# Finite group, representation of a

A homomorphism of a finite group into the group of non-singular linear mappings of a vector space into itself over a field . The representation theory of finite groups is the most highly developed (and is a most important) part of the representation theory of groups.

The representation theory of finite groups over is part of the representation theory of compact groups, and all the results of that theory (the Peter–Weyl theorem, the theory of characters, orthogonality relations, etc.) are valid (and simpler to prove) for finite groups. In particular, every representation of a finite group in a topological vector space is a direct sum of irreducible representations. On the other hand, there are some fundamental results in the representation theory of finite groups that use the specific nature of finite groups. For example, for a finite group the number of different equivalence classes of representations is equal to the number of conjugacy classes of ; the sum of the squares of the dimensions of representations representing the different equivalence classes is equal to the order of ; the dimension of every irreducible representation is a divisor of the index of every Abelian normal subgroup of (in particular, is a divisor of ) and does not exceed the index of Abelian subgroups of . If is the character of a representation of a finite group and is the dimension of , then for all and all conjugacy classes , and are algebraic integers. Every character of a group is a linear combination of the characters of the representations induced (see Induced representation) from representations of its cyclic subgroups and an integral linear combination of the characters of representations induced from one-dimensional representations of subgroups. A group is said to be -elementary if it is the product of a group of order a power of the prime number and a cyclic group of order prime to ; is said to be elementary if it is -elementary for some . Every character of a finite group is an integral linear combination of the representations induced from the representations of the elementary subgroups (Brauer's theorem, which can be generalized to the case where the field has arbitrary characteristic). If is supersolvable, that is, if it has a composition sequence consisting of normal subgroups with cyclic factors, then every irreducible representation of is induced from some one-dimensional representation of a subgroup.

When the characteristic of a field does not divide , the theory is only slightly different from the case . In particular, every finite-dimensional representation of a finite group is completely reducible; if is algebraically closed, then the number of equivalence classes of irreducible representations over is the number of conjugacy classes of the group, and the sum of the squares of the dimensions of representations representing the different equivalence classes is equal to the group order. But for a field that is not algebraically closed there may exist representations that are irreducible over but reducible over extensions of ; a field is said to be a splitting field of an irreducible representation if is irreducible over every extension of , and a splitting field for if is a splitting field for every representation of . If is a field of characteristic 0 or a finite field containing the -th roots of unity, where is the least common multiple of the orders of the elements of , then is a splitting field for ; the representation theory of a finite group over a field that is not a splitting field is connected with the Galois group of the extension of the given field obtained by adjoining all -th roots of unity. In particular, the number of classes of irreducible representations of a group over the field of rational numbers equals the number of conjugacy classes of cyclic subgroups of the group. If is a perfect field there exists a splitting field for that is finite over . For every field the character of an arbitrary representation of a finite group takes values in the set of finite sums of roots of unity in , and the analogues of the orthogonality relations and their consequences hold for the matrix entries and characters; in particular, if is a splitting field of characteristic zero for , then a representation with character is irreducible if and only if . If the characteristic of divides the group order , then the group algebra of over is not semi-simple and there exist representations of over that are not completely reducible.

Let be a local field of characteristic zero which is complete for a non-trivial discrete valuation (cf. Discrete norm), and let be the finite residue class field of of characteristic . Then the representations of over are said to be modular. The theory of modular representations establishes deeper connections between the structure of a group and properties of its representations than does representation theory over . The theory is simpler when and contain all -th roots of unity (and are therefore splitting fields); in this case an analogue of the orthogonality relations holds for the matrix entries and the characters. Let be a representation of a finite group over , let be its character, let be a primitive -th root of unity in , and let be its canonical image in ; let be an element of order prime to , that is, a -regular element, and let be the set of -regular elements. Then is diagonalizable and for some integers . The formula defines a function , called the Brauer character of ; it determines the composition factors of over uniquely. Indecomposable two-sided direct factors in the group algebra of over are called blocks; there exists a classification of inequivalent irreducible representations over , of inequivalent indecomposable representations over , and of non-isomorphic components of the decomposition of the left-regular representation of over in into a direct sum of non-zero indecomposabe representations in terms of the blocks. These results can be extended to the case where and are not splitting fields for .

#### References

 [1] C.W. Curtis, I. Reiner, "Representation theory of finite groups and associative algebras" , Interscience (1962) [2] A.A. Kirillov, "Elements of the theory of representations" , Springer (1976) (Translated from Russian) [3] M.A. Naimark, "Theory of group representations" , Springer (1982) (Translated from Russian) [4] S. Lang, "Algebra" , Addison-Wesley (1974) [5] J.-P. Serre, "Linear representations of finite groups" , Springer (1977) (Translated from French)