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 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 Invariance properties for coefficients of symmetric functions

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Emmanuel Briand ◽  
Rosa Orellana ◽  
Mercedes Rosas
Keyword(s):  
Symmetric Functions ◽  
Invariance Properties ◽  
Nous Montrons ◽  
International Audience ◽  
Kronecker Coefficients

International audience We show that several of the main structural constants for symmetric functions (Littlewood-Richardsoncoefficients, Kronecker coefficients, plethysm coefficients, and the Kostka–Foulkes polynomials) share invarianceproperties related to the operations of taking complements with respect to rectangles and adding rectangles. Nous montrons que plusieurs des principales constantes de structure de la théorie des fonctions symétriques(les coefficients de Littlewood–Richardson, les coefficients de Kronecker, les coefficients du pléthysme, et les polynômesde Kostka–Foulkes) ont en commun des symétries décrites par des opérations de complémentation dans des rectangleset d’ajout de rectangles pour les partitions qui les étiquettent.

scholarly journals The complexity of computing Kronecker coefficients

2008 ◽  
Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽  
Author(s):  
Peter Bürgisser ◽  
Christian Ikenmeyer
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Irreducible Representations ◽  
Schur Polynomials ◽  
Product Decomposition ◽  
Nous Montrons ◽  
Produit Tensoriel ◽  
International Audience ◽  
Kronecker Coefficients

International audience Kronecker coefficients are the multiplicities in the tensor product decomposition of two irreducible representations of the symmetric group $S_n$. They can also be interpreted as the coefficients of the expansion of the internal product of two Schur polynomials in the basis of Schur polynomials. We show that the problem $\mathrm{KRONCOEFF}$ of computing Kronecker coefficients is very difficult. More specifically, we prove that $\mathrm{KRONCOEFF}$ is #$\mathrm{P}$-hard and contained in the complexity class $\mathrm{GapP}$. Formally, this means that the existence of a polynomial time algorithm for $\mathrm{KRONCOEFF}$ is equivalent to the existence of a polynomial time algorithm for evaluating permanents. Les coefficients de Kronecker sont les multiplicités dans la décomposition du produit tensoriel de deux représentations irréductibles du groupe symétrique. On peut aussi les voir comme les coefficients du développement du produit interne des polynômes de Schur. Nous montrons que le problème $\mathrm{KRONCOEFF}$ de calculer les coefficients de Kronecker est très difficile. Plus précisément, nous prouvons que $\mathrm{KRONCOEFF}$ est #$\mathrm{P}$-dur et que $\mathrm{KRONCOEFF}$ est dans la classe de complexité $\mathrm{GapP}$. Cela veut dire qu'il existe un algorithme pour $\mathrm{KRONCOEFF}$ s'exécutant en temps polynomial si et seulement s'il existe un algorithme pour l'évaluation du permanent s'exécutant en temps polynomial.

scholarly journals A diagrammatic approach to Kronecker squares

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Ernesto Vallejo
Keyword(s):  
Polynomial Function ◽  
Young Tableaux ◽  
Piecewise Polynomial ◽  
Piecewise Polynomial Function ◽  
Isomorphism Classes ◽  
Nous Montrons ◽  
Il Y A ◽  
International Audience ◽  
Kronecker Coefficients ◽  
Standard Young Tableaux

International audience In this paper we improve a method of Robinson and Taulbee for computing Kronecker coefficients and show that for any partition $\overline{ν}$ of $d$ there is a polynomial $k_{\overline{ν}}$ with rational coefficients in variables $x_C$, where $C$ runs over the set of isomorphism classes of connected skew diagrams of size at most $d$, such that for all partitions $\lambda$ of $n$, the Kronecker coefficient $\mathsf{g}(\lambda, \lambda, (n-d, \overline{ν}))$ is obtained from $k_{\overline{ν}}(x_C)$ substituting each $x_C$ by the number of $\lambda$-removable diagrams in $C$. We present two applications. First we show that for $\rho_{k} = (k, k-1,\ldots, 2, 1)$ and any partition $\overline{ν}$ of size $d$ there is a piecewise polynomial function $s_{\overline{ν}}$ such that $\mathsf{g}(\rho_k, \rho_k, (|\rho_k| - d, \overline{ν})) = s_{\overline{ν}} (k)$ for all $k$ and that there is an interval of the form $[c, \infty)$ in which $s_{\overline{ν}}$ is polynomial of degree $d$ with leading coefficient the number of standard Young tableaux of shape $\overline{ν}$. The second application is new stability property for Kronecker coefficients. Dans ce papier nous améliorons une méthode de Robinson-Taulbee pour calculer les coefficients de Kronecker et montrons que pour toute partition $\overline{ν}$ de $d$ il y a un polynôme $k_{\overline{ν}}$ avec coefficients rationnels dans les variables $x_C$, où $C$ est dans l’ensemble de classes d’isomorphisme des diagrammes gauches connexes de taille non plus que $d$, tel que pour toute partition $\lambda$ de $n$, le coefficient de Kronecker $\mathsf{g}(\lambda, \lambda, (n-d, \overline{ν}))$ est obtenu de $k_{\overline{ν}}(x_C)$ en substituant chaque $x_C$ pour le nombre de diagrammes $\lambda$-removables en $C$. Nous présentons deux applications. Premièrement nous montrons que pour $\rho_{k} = (k, k-1,\ldots, 2, 1)$ et une partition $\overline{ν}$ de taille $d$ il y a une fonction polynôme par morceaux $s_{\overline{ν}}$ tel que pour toute $k$ on a $\mathsf{g}(\rho_k, \rho_k, (|\rho_k| - d, \overline{ν})) = s_{\overline{ν}} (k)$ et qu'il y a une intervalle de la forme $[c, \infty)$ dans laquelle $s_{\overline{ν}}$ est polynôme de degré $d$ avec coefficient principal le nombre de tableaux de Young standard de forme $\overline{ν}$. La seconde application est une nouveau propriété de stabilité des coefficients de Kronecker.

