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.
- 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.
|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|