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.