Prove2Me

`BookProof.ChapterSchurRepresentation.schur_unitary_representation_apply` {G : Type*} [Group G] (U : G →* (V ≃ₗᵢ[ℂ] V)) (hirr : (repSystem U).IsIrreducible) {S : V →L[ℂ] V} (hcomm

Proved
BookProof.ChapterSchurRepresentation.schur_unitary_representation_apply

by leonardopedro · Oct 17, 2026 · Mathlib 0df444a (Lean v4.33.1)

theoremstimepiece

Prove the following Lean 4 theorem from ChapterSchurRepresentation.

BookProof.ChapterSchurRepresentation.schur_unitary_representation_apply {G : Type*} [Group G] (U : G →* (V ≃ₗᵢ[ℂ] V)) (hirr : (repSystem U).IsIrreducible) {S : V →L[ℂ] V} (hcomm : ∀ (g : G) (x : V), S (U g x) = U g (S x)) : ∃ c : ℂ, ∀ x, S x = c • x

Formalization note: Lean 4 identifier BookProof.ChapterSchurRepresentation.schur_unitary_representation_apply.

Preamble
-- Generated from ChapterSchurRepresentation.lean — theorem BookProof.ChapterSchurRepresentation.schur_unitary_representation_apply
import Definitions.Def_ChapterSchurIrreducible
import Mathlib
import Definitions.Def_ChapterSchurRepresentation
import Definitions.Def_ChapterA
open BookProof.ChapterA
open BookProof.ChapterA.System
open BookProof.ChapterSchurRepresentation


open scoped ComplexConjugate InnerProductSpace


open BookProof.ChapterA BookProof.ChapterA.System BookProof.ChapterSchurIrreducible

variable {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℂ V] [CompleteSpace V]
Formal statement
theorem BookProof.ChapterSchurRepresentation.schur_unitary_representation_apply {G : Type*} [Group G] (U : G →* (V ≃ₗᵢ[ℂ] V))
    (hirr : (repSystem U).IsIrreducible) {S : V →L[ℂ] V}
    (hcomm : ∀ (g : G) (x : V), S (U g x) = U g (S x)) :
    ∃ c : ℂ, ∀ x, S x = c • x := by sorry

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