Group representation

In abstract algebra, a representation of a finite group G is a group homomorphism from G to the general linear group GL(n,C) of invertible complex n-by-n matrices. The study of such representations is called representation theory.

Table of contents 1 Example 2 Equivalence of representations 3 Group actions 4 Reducibility 5 Character theory

Example

Consider the complex number u = exp(2πi/3) which has the property u³ = 1. The cyclic group C₃ = {1, u, u²} has a representation ρ given by:

<math>

\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \qquad \begin{bmatrix} 1 & 0 \\ 0 & u \\ \end{bmatrix} \qquad \begin{bmatrix} 1 & 0 \\ 0 & u^2 \\ \end{bmatrix} </math>

(the three matrices are ρ(1), ρ(u) and ρ(u²) respectively).

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, i.e. if there exists A in GL(n,C) such that for all x in G: ρ₁(x) = Aρ₂(x)A^-1. For example, the representation of C₃ given by the matrices:

<math>

\begin{bmatrix} 1 & 0 \\ 0 & 1 \\ \end{bmatrix} \qquad \begin{bmatrix} u & 0 \\ 0 & 1 \\ \end{bmatrix} \qquad \begin{bmatrix} u^2 & 0 \\ 0 & 1 \\ \end{bmatrix} </math>

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 -> GL_n 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 -> GL_n is the function χ : G -> C 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(1) = 2, χ(u) = 1 + u, χ(u²) = 1 + u².

If g and h are members of G in the same conjugacy class, then χ(g) = χ(h) for any character; the values of a character therefore have to be specified only for the different conjugacy classes of G. Moreover, equivalent representations have the same characters. If a representation is the direct sum of subrepresentations, then the corresponding character is the sum of the subrepresentations' 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 C₃:

      (1)  (u)  (u2)
   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 the standard inner product on C^m, which allows one to compute character tables 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 its 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 non-trivial 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.