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.
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.
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.
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)}).
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 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.