Standard representation of the symmetric group

Definition

The standard representation of a symmetric group on a finite set of degree $n$ is an irreducible representation of degree $n - 1$ (over a field whose characteristic does not divide $n!$ ) defined in the following equivalent ways:

Take a representation of degree $n$ obtained by the usual action of the symmetric group on the basis set of a vector space. Now, look at the $n - 1$ -dimensional subspace of vectors whose sum of coordinates in the basis is zero. The representation restricts to an irreducible representation of degree $n - 1$ on this subspace. This is the standard representation.
Take a representation of degree $n$ obtained by the usual action of the symmetric group on the basis set of a vector space. Consider the subspace spanned by the sum of the basis vectors. This is a subrepresentation of degree one. Consider the quotient space by this subspace. The representation descends naturally to a representation on the quotient space. This is the standard representation.

Facts

The standard representation is the representation corresponding to the partition $n = (n - 1) + 1$ . We see that the hook-length formula gives us a degree:

$\frac{n!}{(n) (n - 2) \dots (2) (1) (1)} = \frac{n!}{n (n - 2)!} = n - 1$

which is the same as the degree we expect.

The matrices for the standard representation (using method (1) or method (2)) can be written using elements in the set ${0, 1, - 1}$ . In fact, using method (2), we can obtain matrices where every column either has exactly one $1$ and everything else a $0$ , or has all $- 1$ s.

Particular cases

$n$	Symmetric group $S_{n}$	Standard representation	Degree of standard representation (= $n - 1$ )	Linear representation theory of group
2	cyclic group:Z2	nontrivial one-dimensional representation, sending the non-identity element to $- 1$	1	linear representation theory of cyclic group:Z2
3	symmetric group:S3	standard representation of symmetric group:S3	2	linear representation theory of symmetric group:S3
4	symmetric group:S4	standard representation of symmetric group:S4	3	linear representation theory of symmetric group:S4
5	symmetric group:S5	standard representation of symmetric group:S5	4	linear representation theory of symmetric group:S5