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
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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 |