Young associé. Grâce à une approche combinatoire à base de cartes, nous prouvons dans cet article un résultat de positivité sur leurs coefficients, qui généralise une conjecture de S. Kerov.

scholarly journals Characters of symmetric groups in terms of free cumulants and Frobenius coordinates

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Maciej Dolega ◽  
Valentin Féray ◽  
Piotr Sniady
Keyword(s):  
Young Diagram ◽  
Symmetric Groups ◽  
Main Tool ◽  
Combinatorial Formula ◽  
Young Diagrams ◽  
Free Cumulants ◽  
International Audience

International audience Free cumulants are nice and useful functionals of the shape of a Young diagram, in particular they give the asymptotics of normalized characters of symmetric groups $\mathfrak{S}(n)$ in the limit $n \to \infty$. We give an explicit combinatorial formula for normalized characters of the symmetric groups in terms of free cumulants. We also express characters in terms of Frobenius coordinates. Our formulas involve counting certain factorizations of a given permutation. The main tool are Stanley polynomials which give values of characters on multirectangular Young diagrams. Les cumulants libres sont des fonctions agréables et utiles sur l'ensemble des diagrammes de Young, en particulier, ils donnent le comportement asymptotiques des caractères normalisés du groupe symétrique $\mathfrak{S}(n)$ dans la limite $n \to \infty$. Nous donnons une formule combinatoire explicite pour les caractères normalisés du groupe symétrique en fonction des cumulants libres. Nous exprimons également les caractères en fonction des coordonnées de Frobenius. Nos formules font intervenir le nombre de certaines factorisations d'une permutation donnée. L'outil principal est la famille de polynômes de Stanley donnant les valeurs des caractères sur les diagrammes de Young multirectangulaires.

scholarly journals Path tableaux and combinatorial interpretations of immanants for class functions on $S_n$

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Sam Clearman ◽  
Brittany Shelton ◽  
Mark Skandera
Keyword(s):  
Irreducible Character ◽  
Symmetric Function ◽  
Nonnegative Matrix ◽  
Schur Function ◽  
Combinatorial Interpretation ◽  
Combinatorial Interpretations ◽  
Characteristic Map ◽  
International Audience ◽  
Totally Nonnegative Matrix

International audience Let $χ ^λ$ be the irreducible $S_n$-character corresponding to the partition $λ$ of $n$, equivalently, the preimage of the Schur function $s_λ$ under the Frobenius characteristic map. Let $\phi ^λ$ be the function $S_n →ℂ$ which is the preimage of the monomial symmetric function $m_λ$ under the Frobenius characteristic map. The irreducible character immanant $Imm_λ {(x)} = ∑_w ∈S_n χ ^λ (w) x_1,w_1 ⋯x_n,w_n$ evaluates nonnegatively on each totally nonnegative matrix $A$. We provide a combinatorial interpretation for the value $Imm_λ (A)$ in the case that $λ$ is a hook partition. The monomial immanant $Imm_{{\phi} ^λ} (x) = ∑_w ∈S_n φ ^λ (w) x_1,w_1 ⋯x_n,w_n$ is conjectured to evaluate nonnegatively on each totally nonnegative matrix $A$. We confirm this conjecture in the case that $λ$ is a two-column partition by providing a combinatorial interpretation for the value $Imm_{{\phi} ^λ} (A)$. Soit $χ ^λ$ le caractère irréductible de $S_n$ qui correspond à la partition λ de l'entier n, ou de manière équivalente, la préimage de la fonction de Schur $s_λ$ par l'application caractéristique de Frobenius. Soit $\phi ^λ$ la fonction $S_n →ℂ$ qui est la préimage de la fonction symétrique monomiale m_λ . La valeur du caractère irréductible immanent $Imm_λ {(x)} = ∑_w ∈S_n χ ^λ (w) x_1,w_1 ⋯x_n,w_n$ est non négative pour chaque matrice totalement non négative. Nous donnons une interprétation combinatoire de la valeur $Imm_λ (A)$ lorsque $λ$ est une partition en équerre. Stembridge a conjecturé que la valeur de l'immanent monomial $Imm_{{\phi} ^λ} (x) = ∑_w ∈S_n φ ^λ (w) x_1,w_1 ⋯x_n,w_n$ de $\phi ^λ$ est elle aussi non négative pour chaque matrice totalement non négative. Nous confirmons cette conjecture quand λ satisfait $λ _1 ≤2$, et nous donnons une interprétation combinatoire de $Imm_{{\phi} ^λ} (A)$ dans ce cas.

scholarly journals A reciprocity approach to computing generating functions for permutations with no pattern matches

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Miles Eli Jones ◽  
Jeffrey Remmel
Keyword(s):  
Symmetric Group ◽  
Generating Functions ◽  
New Method ◽  
Combinatorial Interpretation ◽  
Set Of Permutations ◽  
International Audience ◽  
Nous Utilisons

