schur's lemma

Consider two irreducible representations of a group (algebra) \(G\):

• \(\rho_V\) on the vector space \(V\)
• \(\rho_W\) on the vector space \(W\)

Consider a map \(f\) from \(V\) to \(W\). We say that \(f\) is \(G\) -equivariant or \(G\) -linear (see group invariant and group equivariant functions) if \(f \circ \rho_V(g) (v) = \rho_W \circ f (v)\) for all \(v\in V\) and \(g\in G\). Or equivalently, \(f\) can be represented by a matrix such that \(f\rho_V(g) = \rho_W(g)f\).

Then,

1. If \(V\) and \(W\) are not isomorphic, then there are no nontrivial \(G\) -linear maps between them.
2. If \(V=W\) is finite dimensional and \(\rho_V = \rho_W\) then the only nontrivial G-linear maps are constant matrices – multiples of the identity