Standard representation

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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 -1s.

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