International audience In this paper, we develop a new method to compute generating functions of the form $NM_τ (t,x,y) = \sum\limits_{n ≥0} {\frac{t^n} {n!}}∑_{σ ∈\mathcal{lNM_{n}(τ )}} x^{LRMin(σ)} y^{1+des(σ )}$ where $τ$ is a permutation that starts with $1, \mathcal{NM_n}(τ )$ is the set of permutations in the symmetric group $S_n$ with no $τ$ -matches, and for any permutation $σ ∈S_n$, $LRMin(σ )$ is the number of left-to-right minima of $σ$ and $des(σ )$ is the number of descents of $σ$ . Our method does not compute $NM_τ (t,x,y)$ directly, but assumes that $NM_τ (t,x,y) = \frac{1}{/ (U_τ (t,y))^x}$ where $U_τ (t,y) = \sum_{n ≥0} U_τ ,n(y) \frac{t^n}{ n!}$ so that $U_τ (t,y) = \frac{1}{ NM_τ (t,1,y)}$. We then use the so-called homomorphism method and the combinatorial interpretation of $NM_τ (t,1,y)$ to develop recursions for the coefficient of $U_τ (t,y)$. Dans cet article, nous développons une nouvelle méthode pour calculer les fonctions génératrices de la forme $NM_τ (t,x,y) = \sum\limits_{n ≥0} {\frac{t^n} {n!}}∑_{σ ∈\mathcal{lNM_{n}(τ )}} x^{LRMin(σ)} y^{1+des(σ )}$ où τ est une permutation, $\mathcal{NM_n}(τ )$ est l'ensemble des permutations dans le groupe symétrique $S_n$ sans $τ$-matches, et pour toute permutation $σ ∈S_n$, $LRMin(σ )$ est le nombre de minima de gauche à droite de $σ$ et $des(σ )$ est le nombre de descentes de $σ$ . Notre méthode ne calcule pas $NM_τ (t,x,y)$ directement, mais suppose que $NM_τ (t,x,y) = \frac{1}{/ (U_τ (t,y))^x}$ où $U_τ (t,y) = \sum_{n ≥0} U_τ ,n(y) \frac{t^n}{ n!}$ de sorte que $U_τ (t,y) = \frac{1}{ NM_τ (t,1,y)}$. Nous utilisons ensuite la méthode dite "de l'homomorphisme'' et l'interprétation combinatoire de $NM_τ (t,1,y)$ pour développer des récursions sur le coefficient de $U_τ (t,y)$.

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 Character Condition for Quadruple Transitivity

2011 ◽  
Vol 2011 ◽  
pp. 1-13
Author(s):  
S. Aldhafeeri ◽  
R. T. Curtis
Keyword(s):  
Symmetric Group ◽  
Permutation Group ◽  
Irreducible Character

Let be a permutation group of degree viewed as a subgroup of the symmetric group . We show that if the irreducible character of corresponding to the partition of into subsets of sizes and 2, that is, to say the character often denoted by , remains irreducible when restricted to , then = 4, 5 or 9 and , A5, or PΣL2(8), respectively, or is 4-transitive.

scholarly journals Search for Young diagrams with large dimensions

2019 ◽  
pp. 33-43
Author(s):  
Vasilii S. Duzhin ◽  
◽  
Anastasia A. Chudnovskaya ◽  
Keyword(s):  
Symmetric Group ◽  
Young Diagram ◽  
Numerical Experiments ◽  
Irreducible Representations ◽  
Young Diagrams ◽  
Maximum Dimension ◽  
Asymptotic Combinatorics

Search for Young diagrams with maximum dimensions or, equivalently, search for irreducible representations of the symmetric group $S(n)$ with maximum dimensions is an important problem of asymptotic combinatorics. In this paper, we propose algorithms that transform a Young diagram into another one of the same size but with a larger dimension. As a result of massive numerical experiments, the sequence of $10^6$ Young diagrams with large dimensions was constructed. Furthermore, the proposed algorithms do not change the first 1000 elements of this sequence. This may indicate that most of them have the maximum dimension. It has been found that the dimensions of all Young diagrams of the resulting sequence starting from the 75778th exceed the dimensions of corresponding diagrams of the greedy Plancherel sequence.

On the Modular Representations of the Symmetric Group

1954 ◽  
Vol 6 ◽  
pp. 486-497 ◽  
Author(s):  
G. de B. Robinson
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Young Diagram ◽  
Modular Representation ◽  
Modular Representations ◽  
Modular Representation Theory

The study of the modular representation theory of the symmetric group has been greatly facilitated lately by the introduction of the graph (9, III ), the q-graph (5) and the hook-graph (4) of a Young diagram [λ]. In the present paper we seek to coordinate these ideas and relate them to the r-inducing and restricting processes (9, II ).

