No description available.

We provide a combinatorial interpretation of the frequency of any irreducible representation of Sn in representations of Sn arising from group actions on words. Recognizing that representations arising from group actions naturally split across orbits yields combinatorial interpretations of the irreducible decompositions of representations from similar group actions. The generalization from group actions on words to group actions on matrices gives rise to representations that prove to be much less transparent. We share the progress made thus far on the open problem of determining the irreducible decomposition of certain representations of Sm × Sn arising from group actions on matrices.

No description available.

No description available.

Disertacijoje nagrinėjamos atsitiktinių keitinių problemos yra priskirtinos tikimybinei kombinatorikai. Gauti rezultatai aprašo visiškai adityviųjų funkcijų, apibrėžtų simetrinėje grupėje, reikšmių asimptotinius skirstinius Evenso tikimybinio mato atžvilgiu, kai grupės eilė neaprėžtai didėja. Išvestos adityviųjų funkcijų laipsninių ir faktorialinių momentų formulės. Funkcijų, išreiškiančių atsitiktinio keitinio ciklų su bet kokiais apribojimais skaičius, atveju rastos būtinos ir pakankamos ribinių tikimybinių dėsnių egzistavimo sąlygos. Išsamiai išnagrinėtas konvergavimas į Puasono, quasi-Puasono, Bernulio, binominio ir kitus skirstinius, sukoncentruotus sveikųjų neneigiamų skaičių aibėje. Rezultatai apibendrinti sveikareikšmių visiškai adityviųjų funkcijų klasėje. Darbe įrodytas bendras silpnasis didžiųjų skaičių dėsnis, rastos būtinos ir pakankamos adityviųjų funkcijų sekų pasiskirstymo funkcijų konvergavimo į išsigimusį nuliniame taške dėsnį egzistavimo sąlygos. Sprendžiamos problemos yra susietos su tikimybiniais vektorių, turinčių sveikąsias neneigiamas koordinates, uždaviniais. Adicinėje tokių vektorių pusgrupėje išnagrinėti multiplikatyviųjų funkcijų vidurkiai tikimybinio mato, vadinamo Evanso atrankos formule, atžvilgiu. Gauti tikslūs viršutinieji ir apatinieji įverčiai. Iš jų išplaukia svarbios atsitiktinių keitinių tikimybių savybės. Disertacijoje plėtojami faktorialinių momentų ir kiti kombinatoriniai bei tikimybiniai metodai. / In the thesis the examining problems of random permutations are attributed to the probabilistic combinatorics. Obtained results describe asymptotical distributions of completely additive functions values defined on a symmetric group with respect to Ewens probability measure, if the group order unbounded increases. Power and factorial moments formulae of additive functions are derived. There are established necessary and sufficient conditions under which the distributions of a number of cycles with restricted lengths obey the limit probability laws. The convergence to the Poisson, quasi-Poisson, Bernoulli, binomial and other distributions, defined on the positive whole - number set are exhaustively investigated. The results are generalized on the class of whole - number completely additive functions. The general weak law of large numbers is proved in the thesis, necessary and sufficient existence conditions, under which the distributions of the sequences of additive functions converge to the degenerate at the point zero limit law are established. Examining problems are related to the probability tasks of the vectors, which have whole - numbered nonnegative coordinates. The mean values of multiplicative functions defined on those vectors’ additive semigroup with respect to the Ewens measure, called Ewens Sampling Formula, and investigated. Lower and upper sharp estimates are obtained. From the latter results follow important probabilities’ properties of random... [to full text]

