Keywords: standard representation, symmetric group, casimir invariant
Received August 14, 2015; Revised September 25, 2015; Accepted October 03, 2015
Copyright © 2015 Science and Education Publishing. All Rights Reserved.
1. Introduction
The symmetric group S_{n}, whose elements consist of the set of all permutations on n symbols is of central importance to mathematics and physics ^{[1]}. Cayley’s theorem states that every group is isomorphic to a subgroup of the symmetric group on that group. In physics the classification of atomic and nuclear states depends essentially on the properties of S_{n} ^{[1]}. The representation theory of the symmetric group is a well studied subject ^{[1, 2, 3]}. The partitions of n or equivalently Young diagrams of size n are the natural ways in which to parametrize the irreducible representations of S_{n} ^{[4]}. This paper is concerned, not with the general irreducible representations of S_{n}, but, more specifically, with the socalled standard representation of the symmetric group, formally obtained from the n1 dimensional subspace of vectors whose sum of coordinates is zero in the basis set of a permutation representation. The path taken in this work shall however be nongroup theoretic. For example, we will not be concerned with Young diagrams.
The standard representation of S_{n} is important for the following reason: For n ≥ 7, the permutation representation, the trivial (identical) representation, the sign representation, the standard representation and another n1 dimensional irreducible representation found by tensoring with the sign representation are the only lowestdimensional irreducible representations of S_{n} ^{[4]}. All other irreducible representations have dimension at least n. While it is a fact that all irreducible representations of S_{n} can be found, using Frobenius formula (^{[1]}, pp 189), for example, there are no known explicit formulas for the standard representation. The main result of this paper is the derivation of such formulas, which now make it possible to write down the standard representation matrices directly from those of the permutation representation.
2. The Permutation Representation
Denote the n! elements of S_{n} by , such that, in usual notation,
 (1) 
and , all being distinct. For simplicity, and since no ambiguity can result, we will use the same symbol for the representation matrices. Then, in the permutation representation, the matrices are given, through their elements, by:
 (2) 
which is clearly a unitary representation, since
2.1. A Casimir Invariant for S_{n} in the Permutation RepresentationTheorem 1. The matrix C, with elements is a Casimir Invariant of S_{n} in the permutation representation.
Proof We require to prove that C commutes with every .
 (3) 
A similar calculation gives,
 (4) 
We see, therefore, that so that C is a Casimir Invariant of the symmetric group.
3. The Standard Representation
Since the Casimir invariant C, obtained in the previous section, is not proportional to the identity, Schur’s lemma tells us that the permutation representation is not irreducible, a wellknown fact. It therefore remains to find the matrix P which diagonalizes C. First we prove a lemma.
Lemma 1 The nonsingular matrix P with elements has the inverse where
Proof
 (5) 
It is straightforward to write out the terms and evaluate the summation termwise. One merely needs to note that
and
One then finds , which establishes the claim.
3.1. Diagonal form of the Casimir InvariantTheorem 2. The matrix P given in Lemma 1, diagonalizes the Casimir Invariant, C.
Proof. We wish to compute
Now
Substituting the matrix elements, expanding and evaluating the sums, we find after some algebra, that
 (6) 
Thus we see that D is a diagonal matrix, as claimed, with the entry ‘n−1’ in row 1, column 1 and the remaining diagonal elements being −1.
The matrix P, above, which diagonalizes C will blockdiagonalize the matrices .
3.2. Similarity Transformation of A^{k}: The Standard RepresentationUsing the matrix elements of P, P^{−}^{1} and A^{k}, it is not difficult to obtain the interesting result:
 (7) 
We see from (7) that each matrix is block diagonal, being the direct sum of a matrix with entry 1 and an matrix B^{k}, with elements
 (8) 
The matrices correspond to the identical (trivial) representation in which every element of S_{n} is sent to the onedimensional identity matrix, while the B^{k} matrices correspond to the irreducible dimensional standard representation.
4. Conclusion
In this paper we have shown that the operator C with matrix elements is a Casimir Invariant for the symmetric group S_{n}. We also showed that if , are the representation matrices for the elements of S_{n} in the permutation representation, then the matrices B^{k} for the standard representation of S_{n} are given by
References
[1]  M. Hamermesh. Group Theory and Its Application to Physical Problems. Dover Publications, 1989. 
 In article  

[2]  W. Burnside. Theory of groups of finite order. Dover Publications, 1955. 
 In article  PubMed 

[3]  W. Fulton and J. Harris. Representation theory. A first course,. SpringerVerlag, 1991. 
 In article  

[4]  Wikipedia. Representation theory of the symmetric group http://en.wikipedia.org/wiki/Representation_theory_of_the_symmetric_group. 2015. 
 In article  
