| import OAI.Analysis.GroupDeterminants.RationalRepresentation |
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| namespace OAI |
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| noncomputable section |
| open scoped BigOperators Pointwise |
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| namespace GroupRingDeterminant |
| section ProjectionCriterion |
| variable {G r m : Type} [Group G] [Group.FG G] |
| variable [Fintype r] [Fintype m] [DecidableEq r] [DecidableEq m] [Nonempty r] |
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| theorem projection_criterion (D : ℕ) (hodd : Odd D) |
| (P : Matrix m m (MonoidAlgebra ℚ G)) (hP : P * P = P ∧ IsSelfAdjoint P) |
| (htrace : (Fintype.card r : ℚ) < ∑ i, (P i i).coeff 1) |
| (Z : Matrix m m (MonoidAlgebra ℤ G)) (hZ : Z.map toRat = (D : ℚ) • P) |
| (u : Matrix m r (MonoidAlgebra (ZMod 2) G)) |
| (v : Matrix r m (MonoidAlgebra (ZMod 2) G)) (huv : u * v = Z.map (toMod 2)) : |
| MainStatement := by |
| obtain ⟨k, hk, hq, hqD⟩ := exists_binary_modulus D hodd.pos hodd |
| obtain ⟨U, V, A, hA, hAu⟩ := integral_transfer_of_congruence D hodd P hP.1 Z hZ u v huv k hk hqD |
| let e := Fintype.equivFin (r ⊕ m) |
| let A' := Matrix.reindex e e A |
| have hAu' : IsUnit (rationalMatrix A') := reindexed_rational_isUnit A hAu |
| have hqR : (1 : ℝ) < (2 ^ k : ℕ) := by exact_mod_cast hq |
| have hPC : P.map rationalComplexMap * P.map rationalComplexMap = P.map rationalComplexMap := by |
| rw [← Matrix.map_mul, hP.1] |
| have hOp : regularOperator A' = indexedComplexRepresentation |
| (transferMatrix (((2 ^ k : ℕ) : ℚ) : ℝ) (P.map rationalComplexMap) |
| ((U.map toRat).map rationalComplexMap) ((V.map toRat).map rationalComplexMap)) := by |
| rw [← rationalIndexed_integer A, hA, transferMatrix_map_complex] |
| have hlog := transfer_trace_log (((2 ^ k : ℕ) : ℚ) : ℝ) |
| (lt_trans zero_lt_one hqR) (P.map rationalComplexMap) hPC |
| ((U.map toRat).map rationalComplexMap) ((V.map toRat).map rationalComplexMap) |
| dsimp only at hlog |
| rw [← hOp, rationalComplexMap_trace] at hlog |
| have htraceR : (Fintype.card r : ℝ) < ((∑ i, (P i i).coeff 1 : ℚ) : ℝ) := by |
| exact_mod_cast htrace |
| have hn : 2 * ((Fintype.card r : ℝ) - ((∑ i, (P i i).coeff 1 : ℚ) : ℝ)) * |
| Real.log (((2 ^ k : ℕ) : ℚ) : ℝ) < 0 := |
| mul_neg_of_neg_of_pos (mul_neg_of_pos_of_neg (by norm_num) (sub_neg.mpr htraceR)) |
| (Real.log_pos (by simpa using hqR)) |
| refine ⟨G, inferInstance, inferInstance, Fintype.card (r ⊕ m), |
| Fintype.card_pos, A', regularOperator A', hAu', regularOperator_spec A', |
| regularOperator_isUnit hAu', fkDet_pos _, ?_⟩ |
| unfold fkDet |
| apply Real.exp_lt_one_iff.mpr |
| rw [hlog, Complex.ofReal_re] |
| exact div_neg_of_neg_of_pos hn (by norm_num) |
|
|
| end ProjectionCriterion |
| end GroupRingDeterminant |
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| end |
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| end OAI |
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