Submitted by Johnny Rodrigues (johnnyrodrigues@ufcg.edu.br) on 2018-07-27T15:29:31Z No. of bitstreams: 1 ANTONIO IGOR SILVA DE OLIVEIRA - DISSERTAÇÃO PPGMAT 2011..pdf: 599013 bytes, checksum: 2ae31549fdd89221db237ef278b5a688 (MD5) / Made available in DSpace on 2018-07-27T15:29:31Z (GMT). No. of bitstreams: 1 ANTONIO IGOR SILVA DE OLIVEIRA - DISSERTAÇÃO PPGMAT 2011..pdf: 599013 bytes, checksum: 2ae31549fdd89221db237ef278b5a688 (MD5) Previous issue date: 2011-09 / Capes / As ideias de codimensões e cocaracteres de uma PI-álgebra são de grande importância e são centrais nas aplicações das representações dos grupos simétricos à PIteoria (teoria das identidades polinomiais). Os conceitos de codimensão e cocaracter começaram a ser estudados em 1972 por Amitai Regev em seu importante trabalho sobre identidades polinomiais do produto tensorial de PI-álgebras. Ao longo das últimas décadas muitos resultados importantes surgiram com o uso das representações e dos métodos assintóticos na PI-teoria. Neste trabalho apresentaremos inicialmente ideias e resultados básicos da Teoria de Young sobre as representações dos grupos simétricos. De posse desses resultados, estudaremos as sequências limitadas de codimensões e as sequências de cocaracteres de álgebras que satisfazem alguma identidade de Capelli. Apresentaremos também os cálculos das codimensões e dos cocaracteres da álgebra de Grassmann. / The ideas of codimensions and cocharacters of a PI-algebra are of great and central importance in the applications of representations of symmetric groups to PI-theory (theory of the polynomial identities). The study of the concepts of codimensions and cocharacters started in 1972 by Amitai Regev in his important work about polynomial identities of the tensor product of PI-algebras. During the last decades many important results arose with the use of representations and asymptotic methods in PI-theory. In this work we will present firstly ideas and basic results in the Young’s theory about the representations of symmetric groups. With these results we shall study the limited sequences of codimensions and the cocharacter sequences of algebras that satisfy some of the Capelli identity. It will also be presented the calculation of the codimensions and cocharacters of the Grassmann Algebra.

Šis darbas yra skirtas šiuolaikinės atomo trikdžių teorijos matematinio aparato, paremto efektinių operatorių formalizmu, plėtojimui. Darbe nuosekliai ir sistemingai, pradedant nuo pačių bendriausių principų, nagrinėjami Foko erdvės apribojimo į redukavimo grupių neredukuotinus poerdvius metodai bei pateikiama neredukuotinų tenzorinių operatorių, charakterizuojančių fizikines ir efektines sąveikas, klasifikacija bendrais ir tam tikrais atskirais atvejais. Gautos išraiškos ir iš jų išplaukiančios išvados yra grindžiamos matematine kalba. Dauguma esminių rezultatų yra suformuluoti teoremų pavidalu. Disertaciją sudaro 101 puslapis, 5 skyriai, 4 priedai, 40 lentelių ir 9 paveikslėliai. Pagrindiniai rezultatai, pateikti disertacijoje, yra publikuoti fizikos ir matematikos mokslų žurnaluose. / The principal goals of the thesis are subjected to general methods and forms of effective operators by the nowadays demands of theoretical application of many-body perturbation theory to atomic physics. The present theoretical research follows up step by step by systematic observation of various possibilities to restrict the Fock space operators to their irreducible subspaces and the classification of irreducible tensor operators which represent the physical as well as the effective interactions. To ground the results of the thesis, the symbolic preparation of obtained expressions is strictly proved mathematically. Most of the main results are listed in theorems. The doctoral dissertation contains 101 pages, 5 sections, 4 appendices, 40 tables and 9 figures. The main results described in the present dissertation have been published in journals of physical and mathematical sciences.

The principal goals of the thesis are subjected to general methods and forms of effective operators by the nowadays demands of theoretical application of many-body perturbation theory to atomic physics. The present theoretical research follows up step by step by systematic observation of various possibilities to restrict the Fock space operators to their irreducible subspaces and the classification of irreducible tensor operators which represent the physical as well as the effective interactions. To ground the results of the thesis, the symbolic preparation of obtained expressions is strictly proved mathematically. Most of the main results are listed in theorems. The doctoral dissertation contains 101 pages, 5 sections, 4 appendices, 40 tables and 9 figures. The main results described in the present dissertation have been published in journals of physical and mathematical sciences. / Šis darbas yra skirtas šiuolaikinės atomo trikdžių teorijos matematinio aparato, paremto efektinių operatorių formalizmu, plėtojimui. Darbe nuosekliai ir sistemingai, pradedant nuo pačių bendriausių principų, nagrinėjami Foko erdvės apribojimo į redukavimo grupių neredukuotinus poerdvius metodai bei pateikiama neredukuotinų tenzorinių operatorių, charakterizuojančių fizikines ir efektines sąveikas, klasifikacija bendrais ir tam tikrais atskirais atvejais. Gautos išraiškos ir iš jų išplaukiančios išvados yra grindžiamos matematine kalba. Dauguma esminių rezultatų yra suformuluoti teoremų pavidalu. Disertaciją sudaro 101 puslapis, 5 skyriai, 4 priedai, 40 lentelių ir 9 paveikslėliai. Pagrindiniai rezultatai, pateikti disertacijoje, yra publikuoti fizikos ir matematikos mokslų žurnaluose.

