representation theory Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...

, a subrepresentation of a representation

(\pi, V)

of a group ''G'' is a representation

(\pi, _W, W)

such that ''W'' is a vector subspace of ''V'' and

\pi, _W(g) = \pi(g), _W

. A nonzero finite-dimensional representation always contains a nonzero subrepresentation that is irreducible, the fact seen by induction on dimension. This fact is generally false for infinite-dimensional representations. If

(\pi, V)

is a representation of ''G'', then there is the trivial subrepresentation: :

V^G = \.

References

* Representation theory {{abstract-algebra-stub