The fundamental constraint on maps between irreducible representations
Between irreducible representations, maps that respect the group are either zero or a multiple of the identity. That one fact, Schur's Lemma, drives most of the subject.
Click each case — different irreps, the same irrep, isomorphic irreps — and see which maps survive. The space of intertwiners is always 0- or 1-dimensional.
Schur's Lemma constrains all possible G-equivariant 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 (nonzero maps must be isomorphisms).
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.
Press Start, then Next Step, to follow a vector around the diagram both ways: act then map, or map then act. Both paths land on the same vector.
Step through the commutative diagram. An intertwiner φ makes both paths (down then right, or right then down) give the same result.
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.
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.
Hover over a cell to see the dimension of Hom_G between two irreducibles of S3. The whole grid is the identity matrix.
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.
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.
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.
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