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The term representation of a group is also used in a more general sense to mean any "description" of a group as a group of transformations of some mathematical object. More formally, a "representation" means a homomorphism from the group to the automorphism group of an object. If the object is a vector space we have a linear representation.
A representation is called semisimple or completely reducible if it can be written as a direct sum of irreducible representations. This is analogous to the corresponding definition for a semisimple algebra.
Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but the converse may not hold, e.g. the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices is indecomposable but reducible.
Weyl's theorem implies (in fact is equivalent to) that the enveloping algebra of a finite-dimensional representation is a semisimple ring in the following way.. Given a finite-dimensional Lie algebra representation : (), let be the associative subalgebra of the endomorphism algebra of V generated by ().
The 3! permutations of three objects form a group of order 6, commonly denoted S 3 (the symmetric group of degree three). This group is isomorphic to the point group, consisting of a threefold rotation axis and three vertical mirror planes. The groups have a 2-dimensional irreducible representation (l = 2).
In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to quantum chemistry studies of atoms, molecules and solids. [1] [2]
The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a concise form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on any representative of the respective conjugacy class of G (because characters are class functions).
[2] [3] A systematic modern interpretation has been given by Howe (1995) in the context of his theory of dual pairs. The special case where σ is the trivial representation of H was first used extensively by Hua in his work on the Szegő kernels of bounded symmetric domains in several complex variables, where the Shilov boundary has the form G/H.