Standard representation

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:

1. 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.
2. 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