xi = Σj=mj=1 sij xj (i, j = 1, 2, ..., m),
on a set of m variables, such that if ST = U, then st = u. The linear substitutions s, t, u, ... then constitute a group g with which G is isomorphic; and whether the isomorphism is simple or multiple g is said to give a “representation” of G as a group of linear substitutions. If all the substitutions of g are transformed by the same substitution on the m variables, the (in general) new group of linear substitutions so constituted is said to be “equivalent” with g as a representation of G; and two representations are called “non-equivalent,” or “distinct,” when one is not capable of being transformed into the other.
A group of linear substitutions on m variables is said to be “reducible” when it is possible to choose m′ (< m) linear functions of the variables which are transformed among themselves by every substitution of the group. When this cannot be done the group is called “irreducible.” It can be shown that a group of linear substitutions, of finite order, is always either irreducible, or such that the variables, when suitably chosen, may be divided into sets, each set being irreducibly transformed among themselves. This being so, it is clear that when the irreducible representations of a group of finite order are known, all representations may be built up.
It has been seen at the beginning of this section that every group of finite order N can be presented as a group of permutations (i.e. linear substitutions in a limited sense) on N symbols. This group is obviously reducible; in fact, the sum of the symbols remain unaltered by every substitution of the group. The fundamental theorem in connexion with the representations, as an irreducible group of linear substitutions, of a group of finite order N is the following.
If r is the number of different sets of conjugate operations in the group, then, when the group of N permutations is completely reduced,
(i.) just r distinct irreducible representations occur:
(ii.) each of these occurs a number of times equal to the number of symbols on which it operates:
(iii.) these irreducible representations exhaust all the distinct irreducible representations of the group.
Among these representations what is called the “identical” representation necessarily occurs, i.e. that in which each operation of the group corresponds to leaving a single symbol unchanged. If these representations are denoted by Γ1, Γ2, ..., Γr, then any representation of the group as a group of linear substitutions, or in particular as a group of permutations, may be uniquely represented by a symbol ΣαiΓi, in the sense that the representation when completely reduced will contain the representation Γi just αi times for each suffix i.
A representation of a group of finite order as an irreducible group Group characteristics. of linear substitutions may be presented in an infinite number of equivalent forms. If