representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

, a subrepresentation of a representation

(\pi, V)

of a

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic iden ...

''G'' is a representation

(\pi, _W, W)

such that ''W'' is a

vector subspace Vector most often refers to: * Euclidean vector, a quantity with a magnitude and a direction * Disease vector, an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematics a ...

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 = \.

f: V \to W

is an

equivariant map In mathematics, equivariance is a form of symmetry for function (mathematics), functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are Group action ( ...

between two representations, then its kernel is a subrepresentation of

V

and its image is a subrepresentation of

W

References

* Representation theory {{abstract-algebra-stub