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A representation of a group \(G\) is a homomorphism \(\phi:G \to GL(V)\)
(\(GL\) is the general linear group) for some (finite-dimensional)
vector space \(V\). The dimension of \(V\) is called the degree of
\(\phi\).
Note: The trivial representation sends all elements to \(1\).
Two representations \(\phi:G \to GL(V), \psi:G \to GL(W)\) are
equivalent if there exists an isomorphism such that \(\psi_{g} = T\phi
T^{-1}\) for all \(g \in G\). Where \(T:V \to W\).
Notes which link to here
Let \(\phi:G \to GL(V)\) be a representation. A subspace \(W \leq V\) is
\(G\)-invariant if \(\phi_{g}W \in W\quad \forall g \in G\).
A non-zero representation \(\phi:G \to GL(V)\) is said to be
irreducible if the only \(G\)-invariant subspace of \(V\) are \(\{ 0 \}\)
and \(V\).
Notes which link to here
If \(\phi:G \to GL(V)\) is a representation of degree 2 (\(\text{dim}V =
2\)), then \(\phi\) is irreducible-representation if and only if there is
no common eigenvector \(v\) to all \(\phi_{g}\) with \(g \in G\).