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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 | import OAI.Analysis.GroupDeterminants.RationalRepresentation
namespace OAI
noncomputable section
open scoped BigOperators Pointwise
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]
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
end
end OAI
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