Group representation

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A group representation (or more simply, a representation) of a finite group G is a group_homomorphism between G and a the group GLn of invertible n-by-n matrices. The study of such representations is called representation theory.

Example

The cyclic group Z3 = {0, 1, 2} has a representation ρ given by:

 /  1  0  \    /  1  0  \    /  1  0  \ 
 \  0  1  /    \  0  u  /    \  0  u2 /

(the three matrices are ρ(0), ρ(1) and ρ(2) respectively, and u is the complex number exp(2πi/3) which has the property u3 = 1).

This representation is said to be faithful, because ρ is a one-to-one map.

Equivalence of representations

Two representations ρ1 and ρ2 are said to be equivalent if the matrices only differ by a change of basis. For example, the representation of Z/3Z given by the matrices:

 /  1  0  \    /  u  0  \    /  u2  0  \ 
 \  0  1  /    \  0  1  /    \  0  1  /

is an equivalent representation to the one shown above.

Group actions

Every square n-by-n matrix describes a linear map from an n-dimensional vector space V to itself (once a basis for V has been chosen). Therefore, every representation ρ: G -> GLn defines a group action on V given by g.v = (ρ(g))(v) (for g in G, v in V). One may in fact define a representation of a group as an action of that group on some vector space, thereby avoiding the need to choose a basis and the restriction to finite-dimensional vector spaces.

Reducibility

If V has a non-trivial proper subspace W such that GW is contained in W, then the representation is said to be reducible. A reducible representation can be expressed as a direct_sum of subrepresentations (Maschke's theorem).

If V has no such subspaces, it is said to be an irreducible representation.

In the example above, the representation given is reducible into two 1-dimensional subrepresentations (given by span{(1,0)} and span{(0,1)}).

Character theory

The character of a representation ρ : G -> GLn is a function χ which sends g in G to the trace (the sum of the diagonal elements) of the matrix ρ(g). For example, the character of the representation given above is given by: χ(0) = 2, χ(1) = 1 + u, χ(2) = 1 + u2.

If g and h are members of G in the same conjugacy class, then g and h have the same character. Moreover, equivalent representations have the same characters.

The characters of all the irreducible representations of a finite group form a character table, with conjugacy classes of elements as the columns, and characters as the rows. Here is the character table of Z3:

     (0)  (1)  (2)
  1   1    1    1
  χ1  1    u    u2
  χ2  1    u2   u

The character table is always square, and the rows and columns are orthogonal with respect to a particular inner product, which enables character tables to be computed more easily. The first row of the character table always consists of 1s, and corresponds to the trivial representation (the 1-dimensional representation consisting of 1-by-1 matrices containing the entry 1).

Certain properties of the group G can be deduced from the character table:

  • The order of G is given by the sum of (χ(1))2 over the characters in the table.
  • G is abelian if and only if χ(1) = 1 for all characters in the table.
  • G has a non-trivial normal subgroup (i.e. G is not a simple group) if and only if χ(1) = χ(g) for some character χ in the table and some non-identity element g in G.

The character table does not in general determine the group up to isomorphism: for example, the quaternion group Q and the dihedral group of 8 elements (D8) have the same character table.