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TauCeti.Algebra.Lie.E7.Minuscule.Frobenius

The Frobenius of the full-weight type-E7 minuscule carrier #

TauCeti.E7Minuscule.groupScheme is the explicit full-weight type-E₇ Chevalley carrier over ℤ, the Kostant toral closure built from the 56-dimensional minuscule representation and its admissible lattice, and TauCeti.E7Minuscule.points A realizes its A-valued points as a subgroup of GL₅₆(A). Over a commutative ring A of exponential characteristic p, entrywise p ^ k-th powers are a homomorphism of value rings, so the carrier's functoriality turns them into a group endomorphism of its points.

This file names that endomorphism TauCeti.E7Minuscule.frobenius and records its characteristic equations:

F (g)ᵢⱼ = gᵢⱼ ^ (p ^ k),
F (xᵢ(u)) = xᵢ(u ^ (p ^ k)),
F (t(s)) = t(s ^ (p ^ k)).

The zeroth iterate is the identity, exponents add under composition and multiply under taking powers in the endomorphism monoid, and the fixed points are the points of the same carrier over the Frobenius-fixed subring of A.

The E₇ diagram has no nontrivial symmetry, so a Steinberg endomorphism built on this carrier is a field Frobenius alone, with no graph automorphism to compose with. Nothing here asserts reductivity, maximality of the weight torus, an identification of the carrier's root datum, or any finiteness or simplicity statement.

Main declarations #

References #

source
noncomputable def TauCeti.E7Minuscule.frobenius (p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] :
↥(points A) →* ↥(points A)

The p ^ k-power Frobenius endomorphism of the full-weight type-E₇ minuscule carrier, the functorial map on points induced by the iterated Frobenius endomorphism of the value ring.

For p prime, 0 < k and A an algebraic closure of ZMod p, this is the q-power Frobenius of the carrier's points for q = p ^ k.

Equations
Instances For
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    theorem TauCeti.E7Minuscule.coe_frobenius (p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] (g : ↥(points A)) :
    ↑((frobenius p k A) g) = (Matrix.GeneralLinearGroup.map (iterateFrobenius A p k)) ↑g

    The Frobenius endomorphism of the type-E₇ minuscule carrier acts by entrywise Frobenius.

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

    source
    @[simp]
    theorem TauCeti.E7Minuscule.coe_frobenius_apply (p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] (g : ↥(points A)) (i j : Fin 56) :
    ↑↑((frobenius p k A) g) i j = ↑↑g i j ^ p ^ k

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

    source
    @[simp]
    theorem TauCeti.E7Minuscule.frobenius_rootSubgroupPoints (p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] (i : Fin 7 ⊕ Fin 7) (u : Multiplicative A) :
    (frobenius p k A) ((rootSubgroupPoints i A) u) = (rootSubgroupPoints i A) (Multiplicative.ofAdd (Multiplicative.toAdd u ^ p ^ k))

    Frobenius raises the parameter of every numbered type-E₇ simple-root subgroup to its p ^ k-th power, on both the raising and the lowering generators.

    source
    @[simp]
    theorem TauCeti.E7Minuscule.frobenius_weightTorusPoints (p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] (s : Fin 7 → Aˣ) :
    (frobenius p k A) ((weightTorusPoints A) s) = (weightTorusPoints A) (s ^ p ^ k)

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

    source
    @[simp]
    theorem TauCeti.E7Minuscule.frobenius_zero (p : ℕ) (A : Type v) [CommRing A] [ExpChar A p] :
    frobenius p 0 A = MonoidHom.id ↥(points A)

    The zeroth Frobenius iterate is the identity on the type-E₇ minuscule carrier's point group.

    source
    theorem TauCeti.E7Minuscule.frobenius_add (p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] (m : ℕ) :
    frobenius p (k + m) A = (frobenius p k A).comp (frobenius p m A)

    Frobenius iterates add under composition on the type-E₇ minuscule carrier's point group.

    source
    theorem TauCeti.E7Minuscule.frobenius_pow (p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] (m : ℕ) :
    (have this := frobenius p k A; this) ^ m = frobenius p (k * m) A

    Frobenius exponents multiply under taking powers: the m-th power of the p ^ k-power Frobenius of the type-E₇ minuscule carrier, in the endomorphism monoid of its points, is its p ^ (k * m)-power Frobenius.

    source
    @[simp]
    theorem TauCeti.E7Minuscule.frobenius_eq_self_iff (p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] (g : ↥(points A)) :
    (frobenius p k A) g = g ↔ ∀ (i j : Fin 56), ↑↑g i j ∈ frobeniusFixedSubring A p k

    A type-E₇ minuscule carrier point is fixed by Frobenius exactly when all of its matrix entries lie in the Frobenius-fixed subring.

    source
    theorem TauCeti.E7Minuscule.map_subtype_fixedSubgroup_frobenius_eq (p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] :
    Subgroup.map (points A).subtype (fixedSubgroup (frobenius p k A)) = Subgroup.map (Matrix.GeneralLinearGroup.map (frobeniusFixedSubring A p k).subtype) (points ↥(frobeniusFixedSubring A p k))

    The Frobenius-fixed points of the full-weight type-E₇ minuscule carrier are its points over the Frobenius-fixed subring. No finiteness of either side is asserted.