def Lec4Grassmannian.Gr (K : Type u_1) [Field K] (V : Type u_2) [AddCommGroup V] [Module K V] (k : ℕ) : Type u_2

Lecture 4: the Grassmannian Gr(k, V) of k-dimensional subspaces of V.

theorem Lec4Grassmannian.plucker_ambient_finrank (K : Type u_1) [Field K] (V : Type u_2) [AddCommGroup V] [Module K V] [Module.Free K V] [Module.Finite K V] (k : ℕ) : Module.finrank K ↥(⋀[K]^k V) = (Module.finrank K V).choose k

Dimension of the ambient space of the Plücker embedding: dim (⋀^k V) = C(dim V, k).

def Lec4Grassmannian.IsDecomposableTwo {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (ω : ExteriorAlgebra R M) : Prop

A 2-vector ω ∈ ⋀^2 M is decomposable if it can be written as a single wedge v₁ ∧ v₂.

theorem Lec4Grassmannian.ι_anticommute {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (x y : M) : (ExteriorAlgebra.ι R) x * (ExteriorAlgebra.ι R) y = -((ExteriorAlgebra.ι R) y * (ExteriorAlgebra.ι R) x)

Anticommutativity of the canonical map ι into the exterior algebra: x ∧ y = -(y ∧ x).

theorem Lec4Grassmannian.wedge_sq_zero_of_decomp {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (v₁ v₂ : M) : (ExteriorAlgebra.ι R) v₁ * (ExteriorAlgebra.ι R) v₂ * ((ExteriorAlgebra.ι R) v₁ * (ExteriorAlgebra.ι R) v₂) = 0

The wedge square of a decomposable 2-vector vanishes: (v₁ ∧ v₂) ∧ (v₁ ∧ v₂) = 0.

theorem Lec4Grassmannian.lemma6_forward {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {ω : ExteriorAlgebra R M} (h : IsDecomposableTwo ω) : ω * ω = 0

Easy direction of Lecture 4, Lemma 6: any decomposable 2-vector has zero wedge square.

theorem Lec4Grassmannian.ι_pair_commute {K : Type u_1} [Field K] {M : Type u_2} [AddCommGroup M] [Module K M] (a b c d : M) : (ExteriorAlgebra.ι K) a * (ExteriorAlgebra.ι K) b * ((ExteriorAlgebra.ι K) c * (ExteriorAlgebra.ι K) d) = (ExteriorAlgebra.ι K) c * (ExteriorAlgebra.ι K) d * ((ExteriorAlgebra.ι K) a * (ExteriorAlgebra.ι K) b)

Two wedge-products of pairs commute: (a ∧ b) ∧ (c ∧ d) = (c ∧ d) ∧ (a ∧ b).

theorem Lec4Grassmannian.wedge_sq_sum_decomp {K : Type u_1} [Field K] {M : Type u_2} [AddCommGroup M] [Module K M] (v₁ v₂ v₃ v₄ : M) : ((ExteriorAlgebra.ι K) v₁ * (ExteriorAlgebra.ι K) v₂ + (ExteriorAlgebra.ι K) v₃ * (ExteriorAlgebra.ι K) v₄) * ((ExteriorAlgebra.ι K) v₁ * (ExteriorAlgebra.ι K) v₂ + (ExteriorAlgebra.ι K) v₃ * (ExteriorAlgebra.ι K) v₄) = 2 * ((ExteriorAlgebra.ι K) v₁ * (ExteriorAlgebra.ι K) v₂ * ((ExteriorAlgebra.ι K) v₃ * (ExteriorAlgebra.ι K) v₄))

Expansion of the wedge square of a sum of two decomposables: (v₁ ∧ v₂ + v₃ ∧ v₄)^2 = 2 · (v₁ ∧ v₂ ∧ v₃ ∧ v₄).

theorem Lec4Grassmannian.extPower_two_finrank_of_four {K : Type u_1} [Field K] (V : Type u_2) [AddCommGroup V] [Module K V] [Module.Free K V] [Module.Finite K V] (hdim : Module.finrank K V = 4) : Module.finrank K ↥(⋀[K]^2 V) = 6

For a 4-dimensional V, dim (⋀^2 V) = 6.

theorem Lec4Grassmannian.extPower_four_finrank_of_four {K : Type u_1} [Field K] (V : Type u_2) [AddCommGroup V] [Module K V] [Module.Free K V] [Module.Finite K V] (hdim : Module.finrank K V = 4) : Module.finrank K ↥(⋀[K]^4 V) = 1

For a 4-dimensional V, the top exterior power is 1-dimensional: dim (⋀^4 V) = 1.