scholarly journals Stability of Kronecker coefficients via discrete tomography (Extended abstract)

2015 ◽  
Vol DMTCS Proceedings, 27th... (Proceedings) ◽  
Author(s):  
Ernesto Vallejo
Keyword(s):  
Asymptotic Properties ◽  
Sufficient Condition ◽  
Discrete Tomography ◽  
Previous Approach ◽  
General Stability ◽  
Special Cases ◽  
International Audience ◽  
Kronecker Coefficients ◽  
Cette Notion

International audience In this paper we give a sufficient condition for a general stability of Kronecker coefficients, which we call additive stability. Its main ingredient is the property of a matrix of being additive. This notion seems to be an important one: it appears in Discrete Tomography as a sufficient condition to uniqueness; it also appears in Manivel’s study of asymptotic properties of plethysm through Borel-Weil theory. The proof sketched here combines several results of the author on integer matrices motivated by Discrete Tomography with a new idea of Stembridge, that permits to bound some sequences of Kronecker coefficients. The advantage of additivity with respect to the previous approach by Stembridge is that it is very easy to produce new examples of additive matrices and, therefore, to produce many new examples of stability of Kronecker coefficients. We also show that Murnaghan’s stability property and other instances of stability discovered previously by the author are special cases of additive stability. Besides, our approach permits us to disprove a recent conjecture of Stembridge and to give a new characterization of additivity. Dans ce papier nous donnons une condition suffisant pour la stabilité générale des coefficients de Kronecker, que nous appelons stabilité additive. L'ingrédient principal est la propriété d’une matrice d'être additif. Cette notion est apparemment d’importance: elle apparaît en Tomographie Discrète comme une condition suffisant pour unicité; elle apparaît aussi dans l’étude de Manivel de propriétés asymptotiques du pléthysme par moyen de la théorie de Borel-Weil. La démonstration esquissée ici combine plusieurs résultats de l’auteur sur les matrices à coefficients entiers stimulés pour la Tomographie Discrète avec une nouvelle idée de Stembridge, qui permet de borner quelques successions des coefficients de Kronecker. L’avantage de notre méthode sur l’approche de Stembridge est qu’il est très facile de produire nouveaux exemples de matrices additives, et ainsi, de nouveaux exemples de stabilité des coefficients de Kronecker. Nous démontrons aussi que la stabilité de Murnaghan et d’autres exemples de stabilité trouvés antérieurement par l’auteur sont des cas spéciaux de la stabilité additive. En plus, avec notre approche nous réfutons une conjecture de Stembridge et donnons une nouvelle caractérisation d’additivité.

scholarly journals Kronecker coefficients: the tensor square conjecture and unimodality

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Igor Pak ◽  
Greta Panova ◽  
Ernesto Vallejo
Keyword(s):  
Irreducible Representation ◽  
Representation Theory ◽  
Binomial Coefficients ◽  
Sufficient Condition ◽  
International Audience ◽  
Kronecker Coefficients ◽  
Tensor Square ◽  
Nous Utilisons

