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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.

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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 -> 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 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 Cm, 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.



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