scholarly journals The stability of the Kronecker product of Schur functions

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Emmanuel Briand ◽  
Rosa Orellana ◽  
Mercedes Rosas
Keyword(s):  
Kronecker Product ◽  
Schur Functions ◽  
International Audience ◽  
Kronecker Coefficients ◽  
Kronecker Coefficient ◽  
Alpha Beta ◽  
The Stability

International audience In the late 1930's Murnaghan discovered the existence of a stabilization phenomenon for the Kronecker product of Schur functions. For $n$ large enough, the values of the Kronecker coefficients appearing in the product of two Schur functions of degree $n$ do not depend on the first part of the indexing partitions, but only on the values of their remaining parts. We compute the exact value of n when this stable expansion is reached. We also compute two new bounds for the stabilization of a particular coefficient of such a product. Given partitions $\alpha$ and $\beta$, we give bounds for all the parts of any partition $\gamma$ such that the corresponding Kronecker coefficient is nonzero. Finally, we also show that the reduced Kronecker coefficients are structure coefficients for the Heisenberg product introduced by Aguiar, Ferrer and Moreira. Dans les années 30 Murnaghan a découvert une propriété de stabilité pour le produit de Kronecker de fonctions de Schur. En degré assez grand, les valeurs des coefficients qui aparaissent dans le produit de Kronecker de deux fonctions de Schur ne dépendent pas de la première part des partitions en indice, mais seulement des parts suivantes. Dans ce travail nous calculons la valeur exacte du degré à partir duquel ce développement stable est atteint. Nous calculons aussi deux nouvelles bornes supérieures pour la stabilisation d'un coefficient particulier d'un tel produit. Nous donnons en outre, pour $\alpha$ et $\beta$ fixés, des bornes supérieures pour toutes les parts des partition $\gamma$ rendant le coefficient de Kronecker d'indices $\alpha$, $\beta$, $\gamma$ non―nul. Finalement, nous identifions les coefficients de Kronecker réduits comme des constantes de structures pour le produit de Heisenberg de fonctions symétriques défini par Aguiar, Ferrer et Moreira.

scholarly journals On the Kronecker Product $s_{(n-p,p)}\ast s_{\lambda}$

10.37236/1925 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
C. M. Ballantine ◽  
R. C. Orellana
Keyword(s):  
Irreducible Representation ◽  
Tensor Product ◽  
Symmetric Group ◽  
Kronecker Product ◽  
Schur Functions ◽  
Irreducible Representations ◽  
Kronecker Coefficient

The Kronecker product of two Schur functions $s_{\lambda}$ and $s_{\mu}$, denoted $s_{\lambda}\ast s_{\mu}$, is defined as the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group indexed by partitions of $n$, $\lambda$ and $\mu$, respectively. The coefficient, $g_{\lambda,\mu,\nu}$, of $s_{\nu}$ in $s_{\lambda}\ast s_{\mu}$ is equal to the multiplicity of the irreducible representation indexed by $\nu$ in the tensor product. In this paper we give an algorithm for expanding the Kronecker product $s_{(n-p,p)}\ast s_{\lambda}$ if $\lambda_1-\lambda_2\geq 2p$. As a consequence of this algorithm we obtain a formula for $g_{(n-p,p), \lambda ,\nu}$ in terms of the Littlewood-Richardson coefficients which does not involve cancellations. Another consequence of our algorithm is that if $\lambda_1-\lambda_2\geq 2p$ then every Kronecker coefficient in $s_{(n-p,p)}\ast s_{\lambda}$ is independent of $n$, in other words, $g_{(n-p,p),\lambda,\nu}$ is stable for all $\nu$.

scholarly journals The stability of the Kronecker product of Schur functions

2011 ◽  
Vol 331 (1) ◽  
pp. 11-27 ◽  
Author(s):  
Emmanuel Briand ◽  
Rosa Orellana ◽  
Mercedes Rosas
Keyword(s):  
Kronecker Product ◽  
Schur Functions ◽  
The Stability

scholarly journals Schéma SRNHS Analyse et Application d'un schéma aux volumes finis dédié aux systèmes non homogènes

2006 ◽  
Vol Volume 5, Special Issue TAM... ◽  
Author(s):  
Slah SAHMIM ◽  
Fayssal Benkhaldoun
Keyword(s):  
Shallow Water ◽  
Scalar Case ◽  
Finite Volume Scheme ◽  
Two Phase ◽  
Homogeneous Systems ◽  
Selfsimilar Solution ◽  
International Audience ◽  
The Stability ◽  
New Formulation ◽  
Water Problems

