scholarly journals Position Sequences and a $q$-Analogue for the Modular Hook Length Formula

10.37236/8685 ◽  
2019 ◽  
Vol 26 (4) ◽  
Author(s):  
Anthony Mendes ◽  
Sam Lindbloom-Airey
Keyword(s):  
Symmetric Group ◽  
Irreducible Representations ◽  
Combinatorial Interpretation ◽  
Hook Length

We prove a $q$-analogue of the modular hook length formula using position sequences. These position sequences, which correspond to moving the beads in a mathematical abacus, provide a new combinatorial interpretation for the characters of the irreducible representations of the symmetric group.

The symmetric group and the method of diatomics in molecules: an application to small lithium clusters

1973 ◽  
Vol 333 (1592) ◽  
pp. 69-87 ◽  
Keyword(s):  
Alkali Metal ◽  
Symmetric Group ◽  
Metal Clusters ◽  
Basis Functions ◽  
Basis Set ◽  
Irreducible Representations ◽  
Secular Equations ◽  
Alkali Metal Clusters ◽  
Lithium Clusters

A new ‘most economical’ algorithm for the construction of diatomics in molecules secular equations is described. The method does not require the basis functions to be written down explicitly, since overlap may be factored out of the equations entirely. The theory is presented in detail for the particular case of homogeneous alkali metal clusters. A knowledge of the irreducible representations of the symmetric group for the Jahn-Serber basis set is necessary. The irreducible representations are derived by a genealogical procedure. Some preliminary calculations are presented for the molecules Li 3 through Li 6 , Li + 3 and Li + 4 . The lithium clusters are found to be stable with respect to all possible dissociations, and the i.ps of Li 3 and Li 4 are in agreement with the trends for the species Na 3 , Na 4 , K 3 , K 4 , etc., whose i.ps have been measured experimentally.

scholarly journals A $\lambda$-ring Frobenius Characteristic for $G\wr S_n$

10.37236/1809 ◽  
2004 ◽  
Vol 11 (1) ◽  
Author(s):  
Anthony Mendes ◽  
Jeffrey Remmel ◽  
Jennifer Wagner
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Kronecker Product ◽  
Reproducing Kernel ◽  
Irreducible Representations ◽  
Inner Product ◽  
Irreducible Components ◽  
Lambda Ring

A $\lambda$-ring version of a Frobenius characteristic for groups of the form $G \wr S_n$ is given. Our methods provide natural analogs of classic results in the representation theory of the symmetric group. Included is a method decompose the Kronecker product of two irreducible representations of $G\wr S_n$ into its irreducible components along with generalizations of the Murnaghan-Nakayama rule, the Hall inner product, and the reproducing kernel for $G\wr S_n$.

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.

Frobenius Symbols and the Groups SsGL(n), O(n) and Sp(n)

1973 ◽  
Vol 25 (5) ◽  
pp. 941-959 ◽  
Author(s):  
Y. J. Abramsky ◽  
H. A. Jahn ◽  
R. C. King
Keyword(s):  
Symmetric Group ◽  
Irreducible Representations ◽  
Image Position

Frobenius [2; 3] introduced the symbolsto specify partitions and the corresponding irreducible representations of the symmetric group Ss.

On the Disjoint Product of Irreducible Representations of the Symmetric Group

1949 ◽  
Vol 1 (2) ◽  
pp. 166-175 ◽  
Author(s):  
G. de B. Robinson
Keyword(s):  
Symmetric Group ◽  
Linear Group ◽  
Invariant Theory ◽  
Schur Functions ◽  
Irreducible Representations ◽  
Invariant Matrices ◽  
Disjoint Product

The results of the present paper can be interpreted (a) in terms of the theory of the representations of the symmetric group, or (b) in terms of the corresponding theory of the full linear group. In the latter connection they give a solution to the problem of the expression of an invariant matrix of an invariant matrix as a sum of invariant matrices, in the sense of Schur's Dissertation. D. E. Littlewood has pointed out the significance of this problem for invariant theory and has attacked it via Schur functions, i.e. characters of the irreducible representations of the full linear group. We shall confine our attention here to the interpretation (a).

On the minimal dimensions of irreducible representations of symmetric groups

1983 ◽  
Vol 94 (3) ◽  
pp. 417-424 ◽  
Author(s):  
G. D. James
Keyword(s):  
Symmetric Group ◽  
Symmetric Groups ◽  
Irreducible Representations ◽  
Arbitrary Field ◽  
Precise Information

For each integer m, Rasala [6] has shown how to list all the ordinary irreducible representations of the symmetric group n which have degree less than nm, provided that n is large enough, and in this note we shall prove similar results for the irreducible representations of n over an arbitrary field K. Our estimates are very crude, so although we recover Rasala's results, we get nowhere near his precise information on how large n has to be.

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.

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