161 lines
3.7 KiB
Text
161 lines
3.7 KiB
Text
/-
|
||
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
|
||
Released under Apache 2.0 license.
|
||
Authors: Joseph Tooby-Smith
|
||
-/
|
||
import HepLean.AnomalyCancellation.PureU1.Basic
|
||
import Mathlib.Tactic.Polyrith
|
||
import Mathlib.RepresentationTheory.Basic
|
||
/-!
|
||
# Basis of `LinSols`
|
||
|
||
We give a basis of vector space `LinSols`, and find the rank thereof.
|
||
|
||
-/
|
||
|
||
namespace PureU1
|
||
|
||
open BigOperators
|
||
|
||
variable {n : ℕ}
|
||
namespace BasisLinear
|
||
|
||
/-- The basis elements as charges, defined to have a `1` in the `j`th position and a `-1` in the
|
||
last position. -/
|
||
@[simp]
|
||
def asCharges (j : Fin n) : (PureU1 n.succ).charges :=
|
||
(fun i =>
|
||
if i = j.castSucc then
|
||
1
|
||
else
|
||
if i = Fin.last n then
|
||
- 1
|
||
else
|
||
0)
|
||
|
||
lemma asCharges_eq_castSucc (j : Fin n) :
|
||
asCharges j (Fin.castSucc j) = 1 := by
|
||
simp [asCharges]
|
||
|
||
lemma asCharges_ne_castSucc {k j : Fin n} (h : k ≠ j) :
|
||
asCharges k (Fin.castSucc j) = 0 := by
|
||
simp [asCharges]
|
||
split
|
||
rename_i h1
|
||
rw [Fin.ext_iff] at h1
|
||
simp_all
|
||
rw [Fin.ext_iff] at h
|
||
simp_all
|
||
split
|
||
rename_i h1 h2
|
||
rw [Fin.ext_iff] at h1 h2
|
||
simp at h1 h2
|
||
have hj : j.val < n := by
|
||
exact j.prop
|
||
simp_all
|
||
rfl
|
||
|
||
/-- The basis elements as `LinSols`. -/
|
||
@[simps!]
|
||
def asLinSols (j : Fin n) : (PureU1 n.succ).LinSols :=
|
||
⟨asCharges j, by
|
||
intro i
|
||
simp at i
|
||
match i with
|
||
| 0 =>
|
||
simp only [ Fin.isValue, PureU1_linearACCs, accGrav,
|
||
LinearMap.coe_mk, AddHom.coe_mk, Fin.coe_eq_castSucc]
|
||
rw [Fin.sum_univ_castSucc]
|
||
rw [Finset.sum_eq_single j]
|
||
simp
|
||
have hn : ¬ (Fin.last n = Fin.castSucc j) := by
|
||
have hj := j.prop
|
||
rw [Fin.ext_iff]
|
||
simp
|
||
omega
|
||
split
|
||
rename_i ht
|
||
exact (hn ht).elim
|
||
rfl
|
||
intro k _ hkj
|
||
exact asCharges_ne_castSucc hkj.symm
|
||
intro hk
|
||
simp at hk⟩
|
||
|
||
|
||
lemma sum_of_vectors {n : ℕ} (f : Fin k → (PureU1 n).LinSols) (j : Fin n) :
|
||
(∑ i : Fin k, (f i)).1 j = (∑ i : Fin k, (f i).1 j) := by
|
||
induction k
|
||
simp
|
||
rfl
|
||
rename_i k l hl
|
||
rw [Fin.sum_univ_castSucc, Fin.sum_univ_castSucc]
|
||
have hlt := hl (f ∘ Fin.castSucc)
|
||
erw [← hlt]
|
||
simp
|
||
|
||
/-- The coordinate map for the basis. -/
|
||
noncomputable
|
||
def coordinateMap : ((PureU1 n.succ).LinSols) ≃ₗ[ℚ] Fin n →₀ ℚ where
|
||
toFun S := Finsupp.equivFunOnFinite.invFun (S.1 ∘ Fin.castSucc)
|
||
map_add' S T := by
|
||
simp
|
||
ext
|
||
simp
|
||
map_smul' a S := by
|
||
simp
|
||
ext
|
||
simp
|
||
rfl
|
||
invFun f := ∑ i : Fin n, f i • asLinSols i
|
||
left_inv S := by
|
||
simp
|
||
apply pureU1_anomalyFree_ext
|
||
intro j
|
||
rw [sum_of_vectors]
|
||
simp only [HSMul.hSMul, SMul.smul, PureU1_numberCharges,
|
||
asLinSols_val, Equiv.toFun_as_coe,
|
||
Fin.coe_eq_castSucc, mul_ite, mul_one, mul_neg, mul_zero, Equiv.invFun_as_coe]
|
||
rw [Finset.sum_eq_single j]
|
||
simp
|
||
intro k _ hkj
|
||
rw [asCharges_ne_castSucc hkj]
|
||
simp
|
||
simp
|
||
right_inv f := by
|
||
simp
|
||
ext
|
||
rename_i j
|
||
simp
|
||
rw [sum_of_vectors]
|
||
simp only [HSMul.hSMul, SMul.smul, PureU1_numberCharges,
|
||
asLinSols_val, Equiv.toFun_as_coe,
|
||
Fin.coe_eq_castSucc, mul_ite, mul_one, mul_neg, mul_zero, Equiv.invFun_as_coe]
|
||
rw [Finset.sum_eq_single j]
|
||
simp
|
||
intro k _ hkj
|
||
rw [asCharges_ne_castSucc hkj]
|
||
simp
|
||
simp
|
||
|
||
/-- The basis of `LinSols`.-/
|
||
noncomputable
|
||
def asBasis : Basis (Fin n) ℚ ((PureU1 n.succ).LinSols) where
|
||
repr := coordinateMap
|
||
|
||
instance : Module.Finite ℚ ((PureU1 n.succ).LinSols) :=
|
||
Module.Finite.of_basis asBasis
|
||
|
||
lemma finrank_AnomalyFreeLinear :
|
||
FiniteDimensional.finrank ℚ (((PureU1 n.succ).LinSols)) = n := by
|
||
have h := Module.mk_finrank_eq_card_basis (@asBasis n)
|
||
simp_all
|
||
simp [FiniteDimensional.finrank]
|
||
rw [h]
|
||
simp_all only [Cardinal.toNat_natCast]
|
||
|
||
|
||
end BasisLinear
|
||
|
||
|
||
end PureU1
|