File: Mathlib/RepresentationTheory/Maschke.lean

full_name: LinearMap.conjugate
code: /-- We define the conjugate of `π` by `g`, as a `k`-linear map. -/
def conjugate (g : G) : W →ₗ[k] V 
goal: W →ₗ[k] V

full_name: LinearMap.conjugate_apply
code: theorem conjugate_apply (g : G) (v : W) :
    π.conjugate g v = MonoidAlgebra.single g⁻¹ (1 : k) • π (MonoidAlgebra.single g (1 : k) • v) 
goal: (π.conjugate g) v = MonoidAlgebra.single g⁻¹ 1 • π (MonoidAlgebra.single g 1 • v)

full_name: LinearMap.conjugate_i
code: theorem conjugate_i (g : G) (v : V) : (conjugate π g : W → V) (i v) = v 
goal: (π.conjugate g) (i v) = v

full_name: LinearMap.sumOfConjugates
code: /-- The sum of the conjugates of `π` by each element `g : G`, as a `k`-linear map.

(We postpone dividing by the size of the group as long as possible.)
-/
def sumOfConjugates : W →ₗ[k] V 
goal: W →ₗ[k] V

full_name: LinearMap.sumOfConjugates_apply
code: lemma sumOfConjugates_apply (v : W) : π.sumOfConjugates G v = ∑ g : G, π.conjugate g v 
goal: (LinearMap.sumOfConjugates G π) v = ∑ g : G, (π.conjugate g) v

full_name: LinearMap.sumOfConjugatesEquivariant
code: /-- In fact, the sum over `g : G` of the conjugate of `π` by `g` is a `k[G]`-linear map.
-/
def sumOfConjugatesEquivariant : W →ₗ[MonoidAlgebra k G] V 
goal: W →ₗ[MonoidAlgebra k G] V

full_name: LinearMap.sumOfConjugatesEquivariant_apply
code: theorem sumOfConjugatesEquivariant_apply (v : W) :
    π.sumOfConjugatesEquivariant G v = ∑ g : G, π.conjugate g v 
goal: (LinearMap.sumOfConjugatesEquivariant G π) v = ∑ g : G, (π.conjugate g) v

full_name: LinearMap.equivariantProjection
code: /-- We construct our `k[G]`-linear retraction of `i` as
$$ \frac{1}{|G|} \sum_{g \in G} g⁻¹ • π(g • -). $$
-/
def equivariantProjection : W →ₗ[MonoidAlgebra k G] V 
goal: W →ₗ[MonoidAlgebra k G] V

full_name: LinearMap.equivariantProjection_apply
code: theorem equivariantProjection_apply (v : W) :
    π.equivariantProjection G v = ⅟(Fintype.card G : k) • ∑ g : G, π.conjugate g v 
goal: (LinearMap.equivariantProjection G π) v = ⅟↑(Fintype.card G) • ∑ g : G, (π.conjugate g) v

full_name: LinearMap.equivariantProjection_condition
code: theorem equivariantProjection_condition (v : V) : (π.equivariantProjection G) (i v) = v 
goal: (LinearMap.equivariantProjection G π) (i v) = v

full_name: MonoidAlgebra.exists_leftInverse_of_injective
code: theorem exists_leftInverse_of_injective (f : V →ₗ[MonoidAlgebra k G] W)
    (hf : LinearMap.ker f = ⊥) :
    ∃ g : W →ₗ[MonoidAlgebra k G] V, g.comp f = LinearMap.id 
goal: ∃ g, g ∘ₗ f = LinearMap.id

full_name: MonoidAlgebra.Submodule.exists_isCompl
code: theorem exists_isCompl (p : Submodule (MonoidAlgebra k G) V) :
    ∃ q : Submodule (MonoidAlgebra k G) V, IsCompl p q 
goal: ∃ q, IsCompl p q

full_name: MonoidAlgebra.Submodule.complementedLattice
code: /-- This also implies instances `IsSemisimpleModule (MonoidAlgebra k G) V` and
`IsSemisimpleRing (MonoidAlgebra k G)`. -/
instance complementedLattice : ComplementedLattice (Submodule (MonoidAlgebra k G) V) 
goal: ComplementedLattice (Submodule (MonoidAlgebra k G) V)

full_name: MonoidAlgebra.Submodule.instIsSemisimpleRingAddMonoidAlgebra
code: instance [AddGroup G] : IsSemisimpleRing (AddMonoidAlgebra k G) 
goal: IsSemisimpleRing (AddMonoidAlgebra k G)