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For n = 3 the obvious analogue of the (n − 1)-dimensional representation is reducible – the permutation representation coincides with the regular representation, and thus breaks up into the three one-dimensional representations, as A 3 ≅ C 3 is abelian; see the discrete Fourier transform for representation theory of cyclic groups.
In aqueous solution, ammonia deprotonates a small fraction of the water to give ammonium and hydroxide according to the following equilibrium: . NH 3 + H 2 O ⇌ NH + 4 + OH −.. In a 1 M ammonia solution, about 0.42% of the ammonia is converted to ammonium, equivalent to pH = 11.63 because [NH +
Let V be a representation of a group G; or more generally, let V be a vector space with a set of linear endomorphisms acting on it. In general, a vector space acted on by a set of linear endomorphisms is said to be simple (or irreducible) if the only invariant subspaces for those operators are zero and the vector space itself; a semisimple representation then is a direct sum of simple ...
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.
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).
If has exactly two subrepresentations, namely the trivial subspace {0} and itself, then the representation is said to be irreducible; if has a proper nontrivial subrepresentation, the representation is said to be reducible. [19] The definition of an irreducible representation implies Schur's lemma: an equivariant map : (,) (′, ′) between ...
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 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.