The fundamental constraint on maps between irreducible representations
Schur's Lemma is the cornerstone of representation theory. It states two facts: (1) any G-equivariant map between non-isomorphic irreducible representations must be zero, and (2) any G-equivariant endomorphism of an irreducible representation (over an algebraically closed field) is a scalar multiple of the identity.
These constraints are astonishingly powerful -- they force the orthogonality of characters, the uniqueness of decompositions, and much more.
Explore the three fundamental cases of Schur's Lemma: maps between different irreducibles (must be zero), self-maps of an irreducible (must be scalar), and maps between isomorphic copies (must be isomorphisms).
Schur's Lemma constrains all possible G-equivariant maps between irreducible representations.
Key insight: Schur's Lemma says that the "space of intertwiners" between irreducibles is either 0-dimensional or 1-dimensional. There is no middle ground.
A linear map phi between representation spaces is an intertwiner(or G-equivariant map) if the commutative diagram phi(rho(g)v) = rho'(g)phi(v) holds for all g in G. Step through the diagram to see both paths.
Step through the commutative diagram. An intertwiner \u03C6 makes both paths (down then right, or right then down) give the same result.
Key insight: The intertwining condition says "applying the group action commutes with the map phi." It does not matter whether you act first then map, or map first then act -- the result is the same.
The space of all intertwiners between two irreducible representations forms a vector space Hom_G(V, W). By Schur's Lemma, its dimension is exactly 1 when V and W are isomorphic and 0 otherwise.
The space of intertwiners Hom_G(V_i, V_j) between irreducible representations is 1-dimensional when V_i and V_j are isomorphic, and 0-dimensional otherwise. This is a direct consequence of Schur's Lemma.
Key insight: This identity matrix pattern is what makes the inner product on characters work: orthogonality of characters is a direct consequence of dim Hom_G(V_i, V_j) = delta_ij.