Schur's lemma

Schur's lemma states that a nonzero homomorphism between irreducible representations is an isomorphism. Over an algebraically closed field, every endomorphism of an irreducible representation is scalar.

Group representation: A representation realizes a group as invertible linear transformations of a vector space. Schur's lemma applies to maps between representations of the same group.
Kernel of a linear map: The subspace of inputs that a linear map sends to zero. For an intertwiner, its kernel is invariant, so irreducibility makes it zero or the whole source.
Character orthogonality relations: Identities expressing orthogonality among characters of irreducible representations of finite groups. Schur's lemma is a key ingredient in deriving these identities.
Real representation: A representation on a vector space over the real numbers. An irreducible real representation can have nonscalar commuting endomorphisms.
Irreducible representation: A nonzero representation with no invariant subspaces other than zero and the whole space. Irreducibility forces every nonzero intertwining map to have trivial kernel and full image.
Image of a linear map: The subspace consisting of all outputs of a linear map. Its invariance forces a nonzero intertwiner between irreducibles to be surjective.
Isotypic component: The sum of all irreducible subrepresentations isomorphic to a fixed irreducible representation. Scalar endomorphisms on irreducibles constrain maps between components of different types.
Complex representation: A representation on a vector space over the complex numbers. Finite-dimensional irreducible complex representations satisfy the scalar-endomorphism conclusion.
Intertwining operator: A linear map between representations that commutes with their group actions. The lemma characterizes such maps when their source and target are irreducible.
Eigenvalue: A scalar by which a linear operator multiplies some nonzero vector. Over an algebraically closed field, an endomorphism has an eigenvalue whose subtraction yields a noninvertible intertwiner.

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