- In
representation theory, a
subrepresentation of a
representation ( π , V ) {\displaystyle (\pi ,V)} of a
group G is a
representation ( π | W , W ) {\displaystyle...
- {\displaystyle A} is a
nonzero representation that has no
proper nontrivial subrepresentation ( ρ | W , W ) {\displaystyle (\rho |_{W},W)} , with W ⊂ V {\displaystyle...
- both the
subrepresentation and the
quotient have
smaller dimension.
There are
counterexamples where a
representation has a
subrepresentation, but only...
-
invariant under the
group action is
called a
subrepresentation. If V has
exactly two
subrepresentations,
namely the zero-dimensional
subspace and V itself...
-
properties seem tautologous, it is a
fundamental object of the theory. A
subrepresentation is
equivalent to a
trivial representation, for example, if it consists...
- V is a sum of
simple subrepresentations. Each
subrepresentation W of V
admits a
complementary representation: a
subrepresentation W' such that V = W ⊕...
- if and only if
every irreducible representation of G
occurs as a
subrepresentation of SnV (the n-th
symmetric power of the
representation V) for a sufficiently...
-
representation a
subrepresentation of V.
Every representation of G has
itself and the zero
vector space as
trivial subrepresentations. A representation...
-
theorem that,
while the
decomposition into a
direct sum of
irreducible subrepresentations may not be unique, the
irreducible pieces have well-defined multiplicities...
-
invariant with
respect to all
these transformations, then it is a
subrepresentation and the
group G acts on W in a
natural way. The same construction...
