subrepresentation

In representation theory, a subrepresentation of a representation $(\backslash pi,\; V)$ of a group G is a representation $(\backslash pi|\_W,\; W)$ such that W is a vector subspace of V and $\backslash pi|\_W(g)\; =\; \backslash pi(g)|\_W$.

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 $(\backslash pi,\; V)$ is a representation of G, then there is the trivial subrepresentation:

:$V^G\; =\; \backslash \{\; v\; \backslash in\; V\; \backslash mid\; \backslash pi(g)v\; =\; v,\; \backslash ,\; g\; \backslash in\; G\; \backslash \}.$

If $f:\; V\; \backslash to\; W$ is an equivariant map between two representations, then its kernel is a subrepresentation of $V$ and its image is a subrepresentation of $W$.