International audience This article is devoted to the analysis, and improvement of a finite volume scheme proposed recently for a class of non homogeneous systems. We consider those for which the corressponding Riemann problem admits a selfsimilar solution. Some important examples of such problems are Shallow Water problems with irregular topography and two phase flows. The stability analysis of the considered scheme, in the homogeneous scalar case, leads to a new formulation which has a naturel extension to non homogeneous systems. Comparative numerical experiments for Shallow Water equations with sourec term, and a two phase problem (Ransom faucet) are presented to validate the scheme. Cet article concerne l'analyse et l'application, d'un schéma proposé récemment por une classe de systèmes non homogènes. Nous considérons ceux pour lesquels le problème de Riemann correpondant admet une solution autosimilaire. Deux exemples importants de tels problèmes sont l'écoulement d'eau peu profonde au-dessus d'un fond non plat et les problèmes diphasiques. l'analyse de stabilité du schéma, dans le cas scalaire homogène, amène à une nouvelle écriture qui a une extension naturelle pour le cas non homogène. Des expériences numériques comparatives pour des équations de saint-Venant avec topographie variable, et un problème diphasique (Robinet de Ransom) sont présentés pour évaluer l'efficacité du schéma.

scholarly journals Quasipolynomial formulas for the Kronecker coefficients indexed by two two―row shapes (extended abstract)

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Emmanuel Briand ◽  
Rosa Orellana ◽  
Mercedes Rosas
Keyword(s):  
International Audience ◽  
Kronecker Coefficients

International audience We show that the Kronecker coefficients indexed by two two―row shapes are given by quadratic quasipolynomial formulas whose domains are the maximal cells of a fan. Simple calculations provide explicitly the quasipolynomial formulas and a description of the associated fan. These new formulas are obtained from analogous formulas for the corresponding reduced Kronecker coefficients and a formula recovering the Kronecker coefficients from the reduced Kronecker coefficients. As an application, we characterize all the Kronecker coefficients indexed by two two-row shapes that are equal to zero. This allowed us to disprove a conjecture of Mulmuley about the behavior of the stretching functions attached to the Kronecker coefficients. Nous démontrons que les coefficients de Kronecker indexés par deux partitions de longueur au plus 2 sont donnés par des formules quasipolynomiales quadratiques dont les domaines de validité sont les cellules maximales d'un éventail. Des calculs simples nous donnent une description explicite des formules quasipolynomiales et de l'éventail associé. Ces nouvelles formulas sont obtenues de formules analogues pour les coefficients de Kronecker réduits correspondants et au moyen d'une formule reconstruisant les coefficients de Kronecker à partir des coefficients de Kronecker réduits. Une application est la caractérisation exacte de tous les coefficients de Kronecker non―nuls indexés par deux partitions de longueur au plus deux. Ceci nous a permis de réfuter une conjecture de Mulmuley au sujet des fonctions de dilatations associées aux coefficients de Kronecker.

scholarly journals Pieri rules for Schur functions in superspace

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Miles Eli Jones ◽  
Luc Lapointe
Keyword(s):  
Generating Functions ◽  
Schur Functions ◽  
Macdonald Polynomials ◽  
International Audience ◽  
Pieri Rules

International audience The Schur functions in superspace $s_\Lambda$ and $\overline{s}_\Lambda$ are the limits $q=t= 0$ and $q=t=\infty$ respectively of the Macdonald polynomials in superspace. We present the elementary properties of the bases $s_\Lambda$ and $\overline{s}_\Lambda$ (which happen to be essentially dual) such as Pieri rules, dualities, monomial expansions, tableaux generating functions, and Cauchy identities. Les fonctions de Schur dans le superespace $s_\Lambda$ et $\overline{s}_\Lambda$ sont les limites $q=t= 0$ et $q=t=\infty$ respectivement des polynômes de Macdonald dans le superespace. Nous présentons les propriétés élémentaires des bases $s_\Lambda$ et $\overline{s}_\Lambda$ (qui sont essentiellement duales l'une de l'autre) tels que les règles de Pieri, la dualité, le développement en fonctions monomiales, les fonctions génératrices de tableaux et les identités de Cauchy.

scholarly journals Character Polynomials, their q-Analogs and the Kronecker Product