Semilinear preservers of the immanants in the set of the doubly stochastic matrices

2017 ◽  
Vol 32 ◽  
pp. 76-97
Author(s):  
M. Antonia Duffner ◽  
Rosario Fernandes
Keyword(s):  
Symmetric Group ◽  
Real Number ◽  
Irreducible Character ◽  
Complex Field ◽  
Stochastic Matrices ◽  
Doubly Stochastic ◽  
Doubly Stochastic Matrices

Let $S_n$ denote the symmetric group of degree $n$ and $M_n$ denote the set of all $n$-by-$n$ matrices over the complex field, $\IC$. Let $\chi: S_n\rightarrow \IC$ be an irreducible character of degree greater than $1$ of $S_n$. The immanant $\dc: M_n \rightarrow \IC$ associated with $\chi$ is defined by $$ \dc(X) = \sum_{\sigma \in S_n} \chi(\sigma) \prod_{j=1}^n X_{j\sigma(j)} , \quad X = [X_{jk}] \in M_n. $$ Let $\Omega_n$ be the set of all $n$-by-$n$ doubly stochastic matrices, that is, matrices with nonnegative real entries and each row and column sum is one. We say that a map $T$ from $\Omega_n$ into $\Omega_n$ \begin{itemize} \item is semilinear if $T(\lambda S_1+(1-\lambda )S_2)=\lambda T(S_1)+(1-\lambda )T(S_2)$ for all $S_1,\ S_2\in \Omega_n$ and for all real number $\lambda$ such that $0\leq \lambda\leq 1$; \item preserves $d_{\chi }$ if $d_{\chi }(T(S))=d_{\chi }(S)$ for all $S\in\Omega_n$. \end{itemize} We characterize the semilinear surjective maps $T$ from $\Omega_n $ into $\Omega_n$ that preserve $\dc$, when the degree of $\chi$ is greater than one.

scholarly journals Minimal factorizations of a cycle: a multivariate generating function

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Philippe Biane ◽  
Matthieu Josuat-Vergès
Keyword(s):  
Symmetric Group ◽  
Generating Function ◽  
Simple Formula ◽  
Hook Length ◽  
Noncrossing Partitions ◽  
Long Cycle ◽  
Number Of Factors ◽  
Multivariate Generalization ◽  
International Audience

International audience It is known that the number of minimal factorizations of the long cycle in the symmetric group into a product of k cycles of given lengths has a very simple formula: it is nk−1 where n is the rank of the underlying symmetric group and k is the number of factors. In particular, this is nn−2 for transposition factorizations. The goal of this work is to prove a multivariate generalization of this result. As a byproduct, we get a multivariate analog of Postnikov's hook length formula for trees, and a refined enumeration of final chains of noncrossing partitions.

scholarly journals Mixed volumes of hypersimplices

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Gaku Liu
Keyword(s):  
Catalan Numbers ◽  
Binomial Coefficients ◽  
Type B ◽  
Combinatorial Interpretation ◽  
Mixed Volumes ◽  
Eulerian Numbers ◽  
Eulerian Number ◽  
Nous Montrons ◽  
Set Of Permutations ◽  
International Audience

International audience In this extended abstract we consider mixed volumes of combinations of hypersimplices. These numbers, called mixed Eulerian numbers, were first considered by A. Postnikov and were shown to satisfy many properties related to Eulerian numbers, Catalan numbers, binomial coefficients, etc. We give a general combinatorial interpretation for mixed Eulerian numbers and prove the above properties combinatorially. In particular, we show that each mixed Eulerian number enumerates a certain set of permutations in $S_n$. We also prove several new properties of mixed Eulerian numbers using our methods. Finally, we consider a type $B$ analogue of mixed Eulerian numbers and give an analogous combinatorial interpretation for these numbers. Dans ce résumé étendu nous considérons les volumes mixtes de combinaisons d’hyper-simplexes. Ces nombres, appelés les nombres Eulériens mixtes, ont été pour la première fois étudiés par A. Postnikov, et il a été montré qu’ils satisfont à de nombreuses propriétés reliées aux nombres Eulériens, au nombres de Catalan, aux coefficients binomiaux, etc. Nous donnons une interprétation combinatoire générale des nombres Eulériens mixtes, et nous prouvons combinatoirement les propriétés mentionnées ci-dessus. En particulier, nous montrons que chaque nombre Eulérien mixte compte les éléments d’un certain sous-ensemble de l’ensemble des permutations $S_n$. Nous établissons également plusieurs nouvelles propriétés des nombres Eulériens mixtes grâce à notre méthode. Pour finir, nous introduisons une généralisation en type $B$ des nombres Eulériens mixtes, et nous en donnons une interprétation combinatoire analogue.

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