openAI-math-dataset / lean /OAI /RepresentationTheory /PartialPermutation /RegularRepresentation.lean
| import Mathlib | |
| import OAI.RepresentationTheory.PartialPermutation.Model | |
| namespace OAI | |
| section | |
| open scoped Classical | |
| open scoped BigOperators ComplexConjugate MonoidAlgebra | |
| open scoped BigOperators ComplexConjugate | |
| open scoped MonoidAlgebra BigOperators | |
| open scoped BigOperators MonoidAlgebra Classical | |
| namespace PartialPermutation | |
| noncomputable section | |
| variable (G : Type*) [Group G] [Fintype G] | |
| omit [Fintype G] in | |
| @[simp] lemma regularRep_apply (g h : G) (f : EuclideanSpace ℂ G) : | |
| regularRep G g f h = f (g⁻¹ * h) := rfl | |
| lemma regularRep_unitary (g : G) (f k : EuclideanSpace ℂ G) : | |
| inner ℂ (regularRep G g f) (regularRep G g k) = inner ℂ f k := by | |
| simp only [PiLp.inner_apply, regularRep_apply] | |
| exact Fintype.sum_equiv (Equiv.mulLeft g⁻¹) _ _ (fun h => rfl) | |
| def deltaOne : EuclideanSpace ℂ G := WithLp.toLp 2 (fun g => if g = 1 then 1 else 0) | |
| omit [Fintype G] in | |
| lemma regularRep_delta (g h : G) : | |
| regularRep G g (deltaOne G) h = if h = g then 1 else 0 := by | |
| change (if g⁻¹ * h = 1 then (1 : ℂ) else 0) = if h = g then 1 else 0 | |
| simp [inv_mul_eq_one, eq_comm] | |
| end | |
| end PartialPermutation | |
| end | |
| end OAI | |