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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).
The representation is said to be irreducible if the only invariant subspaces of V are the zero space and V itself. For certain types of Lie groups, namely compact [2] and semisimple [3] groups, every finite-dimensional representation decomposes as a direct sum of irreducible representations, a property known as complete reducibility. For such ...
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.
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 ...
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 identity component of PGL(2,R) (sometimes called PSL(2,R)) is a real reductive group that cannot be viewed as an algebraic group. Similarly, SL(2) is simply connected as an algebraic group over any field, but the Lie group SL(2,R) has fundamental group isomorphic to the integers Z, and so SL(2,R) has nontrivial covering spaces.
More precisely, the set of irreducible characters of a given group G into a field F form a basis of the F-vector space of all class functions G → F. Isomorphic representations have the same characters. Over a field of characteristic 0, two representations are isomorphic if and only if they have the same character. [1]
In mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, hence the name. More concretely, a Lie algebra is reductive if it is a direct sum of a semisimple Lie algebra and an abelian Lie algebra : g = s ⊕ a ; {\displaystyle {\mathfrak {g}}={\mathfrak {s}}\oplus {\mathfrak {a}};} there are alternative ...