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Over an algebraically closed field, every endomorphism of an irreducible representation is scalar
Statement
Let be an algebraically closed field, let be a group, and let be an irreducible representation of over . Then every is of the form for some .
Facts & Assumptions
Given: An algebraically closed field , a group , an irreducible representation of over , and an endomorphism .
Every finite-dimensional endomorphism over an algebraically closed field is triangularisable (Every finite-dimensional endomorphism over an algebraically closed field is triangularisable).
For an irreducible representation, every nonzero -endomorphism is an isomorphism and is a division ring (Schur's lemma for irreducible representations: a nonzero intertwiner is an isomorphism, and is a division ring).
Proof
The representation is nonzero because it is irreducible, so [L1] applies to and yields a basis in which is upper triangular. Let be one diagonal entry. Then is upper triangular with a zero diagonal entry, hence is not invertible.
For every , the identity map commutes with the action of , so is still a -endomorphism. By [L2], a nonzero -endomorphism of an irreducible representation must be invertible. Therefore the noninvertible endomorphism is zero.
Hence , as required.
Depends on
- Every finite-dimensional endomorphism over an algebraically closed field is triangularisable
- Schur's lemma for irreducible representations: a nonzero intertwiner is an isomorphism, and $\operatorname{End}_G(V)$ is a division ring
- An algebraically closed field: every nonconstant polynomial has a root in the field
Used by
- Normal subgroup induction criterion Corollary
- Over an algebraically closed field of characteristic 0, every element of finite order acts diagonalisably in a finite-dimensional representation Corollary
- Fourier series on a torus as Peter–Weyl Example
- Over ℂ, a cyclic group of order n has exactly n irreducible representations up to equivalence, represented by the characters g↦ λ with λⁿ=1 Example
- The irreducible complex characters of a finite cyclic group are the n powers of a primitive nth root Example
- Peter–Weyl gives density, not finite equality False statement
- Additive characters are exactly one-dimensional complex representation characters Lemma
- An invariant irreducible normal representation yields projective inertia operators Lemma
- Isotypical evaluation and multiplicity subspaces Lemma
- The transposition class sum acts on a complex Specht module by total content Lemma
- Class sums act on an irreducible representation by central-character scalars Proposition
- A finite group is abelian if and only if all its irreducible complex characters have degree 1 Theorem
- Absolute irreducibility via the endomorphism division algebra Theorem
- Existence and uniqueness of cuspidal support Theorem
- If k is algebraically closed and char k ∤ |G|, then k[G]≅∏ᵢ₌₁ʳ M_nᵢ(k) Theorem
- Multiplicity-free complex Specht induction Theorem
- Schur-Weyl decomposition and highest weights Theorem
- Symmetric polynomials in the Jucys-Murphy elements give exactly the centre Theorem
- The first orthogonality relation for irreducible complex characters Theorem
- The relative centralizer is generated by the previous centre and the last Jucys-Murphy element Theorem
- The Schur-Weyl mutual centralizer theorem on tensor powers Theorem
- Transitivity and parabolic independence of Harish-Chandra induction Theorem
- Young's rule for complex permutation modules Theorem
Dependency tree · two levels
11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Pavel Etingof et al., Introduction to Representation Theory, Corollary 1.17 (standard reference, not scraped)