Cette thèse se situe dans le domaine de la combinatoire algébrique et porte sur l'étude et les applications de trois ordres sur les permutations : les deux ordres faibles (gauche et droit) et l'ordre fort ou de Bruhat. Dans un premier temps, nous étudions l'action du groupe symétrique sur les polynômes multivariés. En particulier, les opérateurs de emph{différences divisées} permettent de définir des bases de l'anneau des polynômes qui généralisent les fonctions de Schur aussi bien du point de vue de leur construction que de leur interprétation géométrique. Nous étudions plus particulièrement la base des polynômes de Grothendieck introduite par Lascoux et Schützenberger. Lascoux a montré qu'un certain produit de polynômes peut s'interpréter comme un produit d'opérateurs de différences divisées. En développant ce produit, nous ré-obtenons un résultat de Lenart et Postnikov et prouvons de plus que le produit s'interprète comme une somme sur un intervalle de l'ordre de Bruhat. Nous présentons aussi l'implantation que nous avons réalisée sur Sage des polynômes multivariés. Cette implantation permet de travailler formellement dans différentes bases et d'effecteur des changements de bases. Elle utilise l'action des différences divisées sur les vecteurs d'exposants des polynômes multivariés. Les bases implantées contiennent en particulier les polynômes de Schubert, les polynômes de Grothendieck et les polynômes clés (ou caractères de Demazure).Dans un second temps, nous étudions le emph{treillis de Tamari} sur les arbres binaires. Celui-ci s'obtient comme un quotient de l'ordre faible sur les permutations : à chaque arbre est associé un intervalle de l'ordre faible formé par ses extensions linéaires. Nous montrons qu'un objet plus général, les intervalles-posets, permet de représenter l'ensemble des intervalles du treillis de Tamari. Grâce à ces objets, nous obtenons une formule récursive donnant pour chaque arbre binaire le nombre d'arbres plus petits ou égaux dans le treillis de Tamari. Nous donnons aussi une nouvelle preuve que la fonction génératrice des intervalles de Tamari vérifie une certaine équation fonctionnelle décrite par Chapoton. Enfin, nous généralisons ces résultats aux treillis de $m$-Tamari. Cette famille de treillis introduite par Bergeron et Préville-Ratelle était décrite uniquement sur les chemins. Nous en donnons une interprétation sur une famille d'arbres binaires en bijection avec les arbres $m+1$-aires. Nous utilisons cette description pour généraliser les résultats obtenus dans le cas du treillis de Tamari classique. Ainsi, nous obtenons une formule comptant le nombre d'éléments plus petits ou égaux qu'un élément donné ainsi qu'une nouvelle preuve de l'équation fonctionnelle des intervalles de $m$-Tamari. Pour finir, nous décrivons des structures algébriques $m$ qui généralisent les algèbres de Hopf $FQSym$ et $PBT$ sur les permutations et les arbres binaires / This thesis comes within the scope of algebraic combinatorics and studies problems related to three orders on permutations: the two said weak orders (right and left) and the strong order or Bruhat order.We first look at the action of the symmetric group on multivariate polynomials. By using the emph{divided differences} operators, one can obtain some generalisations of the Schur function and form bases of non symmetric multivariate polynomials. This construction is similar to the one of Schur functions and also allows for geometric interpretations. We study more specifically the Grothendieck polynomials which were introduced by Lascoux and Schützenberger. Lascoux proved that a product of these polynomials can be interpreted in terms of a product of divided differences. By developing this product, we reobtain a result of Lenart and Postnikov and also prove that it can be interpreted as a sum over an interval of the Bruhat order. We also present our implementation of multivariate polynomials in Sage. This program allows for formal computation on different bases and also implements many changes of bases. It is based on the action of the divided differences operators. The bases include Schubert polynomials, Grothendieck polynomials and Key polynomials. In a second part, we study the emph{Tamari lattice} on binary trees. This lattice can be obtained as a quotient of the weak order. Each tree is associated with the interval of its linear extensions. We introduce a new object called, emph{interval-posets} of Tamari and show that they are in bijection with the intervals of the Tamari lattice. Using these objects, we give the recursive formula counting the number of elements smaller than or equal to a given tree. We also give a new proof that the generating function of the intervals of the Tamari lattice satisfies some functional equation given by Chapoton. Our final contributions deals with the $m$-Tamari lattices. This family of lattices is a generalization of the classical Tamari lattice. It was introduced by Bergeron and Préville-Ratelle and was only known in terms of paths. We give the description of this order in terms of some family of binary trees, in bijection with $m+1$-ary trees. Thus, we generalize our previous results and obtain a recursive formula counting the number of elements smaller than or equal to a given one and a new proof of the functional equation. We finish with the description of some new $"m"$ Hopf algebras which are generalizations of the known $FQSym$ on permutations and $PBT$ on binary trees

