# Standard representation

From Groupprops

## Definition

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

- Take a representation of degree obtained by the usual action of the symmetric group on the basis set of a vector space. Now, look at the -dimensional subspace of vectors whose sum of coordinates in the basis is zero. The representation restricts to an irreducible representation of degree on this subspace. This is the standard representation.
- Take a representation of degree 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 . We see that the hook-length formula gives us a degree:

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 . In fact, using method (2), we can obtain matrices where every column either has exactly one and everything else a , or has all s.

## Particular cases

Symmetric group | Standard representation | Degree of standard representation (= ) | Linear representation theory of group | |
---|---|---|---|---|

2 | cyclic group:Z2 | nontrivial one-dimensional representation, sending the non-identity element to | 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 |