in mathematics, representation theory is a technique for analyzing abstract groups in terms of groups of linear transformations. see the article on group representations for an introduction. this article discusses the representation theory of groups that have a finite number of elements. ==basic definitions== all the linear representations in this article are finite-dimensional and assumed to be complex unless otherwise stated. a representation of g  is a group homomorphism Ï:g â gl(n,c) from g to the general linear group gl(n,c). thus to specify a representation, we just assign a square matrix to each element of the group, in such a way that the matrices behave in the same way as the group elements when multiplied together. we say that Ï is a real representation of g if the matrices are real, i.e. if Ï(g) â gl(n,r). ==other formulations == a representation Ï: g â gl(n,c) defines a group action of g on the vector space cn. moreover this action completely determines Ï. hence to specify a representation it is enough to specify how it acts on its representing vector space. alternatively, the action of a group g on a complex vector space v induces a left action of the group algebra c[g] on the vector space v, and vice-versa. hence representations are equivalent to left c[g]-modules. the group algebra c[g] is a |g|-dimensional algebra over the complex numbers, on which g acts. (see peterâweyl for the case of compact groups.) in fact c[g] is a representation for g&times;g. more specifically, if g1 and g2 are elements of g and h is an element of c[g] corresponding to the element h of g, :(g1,g2)[h]=g1 h g2-1. c[g] can also be considered as a representation of g in three different ways: *conjugation: g[h] = g h gâ1 *as a left action: g[h] = g h (a regular representation) *as a right action: g[h] = h gâ1 (also); these are all to be 'found' inside the g&times;g action. ==example== for many groups it is entirely natural to represent the group through matrices. consider for example the dihedral group d4 of symmetries of a square. this is generated by the two reflection matrices : m = \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix} : n = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} here m is a reflection that maps (x,y) to (&minus; x,y), while n maps (x,y) to (y,x). multiplying these matrices together creates a set of 8 matrices that form the group. as discussed above, we can either think of the representation in terms of the matrices, or in terms of the action on the two-dimens
Representation_theory_of_finite_groups