| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| namespace KKLocalPatching |
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| noncomputable section |
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| variable {G : Type*} [Fintype G] [DecidableEq G] [AddCommGroup G] |
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| def shiftMatrix (g : G) : Matrix G G ℂ := (Equiv.addRight g).permMatrix ℂ |
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| omit [Fintype G] in |
| @[simp] theorem shift_zero : shiftMatrix (0:G)=1 := by |
| have h : Equiv.addRight (0:G)=1 := by ext; simp [Equiv.addRight] |
| simp [shiftMatrix,h] |
| |
| @[simp] theorem shift_add (g h : G) : shiftMatrix (g+h)=shiftMatrix g*shiftMatrix h := by |
| have he : Equiv.addRight (g+h)=(Equiv.addRight h)*(Equiv.addRight g) := by |
| ext x |
| simp [Equiv.addRight,add_assoc] |
| rw [shiftMatrix,he,Matrix.permMatrix_mul] |
| rfl |
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| omit [Fintype G] in |
| @[simp] theorem shift_star (g : G) : star (shiftMatrix g)=shiftMatrix (-g) := by |
| change ((Equiv.addRight g).permMatrix ℂ).conjTranspose=_ |
| rw [Matrix.conjTranspose_permMatrix] |
| congr 1 |
| ext x |
| simp [Equiv.addRight] |
| |
| def shiftUnitary (g : G) : unitary (Matrix G G ℂ) := |
| ⟨shiftMatrix g, by |
| constructor |
| · rw [shift_star,← shift_add,neg_add_cancel,shift_zero] |
| · rw [shift_star,← shift_add,add_neg_cancel,shift_zero]⟩ |
| |
| @[simp] theorem shiftUnitary_val (g:G) : (shiftUnitary g).val=shiftMatrix g := rfl |
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| def shiftHom : Multiplicative G →* unitary (Matrix G G ℂ) where |
| toFun g := shiftUnitary g.toAdd |
| map_one' := Subtype.ext shift_zero |
| map_mul' g h := Subtype.ext (shift_add g.toAdd h.toAdd) |
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| @[simp] theorem shift_zsmul (k : ℤ) (g : G) : shiftUnitary (k • g)=(shiftUnitary g)^k := |
| (shiftHom : Multiplicative G →* unitary (Matrix G G ℂ)).map_zpow _ _ |
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| theorem shift_trace (g : G) (hg : g ≠ 0) : Matrix.trace (shiftMatrix g)=0 := by |
| rw [shiftMatrix,Matrix.trace_permutation] |
| have hf : Function.fixedPoints (Equiv.addRight g)=∅ := by |
| ext x |
| change (x+g=x) ↔ False |
| simp only [add_eq_left] |
| exact iff_false_intro hg |
| rw [hf,Set.ncard_empty,Nat.cast_zero] |
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| end |
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| end KKLocalPatching |
| end |
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| end OAI |
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