International audience We consider two aspects of Kronecker coefficients in the directions of representation theory and combinatorics. We consider a conjecture of Jan Saxl stating that the tensor square of the $S_n$-irreducible representation indexed by the staircase partition contains every irreducible representation of $S_n$. We present a sufficient condition allowing to determine whether an irreducible representation is a constituent of a tensor square and using this result together with some analytic statements on partitions we prove Saxl conjecture for several partition classes. We also use Kronecker coefficients to give a new proof and a generalization of the unimodality of Gaussian ($q$-binomial) coefficients as polynomials in $q$, and extend this to strict unimodality. Nous considérons deux aspects des coefficients de Kronecker dans le domaine de la Théorie des Représentations et le domaine Combinatoire. Nous considérons la conjecture suivante de Jan Saxl: le tenseur au carré de la représentation irréductible du groupe $S_n$ indexée par la partition $S_n (n= \left( \begin{array}{cc} k+1 \\ 2 \end{array} \right))$. Nous présentons une condition suffisante qui permet de déterminer si une représentation irréductible est une constituante d’un tenseur au carré. En utilisant ce résultat avec des résultats analytiques sur les partitions, nous prouvons la conjecture de Saxl pour plusieurs classes de partitions. Nous utilisons aussi les coefficients de Kronecker pour donner une nouvelle preuve et une généralisation de l’unimodalité des coefficients de Gauss ($q$-binomiaux) comme polynômes en $q$ et nous étendons cela à l’unimodalité stricte.

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.

Somethin’ about Scandinavia: Danish Shorts on the Post-war International Scene

2018 ◽  
Author(s):  
C. Claire Thomson
Keyword(s):  
United Nations ◽  
International Collaboration ◽  
Marshall Plan ◽  
The Third ◽  
Animated Film ◽  
International Audience ◽  
Post War ◽  
Nordic Cooperation

Building on the picture of post-war Anglo-Danish documentary collaboration established in the previous chapter, this chapter examines three cases of international collaboration in which Dansk Kulturfilm and Ministeriernes Filmudvalg were involved in the late 1940s and 1950s. They Guide You Across (Ingolf Boisen, 1949) was commissioned to showcase Scandinavian cooperation in the realm of aviation (SAS) and was adopted by the newly-established United Nations Film Board. The complexities of this film’s production, funding and distribution are illustrative of the activities of the UN Film Board in its first years of operation. The second case study considers Alle mine Skibe (All My Ships, Theodor Christensen, 1951) as an example of a film commissioned and funded under the auspices of the Marshall Plan. This US initiative sponsored informational films across Europe, emphasising national solutions to post-war reconstruction. The third case study, Bent Barfod’s animated film Noget om Norden (Somethin’ about Scandinavia, 1956) explains Nordic cooperation for an international audience, but ironically exposed some gaps in inter-Nordic collaboration in the realm of film.

Conclusion

2018 ◽  
Author(s):  
Alistair Fox
Keyword(s):  
National Identity ◽  
New Zealand ◽  
Identity Formation ◽  
Essential Role ◽  
Coming Of Age ◽  
Financial Returns ◽  
Self Recognition ◽  
International Audience ◽  
National Identity Formation

The conclusion reaffirms the essential role played by cinema generally, and the coming-of-age genre in particular, in the process of national identity formation, because of its effectiveness in facilitating self-recognition and self-experience through a process of triangulation made possible, for the most part, by a dialogue with some of the nation’s most iconic works of literature. This section concludes by point out the danger posed, however, by an observable trend toward generic standardization in New Zealand films motivated by a desire to appeal to an international audience out of consideration for the financial returns expected by funding bodies under current regimes.

Early Cinema in Scotland

2018 ◽  
Keyword(s):  
Social History ◽  
Small Towns ◽  
Early Cinema ◽  
Popular Song ◽  
Feature Film ◽  
Local Newspapers ◽  
The Social ◽  
Production History ◽  
History Of ◽  
International Audience

This collection of essays, drawn from a three-year AHRC research project, provides a detailed context for the history of early cinema in Scotland from its inception in 1896 till the arrival of sound in the early 1930s. It details the movement from travelling fairground shows to the establishment of permanent cinemas, and from variety and live entertainment to the dominance of the feature film. It addresses the promotion of cinema as a socially ‘useful’ entertainment, and, distinctively, it considers the early development of cinema in small towns as well as in larger cities. Using local newspapers and other archive sources, it details the evolution and the diversity of the social experience of cinema, both for picture goers and for cinema staff. In production, it examines the early attempts to establish a feature film production sector, with a detailed production history of Rob Roy (United Films, 1911), and it records the importance, both for exhibition and for social history, of ‘local topicals’. It considers the popularity of Scotland as an imaginary location for European and American films, drawing their popularity from the international audience for writers such as Walter Scott and J.M. Barrie and the ubiquity of Scottish popular song. The book concludes with a consideration of the arrival of sound in Scittish cinemas. As an afterpiece, it offers an annotated filmography of Scottish-themed feature films from 1896 to 1927, drawing evidence from synopses and reviews in contemporary trade journals.

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