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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.
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).
Just like representations, a corepresentation is reducible if there is a proper subspace invariant under the operations of the corepresentation. If the corepresentation is given by matrices, it is reducible if it is equivalent to a corepresentation with each matrix in block diagonal form.
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 ...
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.
Restriction is a fundamental construction in representation theory of groups. Often the restricted representation is simpler to understand. Rules for decomposing the restriction of an irreducible representation into irreducible representations of the subgroup are called branching rules, and have important applications in physics.
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 ...
In representation theory, a branch of mathematics, the theorem of the highest weight classifies the irreducible representations of a complex semisimple Lie algebra. [ 1 ] [ 2 ] There is a closely related theorem classifying the irreducible representations of a connected compact Lie group K {\displaystyle K} . [ 3 ]