| import OAI.Analysis.GroupDeterminants.ComplexRepresentation |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators |
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| namespace GroupRingDeterminant |
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| section RegularStar |
| variable {G : Type} [Group G] {n : ℕ} |
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| theorem complexOperator_smul (z : ℂ) |
| (A : Matrix (Fin n) (Fin n) (MonoidAlgebra ℂ G)) : |
| complexOperator (z • A) = z • complexOperator A := by |
| classical |
| unfold complexOperator |
| simp only [Matrix.smul_apply, MonoidAlgebra.coeff_smul] |
| rw [Finset.smul_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| rw [Finset.smul_sum] |
| apply Finset.sum_congr rfl |
| intro j _ |
| rw [Finsupp.sum_smul_index (fun g => zero_smul ℂ (elementaryOperator i j g)), |
| Finsupp.smul_sum] |
| simp only [smul_smul] |
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| def complexAlgebraRepresentation : Matrix (Fin n) (Fin n) (MonoidAlgebra ℂ G) →ₐ[ℂ] |
| (Hilbert G n →L[ℂ] Hilbert G n) := |
| AlgHom.ofLinearMap |
| { toFun := complexOperator |
| map_add' := complexOperator_add |
| map_smul' := complexOperator_smul } |
| complexOperator_one complexOperator_mul |
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| theorem elementaryOperator_star (i j : Fin n) (g : G) : |
| star (elementaryOperator i j g) = elementaryOperator j i g⁻¹ := by |
| classical |
| rw [eq_comm, ContinuousLinearMap.star_eq_adjoint, |
| ContinuousLinearMap.eq_adjoint_iff] |
| intro ξ η |
| rw [lp.inner_eq_tsum, lp.inner_eq_tsum] |
| let e : Fin n × G ≃ Fin n × G := |
| Equiv.prodCongr (Equiv.swap i j) (Equiv.mulLeft g) |
| rw [← e.tsum_eq (fun a => inner ℂ (ξ a) (elementaryOperator i j g η a))] |
| apply tsum_congr |
| rintro ⟨a, h⟩ |
| simp only [elementaryOperator_apply, e, Equiv.prodCongr_apply] |
| by_cases ha : a = j |
| · subst a |
| simp [elementaryOperator_apply] |
| · have hs : Equiv.swap i j a ≠ i := by |
| intro hsi |
| have := (Equiv.swap i j).injective (hsi.trans (Equiv.swap_apply_right i j).symm) |
| exact ha this |
| simp [elementaryOperator_apply, ha, hs] |
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| theorem groupTrace_star (T : Hilbert G n →L[ℂ] Hilbert G n) : |
| groupTrace (star T) = star (groupTrace T) := by |
| simp only [groupTrace, star_sum, ContinuousLinearMap.star_eq_adjoint, |
| ContinuousLinearMap.adjoint_inner_right] |
| apply Finset.sum_congr rfl |
| intro i _ |
| exact (inner_conj_symm (T (identityVector i)) (identityVector i)).symm |
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| end RegularStar |
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| end GroupRingDeterminant |
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| end |
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| end OAI |
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