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230 ALGEBRA

5.8.3. Group Representations

5.8.3-1. Linear representations of groups. Terminology.
n
A linear representation of a group G in the finite-dimensional Euclidean space V is a
n
homomorphism of G to the group of nondegenerate linear transformations of V ;in other
words, a linear representation of G is a mapping D that associates each element a G with
n
a nondegenerate linear transformation D(a)ofthe space V , so that for any a 1 and a 2 in G,
we have D(a 1 a 2 )= D(a 1 )D(a 2 ).
Thus, for any g G, its image D(g) is an element of the group GL(n), and the set D(G)
consisting of all transformations D(g), g G, is a subgroup of GL(n) isomorphic to the
quotient group G/ker D,where ker D is the kernel of the homomorphism D, i.e., the set of
all g such that D(g) is the identity element of the group GL(n).
The subgroup D(G) is often also called a representation of the group G.
n
The space V is called the representation space; n is called the dimension of the
n
representation; and the basis in V is called the representation basis.
The trivial representation of a group is its homomorphic mapping onto the identity
element of the group GL(n).
A faithful representation of a group G is an isomorphism of G onto a subgroup of GL(n).

5.8.3-2. Matrices of linear representations. Equivalent representations.

If D (μ) (G) is a representation of a group G, each g G corresponds to a linear transformation
D (μ) (g), whose matrix in the basis of the representation D (μ) (G) is denoted by [D (μ) (g)].
ij
n
Two representations D (μ 1 ) (G)and D (μ 2 ) (G) of a group G in the same space E are said
to be equivalent if there exists a nondegenerate linear transformation C of the space E n
–1
such that D (μ 1 ) (g)= C D (μ 2 ) (g)C for each g G.
The choice of a basis in the representation space is important, since the matrices
correspondingtothe group elements may have some standard fairly simple form inthat basis,
and this allows one to make important conclusions with regards to a given representation.

5.8.3-3. Reducible and irreducible representations.

n
A subspace V of V is called invariant for a representation D(G) if it is invariant with

respect to each linear operator in D(G).
Suppose that all matrices of some three-dimensional representation D(G) have the form

A 1 A 2 a 11 a 12 a 13
, A 1 ≡ , A 2 ≡ , A 3 ≡ ( a 33 ), O ≡ ( 00 ).
O A 3 a 21 a 22 a 23
The product of such matrices has the form (see Paragraph 5.2.1-10)

A 1 A 2 A 1 A = A A 1 A ,
1
2
2

O A 3 O A O A A 3
3
3
and therefore the structure of the matrices is preserved. Thus, the matrices A 1 form a
two-dimensional representation of the given group G and the matrices A 3 form its one-
dimensional representation. In such cases, one says that D(G)is a reducible representation.