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18 declarations, 0 contain sorry, 0 declared as axioms

Abstract Hecke operators

Generalities on Hecke operators acting on the fixed points of a group action. We define the abstract Hecke operator attached to a double coset U g U acting on R-valued functions invariant under U.

Imports (8)

Imported by 2 files

def FixedPoints.submodule FixedPoints as a submodule line 67 source
Source documentation
FixedPoints as a submodule.

standard axioms: Quot.sound, propext what this means
Statement uses: Monoid, Semiring, AddCommMonoid, Module, DistribMulAction, SMulCommClass, Submodule
Proof uses: AddSubmonoid, FixedPoints.addSubmonoid, Submodule.mk
19 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
def FixedPoints.submodule (G R M : Type*) [Monoid G] [Semiring R] [AddCommMonoid M]
    [Module R M] [DistribMulAction G M] [SMulCommClass G R M] : Submodule R M where
…
```
instance FixedPoints.(instance at line 79) line 79 source
lemma FixedPoints.coe_zero line 83 source
no axioms what this means
Statement uses: Group, AddCommMonoid, DistribMulAction, Eq, Membership.mem, Set, MulAction.fixedPoints, OfNat.ofNat
Proof uses: rfl
24 instances, coercions and auxiliary constants left out.
lemma FixedPoints.coe_add line 86 source
no axioms what this means
Statement uses: Group, AddCommMonoid, DistribMulAction, MulAction.fixedPoints, Eq, Membership.mem, Set, HAdd.hAdd
Proof uses: rfl
24 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
instance FixedPoints.(instance at line 95) line 95 source
lemma FixedPoints.coe_smul line 100 source
no axioms what this means
Statement uses: Group, AddCommMonoid, DistribMulAction, SMul, SMulCommClass, MulAction.fixedPoints, Eq, Membership.mem, Set, HSMul.hSMul
Proof uses: rfl
30 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
instance FixedPoints.(instance at line 103) line 103 source
instance FixedPoints.module Probably this should be a submodule instance and then get module instance for free line 115 source
Source comment
Probably this should be a submodule instance and then get module instance for free

standard axioms: propext what this means
Statement uses: Group, AddCommMonoid, DistribMulAction, Ring, Module, SMulCommClass, MulAction.fixedPoints
Proof uses: Module.mk, DistribMulAction.mk, MulAction.mk, SemigroupAction.mk
38 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
```
instance module [Ring R] [Module R A] [SMulCommClass G R A] : Module R (fixedPoints G A) where
…
```
lemma Set.bijOn_smul line 137 source
standard axioms: Quot.sound, propext what this means
Statement uses: Group, Subgroup, Set, Membership.mem, Set.BijOn, HSMul.hSMul, HMul.hMul
Proof uses: And.intro, Set.MapsTo, And, Set.InjOn, Set.SurjOn, Exists, Eq, Exists.intro, MulMemClass.mul_mem, of_eq_true, Eq.trans, True, congrFun', congrArg, mul_assoc, eq_self, Set.injOn_of_injective, MulAction.injective, Set.image, Inv.inv and 3 more
41 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
lemma AbstractHeckeOperator.eq_finsum_quotient_out_of_bijOn' If a is fixed by V then ∑ᶠ g ∈ s, g • a is independent of the choice s of coset representatives in G for a subset of G ⧸ V line 159 source
Source documentation
If a is fixed by V then ∑ᶠ g ∈ s, g • a is independent of the choice s of coset representatives in G for a subset of G ⧸ V

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, AddCommMonoid, DistribMulAction, Subgroup, MulAction.fixedPoints, Subtype, Membership.mem, Set, HasQuotient.Quotient, Set.BijOn, Eq, finsum, HSMul.hSMul, Set.image, Quotient, QuotientGroup.leftRel, Quotient.out
Proof uses: Set.BijOn.comp, Quot.out, Set.InjOn.bijOn_image, Set.injOn_of_injective, Quotient.out_injective, HMul.hMul, Exists, QuotientGroup.mk_out_eq_mul, Eq.mpr, id, congrArg, SemigroupAction.mul_smul, Subtype.property, Eq.refl, finsum_mem_eq_of_bijOn
54 instances, coercions and auxiliary constants left out.

Tactic trace: 9 steps
Open to load the checked before-and-after goal states.
```
lemma eq_finsum_quotient_out_of_bijOn' (a : fixedPoints V A)
    {X : Set (G ⧸ V)}
    {s : Set G} (hs : s.BijOn (QuotientGroup.mk : G → G ⧸ V) X) :
    ∑ᶠ g ∈ s, g • (a : A) = ∑ᶠ g ∈ Quotient.out '' X, g • (a : A) := by
…
```
def AbstractHeckeOperator.heckeOperator.toFun The Hecke operator T_g = [UgV] : A^V → A^U associated to the double coset UgV line 174 source
Source documentation
The Hecke operator T_g = [UgV] : A^V → A^U associated to the double coset UgV.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, AddCommMonoid, DistribMulAction, Subgroup, Set.Finite, HasQuotient.Quotient, Set.image, HMul.hMul, Set, Singleton.singleton, MulAction.fixedPoints, Subtype, Membership.mem
Proof uses: Subtype.mk, finsum, Quotient, QuotientGroup.leftRel, Quotient.out, HSMul.hSMul
49 instances, coercions and auxiliary constants left out.

Tactic trace: 21 steps
Open to load the checked before-and-after goal states.
```
noncomputable def heckeOperator.toFun (a : fixedPoints V A) : fixedPoints U A :=
…
```
