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 $G\wr S_n$.

A Remark by Philip Hall

1958 ◽  
Vol 1 (1) ◽  
pp. 21-23 ◽  
Author(s):  
G. de B. Robinson
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Linear Group ◽  
Characteristic Zero ◽  
Irreducible Representations ◽  
Young Diagrams ◽  
Linear Transformations ◽  
Modern Physics ◽  
The Right ◽  
The Relationship

The relationship between the representation theory of the full linear group GL(d) of all non-singular linear transformations of degree d over a field of characteristic zero and that of the symmetric group Sn goes back to Schur and has been expounded by Weyl in his classical groups, [4; cf also 2 and 3]. More and more, the significance of continuous groups for modern physics is being pressed on the attention of mathematicians, and it seems worth recording a remark made to the author by Philip Hall in Edmonton.As is well known, the irreducible representations of Sn are obtainable from the Young diagrams [λ]=[λ1, λ2 ,..., λr] consisting of λ1 nodes in the first row, λ2 in the second row, etc., where λ1≥λ2≥ ... ≥λr and Σ λi = n. If we denote the jth node in the ith row of [λ] by (i,j) then those nodes to the right of and below (i,j), constitute, along with the (i,j) node itself, the (i,j)-hook of length hij.

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 $\nu$.

scholarly journals A Specht Module Analog for the Rook Monoid

10.37236/1619 ◽  
2001 ◽  
Vol 9 (1) ◽  
Author(s):  
Cheryl Grood
Keyword(s):  
Representation Theory ◽  
Symmetric Group ◽  
Algebraic Structure ◽  
Irreducible Representations ◽  
Specht Module ◽  
Specht Modules ◽  
Rook Monoid ◽  
Combinatorial Construction

The wealth of beautiful combinatorics that arise in the representation theory of the symmetric group is well-known. In this paper, we analyze the representations of a related algebraic structure called the rook monoid from a combinatorial angle. In particular, we give a combinatorial construction of the irreducible representations of the rook monoid. Since the rook monoid contains the symmetric group, it is perhaps not surprising that the construction outlined in this paper is very similar to the classic combinatorial construction of the irreducible $S_n$-representations: namely, the Specht modules.

scholarly journals Restricting Supercharacters of the Finite Group of Unipotent Uppertriangular Matrices

10.37236/112 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Nathaniel Thiem ◽  
Vidya Venkateswaran
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Finite Field ◽  
Symmetric Group ◽  
Irreducible Representations ◽  
Combinatorial Theory ◽  
Set Partitions ◽  
Triangular Matrices ◽  
Upper Triangular Matrices ◽  
Supercharacter Theory

It is well-known that understanding the representation theory of the finite group of unipotent upper-triangular matrices $U_n$ over a finite field is a wild problem. By instead considering approximately irreducible representations (supercharacters), one obtains a rich combinatorial theory analogous to that of the symmetric group, where we replace partition combinatorics with set-partitions. This paper studies the supercharacter theory of a family of subgroups that interpolate between $U_{n-1}$ and $U_n$. We supply several combinatorial indexing sets for the supercharacters, supercharacter formulas for these indexing sets, and a combinatorial rule for restricting supercharacters from one group to another. A consequence of this analysis is a Pieri-like restriction rule from $U_n$ to $U_{n-1}$ that can be described on set-partitions (analogous to the corresponding symmetric group rule on partitions).

Some combinatorial results involving Young diagrams

1978 ◽  
Vol 83 (1) ◽  
pp. 1-10 ◽  
Author(s):  
G. D. James
Keyword(s):  
Irreducible Representation ◽  
Representation Theory ◽  
Symmetric Group ◽  
Young Diagram ◽  
New Method ◽  
Irreducible Representations ◽  
Young Diagrams ◽  
Sufficient Criterion ◽  
Power Diagram ◽  
Necessary And Sufficient

In the first half of this paper we introduce a new method of examining the q-hook structure of a Young diagram, and use it to prove most of the standard results about q-cores and q-quotients. In particular, we give a quick new proof of Chung's Conjecture (2), which determines the number of diagrams with a given q-weight and says how many of them are q-regular. In the case where q is prime, this tells us how many ordinary and q-modular irreducible representations of the symmetric group there are in a given q-block. None of the results of section 2 is original. In the next section we give a new definition, the p-power diagram, which is closely connected with the p-quotient. This concept is interesting because when p is prime a condition involving the p-power diagram appears to be a necessary and sufficient criterion for the diagram to be p-regular and the corresponding ordinary irreducible representation of to remain irreducible modulo p. In this paper we derive combinatorial results involving the p-power diagram, and in a later article we investigate the relevant representation theory.

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

scholarly journals A Unified Reproducing Kernel Method and Error Estimation for Solving Linear Differential Equation with Functional Constraints

2015 ◽  
Vol 2015 ◽  
pp. 1-9
Author(s):  
Xinjian Zhang ◽  
Xiongwei Liu
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Linear Differential Equation ◽  
Error Estimation ◽  
Reproducing Kernel ◽  
Kernel Method ◽  
Inner Product ◽  
Reproducing Kernel Method ◽  
Linear Differential ◽  
Polynomial Reproducing

A unified reproducing kernel method for solving linear differential equations with functional constraint is provided. We use a specified inner product to obtain a class of piecewise polynomial reproducing kernels which have a simple unified description. Arbitrary order linear differential operator is proved to be bounded about the special inner product. Based on space decomposition, we present the expressions of exact solution and approximate solution of linear differential equation by the polynomial reproducing kernel. Error estimation of approximate solution is investigated. Since the approximate solution can be described by polynomials, it is very suitable for numerical calculation.

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