theorem Lec4Grassmannian.skew_form_normal_form_dim4 {K : Type u_1} [Field K] (V : Type u_2) [AddCommGroup V] [Module K V] [Module.Free K V] [Module.Finite K V] (hdim : Module.finrank K V = 4) (ω : ExteriorAlgebra K V) (hω2 : ω ∈ ⋀[K]^2 V) : (∃ (v₁ : V) (v₂ : V), ω = (ExteriorAlgebra.ι K) v₁ * (ExteriorAlgebra.ι K) v₂) ∨ ∃ (v₁ : V) (v₂ : V) (v₃ : V) (v₄ : V), LinearIndependent K ![v₁, v₂, v₃, v₄] ∧ ω = (ExteriorAlgebra.ι K) v₁ * (ExteriorAlgebra.ι K) v₂ + (ExteriorAlgebra.ι K) v₃ * (ExteriorAlgebra.ι K) v₄

Skew normal form in dimension 4: any 2-form on a 4-dimensional V is either decomposable or a sum of two decomposable terms based on a linearly independent 4-tuple.

theorem Lec4Grassmannian.orderEmb_fin_eq_id (n : ℕ) (f : Fin n ↪o Fin n) (i : Fin n) : f i = i

An order-embedding Fin n ↪o Fin n is necessarily the identity.

theorem Lec4Grassmannian.product_eq_ιMulti {K : Type u_1} [Field K] {V : Type u_2} [AddCommGroup V] [Module K V] (v₁ v₂ v₃ v₄ : V) : (ExteriorAlgebra.ι K) v₁ * (ExteriorAlgebra.ι K) v₂ * ((ExteriorAlgebra.ι K) v₃ * (ExteriorAlgebra.ι K) v₄) = (ExteriorAlgebra.ιMulti K 4) ![v₁, v₂, v₃, v₄]

The product v₁ ∧ v₂ ∧ v₃ ∧ v₄ in the exterior algebra agrees with the multilinear map ιMulti K 4 applied to the 4-tuple.

theorem Lec4Grassmannian.ιMulti_ne_zero_of_linearIndependent {K : Type u_1} [Field K] {V : Type u_2} [AddCommGroup V] [Module K V] (hdim : Module.finrank K V = 4) (v : Fin 4 → V) (hli : LinearIndependent K v) : (ExteriorAlgebra.ιMulti K 4) v ≠ 0

A linearly independent 4-tuple in a 4-dimensional V has nonzero top wedge product.

theorem Lec4Grassmannian.alternating_form_classification_dim4 {K : Type u_1} [Field K] [CharZero K] (V : Type u_2) [AddCommGroup V] [Module K V] [Module.Free K V] [Module.Finite K V] (hdim : Module.finrank K V = 4) (ω : ExteriorAlgebra K V) (hω2 : ω ∈ ⋀[K]^2 V) (hωω : ω * ω = 0) : IsDecomposableTwo ω

In dimension 4, a 2-form with zero wedge square must be decomposable: this is the converse half of the classification used in Lemma 6.

theorem Lec4Grassmannian.lemma6_converse {K : Type u_1} [Field K] [CharZero K] (V : Type u_2) [AddCommGroup V] [Module K V] [Module.Free K V] [Module.Finite K V] (hdim : Module.finrank K V = 4) (ω : ExteriorAlgebra K V) (hω2 : ω ∈ ⋀[K]^2 V) (hωω : ω * ω = 0) : IsDecomposableTwo ω

Converse direction of Lecture 4, Lemma 6: in dimension 4, vanishing of ω ∧ ω forces ω to be decomposable.

theorem Lec4Grassmannian.lemma6_iff {K : Type u_1} [Field K] [CharZero K] (V : Type u_2) [AddCommGroup V] [Module K V] [Module.Free K V] [Module.Finite K V] (hdim : Module.finrank K V = 4) (ω : ExteriorAlgebra K V) (hω2 : ω ∈ ⋀[K]^2 V) : IsDecomposableTwo ω ↔ ω * ω = 0

Lecture 4, Lemma 6 (full statement): in dimension 4, a 2-form is decomposable iff its wedge square vanishes.

theorem Lec4Grassmannian.gr24_ambient_dim (K : Type u_1) [Field K] (V : Type u_2) [AddCommGroup V] [Module K V] [Module.Free K V] [Module.Finite K V] (hdim : Module.finrank K V = 4) : Module.finrank K ↥(⋀[K]^2 V) = 6

Lecture 4, Theorem 4.1 applied to Gr(2, 4): the Plücker ambient space has dimension 6.