Frobenius on the full-weight type-C carrier

TauCeti.SpStd.groupScheme n is the explicit full-weight Chevalley carrier of type C_(n+1), built from the standard representation of sp_(2n+2) and its coordinate integral lattice. For a commutative value ring A of exponential characteristic p, this file equips its point group TauCeti.SpStd.points n 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 weight torus by the same exponent. Its fixed points are exactly the points of the same carrier over the Frobenius-fixed subring.

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 symplectic group scheme, or that any fixed-point group is finite or simple.

Main definitions

• TauCeti.SpStd.frobenius: the p ^ k-power Frobenius endomorphism of the type-C_(n+1) point group.

Main results

• TauCeti.SpStd.coe_frobenius and TauCeti.SpStd.coe_frobenius_apply: the endomorphism acts by entrywise Frobenius.
• TauCeti.SpStd.frobenius_eq_map: Frobenius is the functorial point map induced by the iterated Frobenius endomorphism of the value ring.
• TauCeti.SpStd.frobenius_rootSubgroupPoints and TauCeti.SpStd.frobenius_weightTorusPoints: the equations on the pinned generating root subgroups and split torus.
• TauCeti.SpStd.frobenius_zero, TauCeti.SpStd.frobenius_add and TauCeti.SpStd.frobenius_pow: the iteration laws.
• TauCeti.SpStd.map_subtype_fixedSubgroup_frobenius_eq: the Frobenius-fixed points are the points over the Frobenius-fixed subring.

References

• R. W. Carter, Finite Groups of Lie Type: Conjugacy Classes and Complex Characters, §1.17.
• J. C. Jantzen, Representations of Algebraic Groups, II.1.

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

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

The p ^ k-power Frobenius endomorphism of the full-weight type-C_(n+1) 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 C_(n+1)(p ^ k) Steinberg map.

Equations

• TauCeti.SpStd.frobenius n p k A = (TauCeti.SpStd.pointsPresentation n A).map (TauCeti.SpStd.pointsPresentation n A) (iterateFrobenius A p k)

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

The Frobenius endomorphism of the type-C_(n+1) carrier acts by entrywise Frobenius.

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

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

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

@[simp]

theorem TauCeti.SpStd.coe_frobenius_apply (n p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] (g : ↥(points n A)) (i j : Fin (n + 1 + (n + 1))) : ↑↑((frobenius n 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.

@[simp]

theorem TauCeti.SpStd.frobenius_rootSubgroupPoints (n p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] (i : Fin (n + 1) ⊕ Fin (n + 1)) (u : Multiplicative A) : (frobenius n p k A) ((rootSubgroupPoints n i A) u) = (rootSubgroupPoints n i A) (Multiplicative.ofAdd (Multiplicative.toAdd u ^ p ^ k))

Frobenius raises the parameter of a numbered type-C_(n+1) root subgroup to its p ^ k-th power.

@[simp]

theorem TauCeti.SpStd.frobenius_weightTorusPoints (n p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] (s : Fin (n + 1) → Aˣ) : (frobenius n p k A) ((weightTorusPoints n A) s) = (weightTorusPoints n A) (s ^ p ^ k)

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

@[simp]

theorem TauCeti.SpStd.frobenius_zero (n p : ℕ) (A : Type v) [CommRing A] [ExpChar A p] : frobenius n p 0 A = MonoidHom.id ↥(points n A)

The zeroth Frobenius iterate is the identity on the type-C_(n+1) point group.

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

Frobenius iterates add under composition on the type-C_(n+1) point group.

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

Frobenius exponents multiply under taking powers: the m-th power of the p ^ k-power Frobenius of the type-C_(n+1) point group, in the endomorphism monoid of its points, is its p ^ (k * m)-power Frobenius.

@[simp]

theorem TauCeti.SpStd.frobenius_eq_self_iff (n p k : ℕ) (A : Type v) [CommRing A] [ExpChar A p] (g : ↥(points n A)) : (frobenius n p k A) g = g ↔ ∀ (i j : Fin (n + 1 + (n + 1))), ↑↑g i j ∈ frobeniusFixedSubring A p k

A type-C_(n+1) carrier point is fixed by Frobenius exactly when all of its matrix entries lie in the Frobenius-fixed subring.

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

The Frobenius-fixed points of the full-weight type-C_(n+1) carrier are its points over the Frobenius-fixed subring.