10.37236/85 ◽  
2009 ◽  
Vol 16 (2) ◽  
Author(s):  
A. M. Garsia ◽  
A. Goupil
Keyword(s):  
Numerical Calculation ◽  
Kronecker Product ◽  
Hecke Algebra ◽  
Kronecker Products ◽  
Proper Role ◽  
Special Cases ◽  
Natural Counterpart ◽  
Kronecker Coefficients ◽  
Combinatorial Construction

The numerical calculation of character values as well as Kronecker coefficients can efficently be carried out by means of character polynomials. Yet these polynomials do not seem to have been given a proper role in present day literature or software. To show their remarkable simplicity we give here an "umbral" version and a recursive combinatorial construction. We also show that these polynomials have a natural counterpart in the standard Hecke algebra ${\cal H}_n(q\, )$. Their relation to Kronecker products is brought to the fore, as well as special cases and applications. This paper may also be used as a tutorial for working with character polynomials in the computation of Kronecker coefficients.

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 Plane partitions and the combinatorics of some families of reduced Kronecker coefficients.

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Laura Colmenarejo
Keyword(s):  
Generating Function ◽  
Plane Partitions ◽  
Rate Of Growth ◽  
International Audience ◽  
Kronecker Coefficients

International audience We compute the generating function of some families of reduced Kronecker coefficients. We give a combi- natorial interpretation for these coefficients in terms of plane partitions. This unexpected relation allows us to check that the saturation hypothesis holds for the reduced Kronecker coefficients of our families. We also compute the quasipolynomial that govern these families, specifying the degree and period. Moving to the setting of Kronecker co- efficients, these results imply some observations related to the rate of growth experienced by the families of Kronecker coefficients associated to the reduced Kronecker coefficients already studied.

scholarly journals The down operator and expansions of near rectangular k-Schur functions

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Chris Berg ◽  
Franco Saliola ◽  
Luis Serrano
Keyword(s):  
Weyl Group ◽  
Schur Functions ◽  
Nous Obtenons ◽  
Combinatorial Interpretation ◽  
Affine Weyl Group ◽  
Nous Montrons ◽  
International Audience

International audience We prove that the Lam-Shimozono ``down operator'' on the affine Weyl group induces a derivation of the affine Fomin-Stanley subalgebra of the affine nilCoxeter algebra. We use this to verify a conjecture of Berg, Bergeron, Pon and Zabrocki describing the expansion of k-Schur functions of ``near rectangles'' in the affine nilCoxeter algebra. Consequently, we obtain a combinatorial interpretation of the corresponding k-Littlewood–Richardson coefficients. Nous montrons que l’opérateur ``down'', défini par Lam et Shimozono sur le groupe de Weyl affine, induit une dérivation de la sous-algèbre affine de Fomin-Stanley de l'algèbre affine de nilCoxeter. Nous employons cette dérivation pour vérifier une conjecture de Berg, Bergeron, Pon et Zabrocki sur l'expansion des k-fonctions de Schur indexées par les partitions qui sont ``presque rectangles''. Par conséquent, nous obtenons une interprétation combinatoire des k-coefficients de Littlewood–Richardson correspondants.

scholarly journals Stability of Kronecker Products of Irreducible Characters of the Symmetric Group

10.37236/1471 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Ernesto Vallejo
Keyword(s):  
Lower Bound ◽  
Symmetric Group ◽  
Kronecker Product ◽  
Kronecker Products ◽  
Irreducible Characters ◽  
Long Time ◽  
The Stability

F. Murnaghan observed a long time ago that the computation of the decompositon of the Kronecker product $\chi^{(n-a, \lambda_2, \dots )}\otimes \chi^{(n-b, \mu_2, \dots)}$ of two irreducible characters of the symmetric group into irreducibles depends only on $\overline\lambda=(\lambda_2,\dots )$ and $\overline\mu =(\mu_2,\dots )$, but not on $n$. In this note we prove a similar result: given three partitions $\lambda$, $\mu$, $\nu$ of $n$ we obtain a lower bound on $n$, depending on $\overline\lambda$, $\overline\mu$, $\overline\nu$, for the stability of the multiplicity $c(\lambda,\mu,\nu)$ of $\chi^\nu$ in $\chi^\lambda \otimes \chi^\mu$. Our proof is purely combinatorial. It uses a description of the $c(\lambda,\mu,\nu)$'s in terms of signed special rim hook tabloids and Littlewood-Richardson multitableaux.

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