Dimensions of skew-shifted young diagrams and projective characters of the infinite symmetric group

1999 ◽  
Vol 96 (5) ◽  
pp. 3517-3530 ◽  
Author(s):  
V. N. Ivanov
Keyword(s):  
Symmetric Group ◽  
Young Diagrams ◽  
Infinite Symmetric Group

scholarly journals Fluctuations of central measures on partitions

2012 ◽  
Vol DMTCS Proceedings vol. AR,... (Proceedings) ◽  
Author(s):  
Pierre-Loïc Mèliot
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Symmetric Group ◽  
Central Limit ◽  
Symmetric Groups ◽  
Polynomial Functions ◽  
Young Diagrams ◽  
Random Partitions ◽  
Infinite Symmetric Group ◽  
International Audience

International audience We study the fluctuations of models of random partitions $(\mathbb{P}_n,ω )_n ∈\mathbb{N}$ stemming from the representation theory of the infinite symmetric group. Using the theory of polynomial functions on Young diagrams, we establish a central limit theorem for the values of the irreducible characters $χ ^λ$ of the symmetric groups, with $λ$ taken randomly according to the laws $\mathbb{P}_n,ω$ . This implies a central limit theorem for the rows and columns of the random partitions, and these ``geometric'' fluctuations of our models can be recovered by relating central measures on partitions, generalized riffle shuffles, and Brownian motions conditioned to stay in a Weyl chamber. Nous étudions les fluctuations de modèles de partitions aléatoires $(\mathbb{P}_n,ω )_n ∈\mathbb{N}$ issus de la théorie des représentations du groupe symétrique infini. En utilisant la théorie des fonctions polynomiales sur les diagrammes de Young, nous établissons un théorème central limite pour les valeurs des caractères irréductibles $χ ^λ$ des groupes symétriques, avec $λ$ pris aléatoirement suivant les lois $\mathbb{P}_n,ω$ . Ceci implique un théorème central limite pour les lignes et les colonnes des partitions aléatoires, et ces fluctuations ``géométriques'' de nos modèles peuvent être retrouvées en reliant les mesures centrales sur les partitions, les battages généralisés de cartes, et les mouvements browniens conditionnés à rester dans une chambre de Weyl.

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.

scholarly journals Certain unitary representations of the infinite symmetric group, II

1987 ◽  
Vol 106 ◽  
pp. 143-162 ◽  
Author(s):  
Nobuaki Obata
Keyword(s):  
Symmetric Group ◽  
Discrete Group ◽  
Inductive Limit ◽  
Symmetric Groups ◽  
Discrete Groups ◽  
Type I ◽  
Infinite Symmetric Group ◽  
Locally Finite ◽  
Distinctive Features ◽  
Infinite Dimensional

The infinite symmetric group is the discrete group of all finite permutations of the set X of all natural numbers. Among discrete groups, it has distinctive features from the viewpoint of representation theory and harmonic analysis. First, it is one of the most typical ICC-groups as well as free groups and known to be a group of non-type I. Secondly, it is a locally finite group, namely, the inductive limit of usual symmetric groups . Furthermore it is contained in infinite dimensional classical groups GL(ξ), O(ξ) and U(ξ) and their representation theories are related each other.

Some combinatorial results involving shifted Young diagrams

1986 ◽  
Vol 99 (1) ◽  
pp. 23-31 ◽  
Author(s):  
A. O. Morris ◽  
A. K. Yaseen
Keyword(s):  
Symmetric Group ◽  
Block Structure ◽  
Young Diagrams ◽  
Linear Representations ◽  
Projective Representations

In [6] the first author introduced some combinatorial concepts involving Young diagrams corresponding to partitions with distinct parts and applied them to the projective representations of the symmetric group Sn. A conjecture concerning the p-block structure of the projective representations of Sn was formulated in terms of these concepts which corresponds to the well-known, but long proved, Nakayama ‘conjecture’ for the p-block structure of the linear representations of Sn. This conjecture has recently been proved by Humphreys [1].

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