| import OAI.Analysis.GroupDeterminants.ShearTrace |
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|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Pointwise |
|
|
| namespace GroupRingDeterminant |
| section TransferTrace |
| variable {G |
|
|
| def ⊕ ⊕ ℂ |
| Matrix.fromBlocks |
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| def ℂ |
| Matrix ⊕ ⊕ ℂ |
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|
| def ℂ |
| Matrix ⊕ ⊕ ℂ |
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| def ℂ |
| Matrix ⊕ ⊕ ℂ |
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|
| lemma ℂ |
| simp |
| lemma ℂ |
| simp |
| omit |
| lemma ℂ |
| lowerShear |
| rw ← |
| simp |
| omit |
| lemma ℂ |
| upperShear |
| rw ← |
| simp |
|
|
| lemma ℝ ℂ |
| Matrix.fromBlocks • ℂ |
| (projectionReweight ⁻¹ ℝ |
| weightedPair ⁻¹ |
| apply |
| intro |
| rcases | ;> rcases j with j | j |
| · by_cases hij : i = j <;> |
| simp [weightedPair, topBlockIdentity, bottomBlock, Matrix.fromBlocks, hij] |
| · simp [weightedPair, topBlockIdentity, bottomBlock, Matrix.fromBlocks] |
| · simp [weightedPair, topBlockIdentity, bottomBlock, Matrix.fromBlocks] |
| · simp [weightedPair, topBlockIdentity, bottomBlock, projectionReweight, |
| Matrix.fromBlocks, Matrix.one_apply] |
|
|
| lemma transfer_trace_log (q : ℝ) (hq : 0 < q) |
| (P : Matrix m m (MonoidAlgebra ℂ G)) (hP : P * P = P) |
| (U : Matrix m r (MonoidAlgebra ℂ G)) (V : Matrix r m (MonoidAlgebra ℂ G)) : |
| let M := indexedComplexRepresentation (transferMatrix q P U V) |
| groupTrace (CFC.log (star M * M)) = |
| (2 * ((Fintype.card r : ℝ) - (∑ i, (P i i).coeff 1).re) * Real.log q : ℝ) := by |
| let Q : Matrix (r ⊕ m) (r ⊕ m) (MonoidAlgebra ℂ G) := topBlockIdentity |
| let R : Matrix (r ⊕ m) (r ⊕ m) (MonoidAlgebra ℂ G) := bottomBlock P |
| have hQ : Q * Q = Q := by simp [Q, topBlockIdentity, Matrix.fromBlocks_multiply] |
| have hR : R * R = R := by simp [R, bottomBlock, Matrix.fromBlocks_multiply, hP] |
| have hQR : Q * R = 0 := by |
| simp [Q, R, topBlockIdentity, bottomBlock, Matrix.fromBlocks_multiply, Matrix.fromBlocks_zero] |
| have hRQ : R * Q = 0 := by |
| simp [Q, R, topBlockIdentity, bottomBlock, Matrix.fromBlocks_multiply, Matrix.fromBlocks_zero] |
| have hQs : IsSelfAdjoint Q := by |
| change Matrix.conjTranspose Q = Q |
| simp [Q, topBlockIdentity, Matrix.fromBlocks_conjTranspose] |
| let B := weightedPair Q R q (-q⁻¹) (-1) |
| have hB : IsUnit B := weightedPair_unit Q R hQ hR hQR hRQ _ _ _ |
| hq.ne' (neg_ne_zero.mpr (inv_ne_zero hq.ne')) (by norm_num) |
| let S := lowerNil (q⁻¹ • U) |
| let T := upperNil (q⁻¹ • V) |
| have htS : ∑ i, (S i i).coeff 1 = 0 := by simp [S, lowerNil, Fintype.sum_sum_type] |
| have htT : ∑ i, (T i i).coeff 1 = 0 := by simp [T, upperNil, Fintype.sum_sum_type] |
| have hf : transferMatrix q P U V = (1 + S) * B * (1 + T) := by |
| rw [← transferMatrix_factorization q hq.ne', transfer_diagonal_weightedPair, |
| lowerShear_eq_one_add, upperShear_eq_one_add] |
| have H := indexed_trace_log_shear B S T hB (lowerNil_square _) (upperNil_square _) htS htT |
| have H' := indexed_trace_log_weightedPair Q R hQ hR hQR hRQ hQs (Real.log q) |
| have hexp : Real.exp (Real.log q) = q := Real.exp_log hq |
| have hexpn : Real.exp (-Real.log q) = q⁻¹ := by rw [Real.exp_neg, hexp] |
| dsimp only at H H' ⊢ |
| rw [hf, H] |
| have ht : (∑ i, ((Q - R) i i).coeff 1) = |
| (Fintype.card r : ℂ) - ∑ i, (P i i).coeff 1 := by |
| simp [Q, R, topBlockIdentity, bottomBlock, Fintype.sum_sum_type] |
| rw [hexp, hexpn, ht] at H' |
| calc |
| _ = ((Real.log q * (2 * ((Fintype.card r : ℂ) - ∑ i, (P i i).coeff 1).re) : ℝ) : ℂ) := H' |
| _ = _ := by congr 1; simp only [Complex.sub_re, Complex.natCast_re]; ring |
| |
| end TransferTrace |
| end GroupRingDeterminant |
| |
| end |
| |
| end OAI |
| |
