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 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 In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, li ...

of ''V'' and

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

. A nonzero finite-dimensional representation always contains a nonzero subrepresentation that is

irreducible In philosophy, systems theory, science, and art, emergence occurs when an entity is observed to have properties its parts do not have on their own, properties or behaviors that emerge only when the parts interact in a wider whole. Emergence ...

, the fact seen by

induction Induction, Inducible or Inductive may refer to: Biology and medicine * Labor induction (birth/pregnancy) * Induction chemotherapy, in medicine * Induced stem cells, stem cells derived from somatic, reproductive, pluripotent or other cell t ...

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