bibkey: etingof2009representation
authors: Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina
year: 2009
title: Introduction to representation theory
doi: null
url: https://arxiv.org/abs/0901.0827
claim: Schur's lemma for irreducible representations; finite-group regular representations and Frobenius determinants.
strata_touched:
  - D5/S3/Quantum/Matrix/RecordSymmetryNoGo
license: citation-only
triage: anchor

Schur’s lemma and the triviality of equivariant idempotents

Verified locator

DOI: none assigned; the canonical locator is the arXiv record: https://arxiv.org/abs/0901.0827

Retrieved 2026-09-11: the abstract page gives the title above, the seven authors above, and the identifier arXiv:0901.0827v5 in the primary class math.RT.

The statement this repository depends on is Schur’s lemma in the form given in section 1.3: Proposition 1.16 states that a nonzero intertwining operator between irreducible representations is an isomorphism, and Corollary 1.17 states that over an algebraically closed field every self-intertwining operator of a finite-dimensional irreducible representation is a scalar.

An equivalent statement, phrased directly as the correspondence between invariant subspaces and equivariant orthogonal projections, appears in Sophie Morel, MAT 449: Representation theory (2018), section I.3.4, Theorem I.3.4.1 and Lemma I.3.4.3, printed pages 25 to 26.

Scope of the dependency

The repository module derives from this only the corollary that an equivariant idempotent in an irreducible representation is zero or one, and the converse that a nontrivial such idempotent witnesses reducibility. No claim of a new proof of Schur’s lemma is made, and no priority claim is made about the earliest appearance of the equivalence in the literature.

The repository module states its hypotheses over the complex matrices with self-adjointness. Under algebraic irreducibility the zero-or-one conclusion does not require unitarity, self-adjointness, the complex field, or finite dimensionality; those hypotheses are therefore stronger than necessary and their removal would not make the statement new.

Section 4.2 also uses the regular representation of a finite group in its Frobenius determinant discussion. The elementary cycle computation in theory section 2143.6 specializes that background to one finite-order element; it is derived there and is not attributed to a new representation-theoretic discovery.