| import OAI.Analysis.GroupDeterminants.TransferAlgebra | |
| namespace OAI | |
| noncomputable section | |
| open scoped BigOperators | |
| namespace GroupRingDeterminant | |
| section RegularTrace | |
| variable {G : Type} [Group G] {n : ℕ} | |
| theorem identityVector_apply [DecidableEq G] (i j : Fin n) (g : G) : | |
| identityVector i (j, g) = if j = i ∧ g = 1 then 1 else 0 := by | |
| classical | |
| simp only [identityVector, lp.single_apply, Pi.single_apply, Prod.mk.injEq] | |
| theorem groupTrace_regular {A : Matrix (Fin n) (Fin n) (MonoidAlgebra ℤ G)} | |
| {T : Hilbert G n →L[ℂ] Hilbert G n} (hT : IsRegularOperator A T) : | |
| groupTrace T = ∑ i : Fin n, ((A i i).coeff 1 : ℂ) := by | |
| classical | |
| unfold groupTrace | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| have hi (ξ : Hilbert G n) : inner ℂ (identityVector i) ξ = ξ (i, 1) := by | |
| simp [identityVector, lp.inner_single_left, RCLike.inner_apply] | |
| rw [hi, hT] | |
| rw [Finset.sum_eq_single i] | |
| · rw [(A i i).coeff.sum_eq_single 1] | |
| · simp [identityVector_apply] | |
| · intro g _ hg | |
| simp [identityVector_apply, hg] | |
| · simp | |
| · intro j _ hji | |
| simp [identityVector_apply, hji] | |
| · simp | |
| end RegularTrace | |
| end GroupRingDeterminant | |
| end | |
| end OAI | |