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TauCeti.Algebra.Lie.Orthogonal.TypeD.SpinCarrier.Frobenius

Frobenius on the full-weight type-D spin carrier #

TauCeti.TypeDSpinCarrier.groupScheme n hn is the explicit full-weight Chevalley carrier of type Dₙ, cut out inside GL_(2^n) over ℤ by the split spin representation and its exterior coordinate lattice. For a commutative value ring A of exponential characteristic p, this file equips its point group TauCeti.TypeDSpinCarrier.points n hn A with the p ^ k-power Frobenius endomorphism.

The endomorphism raises every matrix entry to its p ^ k-th power. In particular it satisfies the pinned root-subgroup equation

F (x_i(u)) = x_i(u ^ (p ^ k))

for every Bourbaki-numbered raising or lowering generator, and it raises every coordinate of the split spin weight torus by the same exponent. Its fixed points are exactly the points of the same carrier over the Frobenius-fixed subring of A.

The construction is the carrier's functorial point map at the iterated Frobenius of the value ring. Nothing asserts that the carrier is reductive, that it is the spin group scheme, or that any fixed-point group is finite or simple.

Main definitions #

Main results #

References #

The organization follows the sibling carrier specialization TauCeti.Algebra.Lie.Orthogonal.TypeB.SpinCarrier.Frobenius.

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noncomputable def TauCeti.TypeDSpinCarrier.frobenius (n : ℕ) (hn : 4 ≤ n) (p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] :
↥(points n hn A) →* ↥(points n hn A)

The p ^ k-power Frobenius endomorphism of the full-weight type-Dₙ spin carrier.

For p prime, 0 < k, and A an algebraic closure of ZMod p, this is the Frobenius component intended for a future construction of the Dₙ(p ^ k), ²Dₙ(p ^ k) and ³D₄(p ^ k) Steinberg maps.

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Instances For
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    theorem TauCeti.TypeDSpinCarrier.coe_frobenius (n : ℕ) (hn : 4 ≤ n) (p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] (g : ↥(points n hn A)) :
    ↑((frobenius n hn p k A) g) = (Matrix.GeneralLinearGroup.map (iterateFrobenius A p k)) ↑g

    The Frobenius endomorphism of the type-Dₙ spin carrier acts by entrywise Frobenius.

    This is not a simp lemma because coe_frobenius_apply is the canonical coefficient-level normal form.

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    theorem TauCeti.TypeDSpinCarrier.frobenius_eq_map (n : ℕ) (hn : 4 ≤ n) (p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] :
    frobenius n hn p k A = (pointsPresentation n hn A).map (pointsPresentation n hn A) (iterateFrobenius A p k)

    The carrier Frobenius is the functorial map on points induced by the iterated Frobenius endomorphism of the value ring.

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    @[simp]
    theorem TauCeti.TypeDSpinCarrier.coe_frobenius_apply (n : ℕ) (hn : 4 ≤ n) (p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] (g : ↥(points n hn A)) (r c : Fin (dimension n)) :
    ↑↑((frobenius n hn p k A) g) r c = ↑↑g r c ^ p ^ k

    Entrywise, the Frobenius endomorphism raises each matrix coefficient to its p ^ k-th power.

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    @[simp]
    theorem TauCeti.TypeDSpinCarrier.frobenius_rootSubgroupPoints (n : ℕ) (hn : 4 ≤ n) (p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] (i : Fin n ⊕ Fin n) (u : Multiplicative A) :
    (frobenius n hn p k A) ((rootSubgroupPoints n hn i A) u) = (rootSubgroupPoints n hn i A) (Multiplicative.ofAdd (Multiplicative.toAdd u ^ p ^ k))

    Frobenius raises the parameter of a numbered type-Dₙ root subgroup to its p ^ k-th power, that is, F (x_i(u)) = x_i(u ^ (p ^ k)) on both the raising and the lowering generators.

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    @[simp]
    theorem TauCeti.TypeDSpinCarrier.frobenius_weightTorusPoints (n : ℕ) (hn : 4 ≤ n) (p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] (s : Fin n → Aˣ) :
    (frobenius n hn p k A) ((weightTorusPoints n hn A) s) = (weightTorusPoints n hn A) (s ^ p ^ k)

    Frobenius raises every coordinate of the pinned split spin weight torus to its p ^ k-th power.

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    @[simp]
    theorem TauCeti.TypeDSpinCarrier.frobenius_zero (n : ℕ) (hn : 4 ≤ n) (p : ℕ) (A : Type v) [CommRing A] [ExpChar A p] :
    frobenius n hn p 0 A = MonoidHom.id ↥(points n hn A)

    The zeroth Frobenius iterate is the identity on the type-Dₙ spin carrier's point group.

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    theorem TauCeti.TypeDSpinCarrier.frobenius_add (n : ℕ) (hn : 4 ≤ n) (p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] (m : ℕ) :
    frobenius n hn p (k + m) A = (frobenius n hn p k A).comp (frobenius n hn p m A)

    Frobenius iterates add under composition on the type-Dₙ spin carrier's point group.

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    theorem TauCeti.TypeDSpinCarrier.frobenius_pow (n : ℕ) (hn : 4 ≤ n) (p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] (m : ℕ) :
    (have this := frobenius n hn p k A; this) ^ m = frobenius n hn p (k * m) A

    Frobenius exponents multiply under taking powers: the m-th power of the p ^ k-power Frobenius of the type-Dₙ spin carrier's point group, in the endomorphism monoid of its points, is its p ^ (k * m)-power Frobenius.

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    @[simp]
    theorem TauCeti.TypeDSpinCarrier.frobenius_eq_self_iff (n : ℕ) (hn : 4 ≤ n) (p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] (g : ↥(points n hn A)) :
    (frobenius n hn p k A) g = g ↔ ∀ (r c : Fin (dimension n)), ↑↑g r c ∈ frobeniusFixedSubring A p k

    A type-Dₙ spin carrier point is fixed by Frobenius exactly when all of its matrix entries lie in the Frobenius-fixed subring.

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    theorem TauCeti.TypeDSpinCarrier.map_subtype_fixedSubgroup_frobenius_eq (n : ℕ) (hn : 4 ≤ n) (p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] :
    Subgroup.map (points n hn A).subtype (fixedSubgroup (frobenius n hn p k A)) = Subgroup.map (Matrix.GeneralLinearGroup.map (frobeniusFixedSubring A p k).subtype) (points n hn ↥(frobeniusFixedSubring A p k))

    The Frobenius-fixed points of the full-weight type-Dₙ spin carrier are its points over the Frobenius-fixed subring. For p prime, 0 < k, A an algebraic closure of ZMod p and q = p ^ k this reads the fixed group of the untwisted Dₙ(q) Steinberg map as the carrier's 𝔽_q-points; no finiteness of either side is asserted.