Schur's Lemmas — Group Theory in Physics (MA PH 464)
by Vincent Bouchard · University of Alberta
Section of an open group-theory-for-physicists textbook stating and proving both of Schur's lemmas with explicit matrix arguments, then applying the second lemma to show every irreducible representation of a finite abelian group is one-dimensional.
More resources on Schur’s Lemma
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
nLab: Schur's Lemma
Community-maintained research wiki entry on Schur's lemma, stated for simple modules and irreducible representations and generalized to simple objects in abelian and linear categories, with references to proofs and related results. Shows how the lemma extends beyond group representations.
Etingof's Introduction to Representation Theory
Free lecture notes by Pavel Etingof and co-authors from MIT, introducing representations of algebras, finite groups, quivers, and Lie algebras, with many exercises. Readers learn character theory, Schur-Weyl duality, and Gabriel's theorem, assuming a solid linear algebra and abstract algebra background.
Group Theory and Quantum Mechanics
Classic graduate physics text whose representation-theory chapters build on Schur's lemma to prove the great orthogonality theorem, then apply it to degeneracy of quantum states, selection rules, and the Wigner-Eckart theorem.
Schur's Lemma and Its Applications (Algebra I, Lecture 6)
The best free document devoted entirely to this one lemma. Eight-page lecture note proving Schur's lemma for linear representations over an arbitrary field, deriving the division-algebra structure of endomorphism rings, then specializing to algebraically closed fields and decomposing invariant subspaces of direct sums.
Introduction to Representation Theory (arXiv edition)
Free 108-page lecture-note textbook covering group, Lie algebra, and quiver representations. Section 2 states and proves Schur's lemma over general and algebraically closed fields, then uses it to derive orthogonality relations and character theory.