| import Mathlib |
|
|
| namespace OAI |
|
|
| section |
| open scoped Classical |
| open scoped BigOperators ComplexConjugate MonoidAlgebra |
| open scoped BigOperators ComplexConjugate |
| open scoped MonoidAlgebra BigOperators |
| open scoped BigOperators MonoidAlgebra Classical |
|
|
| attribute [local instance] Classical.propDecidable |
| open scoped BigOperators |
| |
| namespace PartialPermutation |
|
|
| section MatrixTraces |
| variable {V : Type*} [AddCommGroup V] [Module ℂ V] |
|
|
| lemma matrixUnit_trace_eq {d : ℕ} |
| (A : Matrix (Fin d) (Fin d) ℂ →ₐ[ℂ] Module.End ℂ V) |
| (B : Module.End ℂ V) (hB : ∀ X, Commute (A X) B) (i j : Fin d) : |
| LinearMap.trace ℂ V (A (Matrix.single i i 1) * B) = |
| LinearMap.trace ℂ V (A (Matrix.single j j 1) * B) := by |
| have hu (i j k : Fin d) : A (Matrix.single i j 1) * A (Matrix.single j k 1) = |
| A (Matrix.single i k 1) := by |
| rw [← map_mul, Matrix.single_mul_single_same, one_mul] |
| calc |
| _ = LinearMap.trace ℂ V |
| (A (Matrix.single i j 1) * (A (Matrix.single j i 1) * B)) := by |
| rw [← mul_assoc, hu] |
| _ = LinearMap.trace ℂ V |
| ((A (Matrix.single j i 1) * B) * A (Matrix.single i j 1)) := |
| LinearMap.trace_mul_comm ℂ _ _ |
| _ = LinearMap.trace ℂ V |
| (A (Matrix.single j i 1) * A (Matrix.single i j 1) * B) := by |
| rw [mul_assoc, ← (hB _).eq, ← mul_assoc] |
| _ = _ := by rw [hu] |
|
|
| lemma degree_mul_matrixUnit_trace {d : ℕ} |
| (A : Matrix (Fin d) (Fin d) ℂ →ₐ[ℂ] Module.End ℂ V) |
| (B : Module.End ℂ V) (hB : ∀ X, Commute (A X) B) (t : Fin d) : |
| (d : ℂ) * LinearMap.trace ℂ V (A (Matrix.single t t 1) * B) = |
| LinearMap.trace ℂ V B := by |
| classical |
| calc |
| _ = ∑ j : Fin d, LinearMap.trace ℂ V (A (Matrix.single t t 1) * B) := by simp |
| _ = ∑ j : Fin d, LinearMap.trace ℂ V (A (Matrix.single j j 1) * B) := by |
| apply Finset.sum_congr rfl |
| intro j hj |
| exact matrixUnit_trace_eq A B hB t j |
| _ = LinearMap.trace ℂ V ((∑ j : Fin d, A (Matrix.single j j 1)) * B) := by |
| simp [Finset.sum_mul] |
| _ = _ := by rw [← map_sum, Matrix.sum_single_one, map_one, one_mul] |
|
|
| end MatrixTraces |
|
|
| section CommutingDegrees |
| variable {V : Type*} [AddCommGroup V] [Module ℂ V] [FiniteDimensional ℂ V] |
| {ι : Type*} [DecidableEq ι] (d : ι → ℕ) (hd : ∀ i, 0 < d i) |
| (A : ∀ i, Matrix (Fin (d i)) (Fin (d i)) ℂ →ₐ[ℂ] Module.End ℂ V) |
| (hcomm : ∀ i j, i ≠ j → ∀ X Y, Commute (A i X) (A j Y)) |
|
|
| include hd A hcomm |
|
|
| lemma commuting_degrees_dvd_finrank (s : Finset ι) : |
| (∏ i ∈ s, d i) ∣ Module.finrank ℂ V := by |
| classical |
| let p (i : ι) : Module.End ℂ V := |
| A i (Matrix.single (⟨0, hd i⟩ : Fin (d i)) ⟨0, hd i⟩ 1) |
| have hp (i : ι) : IsIdempotentElem (p i) := by |
| dsimp [IsIdempotentElem, p] |
| rw [← map_mul, Matrix.single_mul_single_same, one_mul] |
| have hc (i j : ι) (hij : i ≠ j) : Commute (p i) (p j) := hcomm i j hij _ _ |
| have hcc (s : Finset ι) : (↑s : Set ι).Pairwise (Function.onFun Commute p) := by |
| intro i hi j hj hij |
| exact hc i j hij |
| let P (s : Finset ι) : Module.End ℂ V := |
| s.noncommProd p (hcc s) |
| have hP (s : Finset ι) (i : ι) (hi : i ∉ s) (X) : Commute (A i X) (P s) := by |
| apply Finset.noncommProd_commute |
| intro j hj |
| exact hcomm i j (fun he => hi (he ▸ hj)) _ _ |
| have hid (s : Finset ι) : IsIdempotentElem (P s) := by |
| induction s using Finset.induction_on with |
| | empty => simp [P, Finset.noncommProd_empty, IsIdempotentElem] |
| | @insert i s hi ih => |
| dsimp only [P] |
| rw [Finset.noncommProd_insert_of_notMem _ _ _ _ hi] |
| change IsIdempotentElem (p i * P s) |
| change (p i * P s) * (p i * P s) = p i * P s |
| rw [(hP s i hi _).symm.mul_mul_mul_comm, hp i, ih] |
| have ht (s : Finset ι) : |
| ((∏ i ∈ s, d i : ℕ) : ℂ) * LinearMap.trace ℂ V (P s) = Module.finrank ℂ V := by |
| induction s using Finset.induction_on with |
| | empty => simp [P, Finset.noncommProd_empty, Module.End.one_eq_id, LinearMap.trace_id] |
| | @insert i s hi ih => |
| dsimp only [P] |
| rw [Finset.prod_insert hi, Nat.cast_mul, |
| Finset.noncommProd_insert_of_notMem _ _ _ _ hi] |
| change ((d i : ℂ) * ↑(∏ j ∈ s, d j)) * |
| LinearMap.trace ℂ V (p i * P s) = _ |
| calc |
| _ = (↑(∏ j ∈ s, d j) : ℂ) * |
| ((d i : ℂ) * LinearMap.trace ℂ V (p i * P s)) := by ring |
| _ = (↑(∏ j ∈ s, d j) : ℂ) * LinearMap.trace ℂ V (P s) := by |
| rw [degree_mul_matrixUnit_trace (A i) (P s) (hP s i hi)] |
| _ = _ := ih |
| refine ⟨Module.finrank ℂ (LinearMap.range (P s)), ?_⟩ |
| have he := ht s |
| rw [(LinearMap.IsIdempotentElem.isProj_range (P s) (hid s)).trace] at he |
| exact_mod_cast he.symm |
|
|
| lemma commuting_degrees_le_finrank [Nontrivial V] (s : Finset ι) : |
| (∏ i ∈ s, d i) ≤ Module.finrank ℂ V := |
| Nat.le_of_dvd (Module.finrank_pos (R := ℂ)) |
| (commuting_degrees_dvd_finrank d hd A hcomm s) |
|
|
| end CommutingDegrees |
|
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| end PartialPermutation |
|
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| end |
|
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| end OAI |
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