The Analysis of the Kronecker Product of Irreducible Representations of the Symmetric Group

1938 ◽  
Vol 60 (3) ◽  
pp. 761 ◽  
Author(s):  
F. D. Murnaghan
Keyword(s):  
Symmetric Group ◽  
Kronecker Product ◽  
Irreducible Representations

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 On the Analysis of the Kronecker Product of Irreducible Representations of the Symmetric Group

1949 ◽  
Vol 8 (3) ◽  
pp. 133-137 ◽  
Author(s):  
Ragy H. Makar
Keyword(s):  
Symmetric Group ◽  
Kronecker Product ◽  
Irreducible Representations ◽  
Irreducible Matrix ◽  
Matrix Representations

The Kronecker product of two irreducible matrix representations D(λ), D(μ) of the symmetric group on n letters, furnishes a representation of that group, which is, in general reducible. The question of what irreducible representations will appear in the analysis of such products has been dealt with by Prof. F. D. Murnaghan. Indeed he has obtained the analysis of D(n − p, λ2, …) × D(n − q, μ2, …), for the particular values, p = 1, q = 1, 2, 3, 4, 5; p = 2, q = 2, 3, 4; p = 3, q = 3, 4, applying a method which is a recurrence one, in the sense that to obtain such an analysis we have to look at some other analyses which come first in order.

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 $\nu$.

High-Temperature Series Expansions for the Spin-½ Heisenberg Model by the Method of Irreducible Representations of the Symmetric Group

1964 ◽  
Vol 135 (5A) ◽  
pp. A1272-A1277 ◽  
Author(s):  
George A. Baker ◽  
G. S. Rushbrooke ◽  
H. E. Gilbert
Keyword(s):  
High Temperature ◽  
Symmetric Group ◽  
Heisenberg Model ◽  
Temperature Series ◽  
Irreducible Representations ◽  
Series Expansions

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 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.

Representations Subduced on an Ideal of a Lie Algebra

1962 ◽  
Vol 14 ◽  
pp. 293-303 ◽  
Author(s):  
B. Noonan
Keyword(s):  
Irreducible Representation ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Kronecker Product ◽  
Point Of View ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Representations Of Lie Algebras ◽  
The Ideal

This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).

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).

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