def AbstractHeckeOperator.heckeOperator.addMonoidHom The Hecke operator T_g = [UgV] : A^V → A^U packaged as an additive monoid homomorphism line 195 source
Source documentation
The Hecke operator T_g = [UgV] : A^V → A^U packaged as an additive monoid homomorphism.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, AddCommMonoid, DistribMulAction, Subgroup, Set.Finite, HasQuotient.Quotient, Set.image, HMul.hMul, Set, Singleton.singleton, AddMonoidHom, MulAction.fixedPoints, Subtype, Membership.mem
Proof uses: AddMonoidHom.mk, ZeroHom.mk, AbstractHeckeOperator.heckeOperator.toFun
52 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
noncomputable def heckeOperator.addMonoidHom : fixedPoints V A →+ fixedPoints U A where
…
```
def AbstractHeckeOperator.heckeOperator The Hecke operator T_g = [UgV] : A^V → A^U as an R-linear map, where R is any ring acting on A and commuting with the G-action line 211 source
Source documentation
The Hecke operator T_g = [UgV] : A^V → A^U as an R-linear map, where R is any ring acting on A and commuting with the G-action.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, AddCommMonoid, DistribMulAction, Subgroup, Set.Finite, HasQuotient.Quotient, Set.image, HMul.hMul, Set, Singleton.singleton, Ring, Module, SMulCommClass, LinearMap, RingHom.id, Subtype, Membership.mem, Submodule, FixedPoints.submodule
Proof uses: AddMonoidHom, MulAction.fixedPoints, AbstractHeckeOperator.heckeOperator.addMonoidHom, LinearMap.mk, AddHom.mk, ZeroHom.toFun
75 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
noncomputable def heckeOperator :
    FixedPoints.submodule V R A →ₗ[R] FixedPoints.submodule U R A where
…
```
Mentions: FixedPoints.submodule
lemma AbstractHeckeOperator.heckeOperator_apply line 219 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, AddCommMonoid, DistribMulAction, Subgroup, Set.Finite, HasQuotient.Quotient, Set.image, HMul.hMul, Set, Singleton.singleton, Ring, Module, SMulCommClass, MulAction.fixedPoints, Subtype, Membership.mem, Eq, Submodule, FixedPoints.submodule, LinearMap and 7 more
Proof uses: rfl
78 instances, coercions and auxiliary constants left out.
lemma AbstractHeckeOperator.heckeOperator_apply' line 224 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, AddCommMonoid, DistribMulAction, Subgroup, Set.Finite, HasQuotient.Quotient, Set.image, HMul.hMul, Set, Singleton.singleton, Ring, Module, SMulCommClass, MulAction.fixedPoints, Subtype, Membership.mem, Eq, Submodule, FixedPoints.submodule, LinearMap and 7 more
Proof uses: Eq.mpr, finsum, Quotient, id, congrArg, AbstractHeckeOperator.heckeOperator_apply, Finset, Eq.symm, finsum_mem_coe_finset, finsum_mem_image, Function.Injective.injOn, Quotient.out_injective, eq_of_heq, HEq, Eq.refl, HEq.refl, heq_of_eq, funext, finsum_congr_Prop, outParam and 12 more
88 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
lemma AbstractHeckeOperator.heckeOperator_eq_finsetSum line 231 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, AddCommMonoid, DistribMulAction, Subgroup, Set.Finite, HasQuotient.Quotient, Set.image, HMul.hMul, Set, Singleton.singleton, Ring, Module, SMulCommClass, MulAction.fixedPoints, Subtype, Membership.mem, Finset, Set.BijOn, Eq, Submodule and 6 more
Proof uses: Eq.trans, finsum, Quotient, QuotientGroup.leftRel, Quotient.out, Eq.symm, AbstractHeckeOperator.eq_finsum_quotient_out_of_bijOn', finsum_mem_coe_finset
74 instances, coercions and auxiliary constants left out.
lemma AbstractHeckeOperator.heckeOperator_eq_of_mem_normalizer line 236 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, AddCommMonoid, DistribMulAction, Subgroup, Ring, Module, SMulCommClass, Set.Finite, HasQuotient.Quotient, Set.image, HMul.hMul, Set, Singleton.singleton, MulAction.fixedPoints, Subtype, Membership.mem, Subgroup.normalizer, Eq, Submodule, FixedPoints.submodule and 4 more
Proof uses: Eq.mpr, finsum, Quotient, QuotientGroup.leftRel, Quotient.out, id, congrArg, AbstractHeckeOperator.heckeOperator_apply, AbstractHeckeOperator.eq_finsum_quotient_out_of_bijOn', Set.ext, of_eq_true, Iff, Eq.trans, Inv.inv, True, congr, Set.preimage, congrFun', Set.mul_singleton, Set.image_mul_right and 41 more
108 instances, coercions and auxiliary constants left out.

Tactic trace: 16 steps
Open to load the checked before-and-after goal states.
theorem AbstractHeckeOperator.comm line 257 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Group, AddCommMonoid, DistribMulAction, Subgroup, Ring, Module, SMulCommClass, Set.Finite, HasQuotient.Quotient, Set.image, HMul.hMul, Set, Singleton.singleton, Exists, And, Set.BijOn, Membership.mem, Eq, LinearMap, RingHom.id and 5 more
Proof uses: LinearMap.ext, Subtype.ext, id, Eq.mpr, finsum, Quotient, QuotientGroup.leftRel, Quotient.out, HSMul.hSMul, MulAction.fixedPoints, congrArg, AbstractHeckeOperator.heckeOperator_apply, Eq.symm, AbstractHeckeOperator.eq_finsum_quotient_out_of_bijOn', propext, Set.BijOn.finite_iff_finite, funext, finsum_congr_Prop, Eq.refl, DistribSMul and 5 more
80 instances, coercions and auxiliary constants left out.

Tactic trace: 25 steps
Open to load the checked before-and-after goal states.