Cette thèse est consacrée à l’étude combinatoire, algébrique et homologique des hyperarbres et des partitions semi-pointées. Nous étudions plus précisément des structures algébriques et homologiques construites à partir des hyperarbres, puis des partitions semi-pointées.Après un bref rappel des notions utilisées, nous utilisons la théorie des espèces de structure afin de déterminer l’action du groupe symétrique sur l’homologie du poset des hyperarbres. Cette action s’identifie à l’action du groupe symétrique liée à la structure anti-cyclique de l’opérade PreLie. Nous raffinons ensuite nos calculs sur une graduation de l’homologie, appelée homologie de Whitney. Cette étude motive l'introduction de la notion d’hyperarbre aux arêtes décorées par une espèce. Une bijection des hyperarbres décorés avec des arbres en boîtes et des partitions décorées permet d’obtenir une formule close pour leur cardinal, à l’aide d’un codage de Prüfer. Nous adaptons ensuite les méthodes de calcul de caractères sur les algèbres de Hopf d’incidence, introduites par W. Schmitt dans le cas de familles de posets bornés, à des familles de posets non bornés vérifiant certaines propriétés. Nous appliquons ensuite cette adaptation aux posets des hyperarbres. Enfin, au cours de notre étude une généralisation des posets des partitions et des posets des partitions pointées apparaît : les poset des partitions semi-pointées. Nous montrons que ces posets sont aussi Cohen-Macaulay, avant de déterminer à l’aide de la théorie des espèces une formule close pour la dimension de l’unique groupe d’homologie non trivial de ces posets / This thesis is dedicated to the combinatorial, algebraic and homological study of hypertrees and semi-pointed partitions. More precisely, we study algebraic and homological structures built from hypertrees and semi-pointed partitions. After recalling briefly the notions needed, we use the theory of species of structures to compute the action of the symmetric group on the homology of the hypertree posets. This action is the same as the action of the symmetric group linked with the anticyclic structure of the PreLie operad. We refine our computations on a grading of the homology : Whitney homology. This study is a motivation for the introduction of the notion of edge-decorated hypertrees. A one-to-one correspondence of decorated hypertrees with box trees and decorated partitions enables us to compute a close formula for the cardinality of decorated hypertrees, thanks to a Prüfer code. Moreover, we adapt computation methods of characters on incidence Hopf algebras, introduced by W. Schmitt for families of bounded posets, to families of unbounded posets satisfying some additional properties, called triangle and diamond posets. We apply these results to the hypertree posets. Finally, we unveil a new family of posets : the semi-pointed partition posets, which generalize both partition posets and pointed partition posets. We show the Cohen-Macaulayness of these posets and obtain, thanks to species theory, a closed formula for the dimension of its unique homology group, which extend the ones established for partition posets and pointed partition posets