The F4 carrier endomorphism from the represented quotient #
The middle quotient of the represented adjoint flag defines a morphism from the prime-field
carrier to GL₂₆. Its pinned root and torus equations show that it maps the generating subgroup
schemes back into the carrier. The common-kernel universal property therefore factors it through
the carrier's coordinate Hopf algebra.
quotientIsogeny_comp_self proves the Frobenius square as an equality of coordinate morphisms.
specialIsogenyHom and specialIsogenyHom_comp_self expose the corresponding group-scheme
endomorphism and its square.
specialIsogeny transports this construction to the existing matrix-valued carrier points, with
its numbered root action and Frobenius square. These supply the exceptional endomorphisms used
for the Ree F4 and Tits families.
The carrier is explicit; no identification with the pinned simply connected F4 group scheme, or finiteness or simplicity theorem for its fixed-point candidates, is asserted here.
References #
- R. Steinberg, Endomorphisms of linear algebraic groups, Memoirs AMS 80 (1968), §11.
- R. W. Carter, Simple Groups of Lie Type, §12.3.
The coordinate morphism of the represented middle quotient of the F4 carrier.
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The represented quotient coordinate morphism kills the carrier ideal because its universal root and torus points belong to the carrier.
The endomorphism of the F4 coordinate Hopf algebra induced by its represented quotient.
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Pulling the quotient endomorphism back to the ambient coordinate algebra recovers the matrix coefficient morphism of the represented quotient.
The induced coordinate endomorphism has the prescribed action on every root point.
The induced coordinate endomorphism has the prescribed action on every weight-torus point.
The represented quotient endomorphism squares to Frobenius as a morphism of coordinate Hopf algebras.
The exceptional endomorphism of the explicit F4 carrier group scheme, represented by
quotientIsogeny on its coordinate Hopf algebra.
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The scheme morphism is the spectrum of the quotient-coordinate endomorphism.
The exceptional endomorphism squares to Frobenius as a morphism of group schemes.
The characteristic-two special endomorphism of the matrix-valued F4 carrier. It is induced by the represented quotient, with exponent one on long roots and two on short roots.
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- One or more equations did not get rendered due to their size.
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The special endomorphism is precomposition by its coordinate morphism.
On scheme-valued points, the carrier special isogeny is the named matrix-valued map.
On scheme-valued points, the carrier Frobenius is the named pointwise Frobenius.
The special endomorphism exchanges each signed simple root with its reversed root, using the pinned long/short exponent convention.
The special isogeny applies the exceptional torus parameter map.