Finite group, representation of a

A homomorphism of a finite group $ G $ into the group of non-singular linear mappings of a vector space into itself over a field $ K $. 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 $ \mathbf C $ 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 $ G $ the number of different equivalence classes of representations is equal to the number of conjugacy classes of $ G $; the sum of the squares of the dimensions of representations representing the different equivalence classes is equal to the order $ | G | $ of $ G $; the dimension of every irreducible representation is a divisor of the index of every Abelian normal subgroup of $ G $( in particular, is a divisor of $ | G | $) and does not exceed the index of Abelian subgroups of $ G $. If $ \chi $ is the character of a representation $ \pi $ of a finite group and $ d $ is the dimension of $ \pi $, then for all $ s \in G $ and all conjugacy classes $ Q \subset G $, $ \chi ( s) $ and $ d ^ {-} 1 \sum _ {s \in Q } \chi ( s) $ are algebraic integers. Every character of a group $ G $ 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 $ H $ is said to be $ p $- elementary if it is the product of a group of order a power of the prime number $ p $ and a cyclic group of order prime to $ p $; $ H $ is said to be elementary if it is $ p $- elementary for some $ p $. Every character of a finite group $ G $ is an integral linear combination of the representations induced from the representations of the elementary subgroups $ H \subset G $( Brauer's theorem, which can be generalized to the case where the field has arbitrary characteristic). If $ G $ is supersolvable, that is, if it has a composition sequence consisting of normal subgroups with cyclic factors, then every irreducible representation of $ G $ is induced from some one-dimensional representation of a subgroup.

When the characteristic $ p $ of a field $ K $ does not divide $ | G | $, the theory is only slightly different from the case $ K = \mathbf C $. In particular, every finite-dimensional representation of a finite group is completely reducible; if $ K $ is algebraically closed, then the number of equivalence classes of irreducible representations over $ K $ 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 $ K $ that is not algebraically closed there may exist representations that are irreducible over $ K $ but reducible over extensions of $ K $; a field $ K $ is said to be a splitting field of an irreducible representation $ \pi $ if $ \pi $ is irreducible over every extension of $ K $, and a splitting field for $ G $ if $ K $ is a splitting field for every representation of $ G $. If $ K $ is a field of characteristic 0 or a finite field containing the $ m $- th roots of unity, where $ m $ is the least common multiple of the orders of the elements of $ G $, then $ K $ is a splitting field for $ G $; 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 $ m $- th roots of unity. In particular, the number of classes of irreducible representations of a group $ G $ over the field of rational numbers equals the number of conjugacy classes of cyclic subgroups of the group. If $ K $ is a perfect field there exists a splitting field for $ G $ that is finite over $ K $. For every field $ K $ the character of an arbitrary representation of a finite group takes values in the set of finite sums of roots of unity in $ K $, and the analogues of the orthogonality relations and their consequences hold for the matrix entries and characters; in particular, if $ K $ is a splitting field of characteristic zero for $ G $, then a representation with character $ \chi $ is irreducible if and only if $ \sum _ {g \in G } \chi ( g) \chi ( g ^ {-} 1 ) = | G | $. If the characteristic $ p $ of $ K $ divides the group order $ | G | $, then the group algebra of $ G $ over $ K $ is not semi-simple and there exist representations of $ G $ over $ K $ that are not completely reducible.

Let $ k $ be a local field of characteristic zero which is complete for a non-trivial discrete valuation (cf. Discrete norm), and let $ K $ be the finite residue class field of $ k $ of characteristic $ p $. Then the representations of $ G $ over $ K $ 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 $ \mathbf C $. The theory is simpler when $ k $ and $ K $ contain all $ m $- 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 $ \pi $ be a representation of a finite group over $ K $, let $ \chi $ be its character, let $ \Delta $ be a primitive $ m $- th root of unity in $ k $, and let $ \delta $ be its canonical image in $ K $; let $ s \in G $ be an element of order prime to $ p $, that is, a $ p $- regular element, and let $ G _ { \mathop{\rm reg} } $ be the set of $ p $- regular elements. Then $ \pi ( s) $ is diagonalizable and $ \chi ( s) = \delta ^ {a _ {1} } + \dots + \delta ^ {a _ {n} } $ for some integers $ a _ {1} \dots a _ {n} $. The formula $ \eta ( s) = \Delta ^ {a _ {1} } + \dots + \Delta ^ {a _ {n} } $ defines a function $ \eta : G _ { \mathop{\rm reg} } \rightarrow k $, called the Brauer character of $ \pi $; it determines the composition factors of $ \pi $ over $ K $ uniquely. Indecomposable two-sided direct factors in the group algebra $ K ( G) $ of $ G $ over $ K $ are called blocks; there exists a classification of inequivalent irreducible representations over $ k $, of inequivalent indecomposable representations over $ K $, and of non-isomorphic components of the decomposition of the left-regular representation of $ G $ over $ K $ in $ K ( G) $ into a direct sum of non-zero indecomposabe representations in terms of the blocks. These results can be extended to the case where $ k $ and $ K $ are not splitting fields for $ G $.

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)

Comments

See also Representation of a compact group; Representation of a group; Representation of an infinite group; Representation theory.

About characters see Character of a group; Character of a representation of a group.

Concerning reducibility of a representation see Irreducible representation; Reducible representation. Reference [1] has been superceded by [a1].

References

[a1]	C.W. Curtis, I. Reiner, "Methods of representation theory" , 1–2 , Wiley (Interscience) (1981–1987)
[a2]	W. Feit, "The representation theory of finite groups" , North-Holland (1982)
[a3]	B. Huppert, "Finite groups" , 2–3 , Springer (1982)
[a4]	D.J. Benson, "Modular representation theory: New trends and methods" , Lect. notes in math. , 1081 , Springer (1984)