The short-root carrier of type F4 #
This file feeds the explicit twenty-six-dimensional short-root representation of type F₄, its
admissible coordinate lattice, and its full set of weights into the Kostant toral-closure
construction. The result is an affine group scheme over ℤ: the smallest closed subgroup scheme
of GL₂₆ containing the represented simple root subgroups and the short-root weight torus. It is
cut out by an explicit Hopf ideal, the largest one killed by the coordinate morphisms of those
subgroups; the subgroups themselves have different source schemes, so it is their images, not
their kernels, that the carrier is generated by.
The construction exposes the positive and negative simple root subgroups, the closed rank-four
weight torus, matrix-valued points over every commutative ring, and the scheme-level pinning
equation. Every ingredient is explicit data from
TauCeti.Algebra.Lie.F4.ShortRoot.AdmissibleLattice; no carrier is selected from an existence
theorem. The weights of the module are the short roots and the zero weight twice, and the short
roots generate the root lattice of F₄, which is its weight lattice, so the weight torus is a
closed immersion and the carrier realizes the character lattice of the simply connected group.
The short simple root generators act with nilpotence index three, so their root subgroups have
matrices 1 + u E + u² E⁽²⁾ with E⁽²⁾ the integral divided square; the long ones square to
zero and have matrices 1 + u E.
This carrier is not identified with the pinned simply connected group scheme of type F₄
constructed from the root datum. Nothing here asserts reductivity, identifies the root datum of
the carrier beyond its named simple root subgroups and weight torus, or constructs root subgroups
for nonsimple roots. Constructions made on this carrier, such as its Frobenius or its special
isogeny in characteristic two, transfer to the pinned group scheme only along an identification
of the two, which is not part of this file.
Main definitions #
TauCeti.F4ShortRoot.groupScheme: the short-root toral closure inGL₂₆.TauCeti.F4ShortRoot.rootSubgroup: its eight numbered simple root subgroups.TauCeti.F4ShortRoot.weightTorus: its rank-four split weight torus.TauCeti.F4ShortRoot.points: its matrix-valued points over a commutative ring.TauCeti.F4ShortRoot.rootSubgroupPointsandTauCeti.F4ShortRoot.weightTorusPoints: the numbered simple root subgroups and the weight torus on points.
Main results #
TauCeti.F4ShortRoot.isClosedImmersion_rootSubgroup: each numbered root subgroup is a closed copy of the additive group.TauCeti.F4ShortRoot.isClosedImmersion_weightTorus: the short-root weights make the split torus a closed subgroup of the carrier.TauCeti.F4ShortRoot.coe_rootSubgroupPoints_eq: the matrix1 + u X + u² X⁽²⁾of every numbered simple-root point, withTauCeti.F4ShortRoot.coe_rootSubgroupPoints_inl_of_lt_twoandcoe_rootSubgroupPoints_inr_of_lt_twofor the long simple roots, where the quadratic term vanishes.TauCeti.F4ShortRoot.weightTorus_conj_rootSubgroup: the scheme-level pinning equation.TauCeti.F4ShortRoot.weightTorusPoints_conj_rootSubgroupPoints: the same equation on matrix-valued points.
References #
The construction is the Chevalley--Demazure construction on the admissible lattice of a
representation; see J. E. Humphreys, Linear Algebraic Groups, §26, and R. W. Carter, Simple
Groups of Lie Type, §§4.4 and 7.1. The type-F₄ numbering follows N. Bourbaki, Lie Groups and
Lie Algebras, Chapters 4--6, Plate VIII. The formal construction follows the corresponding
type-E₇ carrier in TauCeti.Algebra.Lie.E7.Minuscule.Carrier.
The short-root lattice is stable under the generic Kostant form generated by the Serre generators. This is the form required by the toral-closure construction.
Root characters and the nonzero root steps #
The Cartan generators act on the numbered simple root generators through their root characters.
The pinned carrier #
The Hopf ideal cutting out the short-root carrier of type F₄ inside GL₂₆.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The defining ideal is the one supplied by the generic Kostant toral-closure construction.
The short-root carrier of type F₄: the smallest closed subgroup scheme of GL₂₆
containing the represented simple root subgroups and the short-root weight torus.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The quotient-spectrum presentation of the short-root carrier.
The canonical inclusion of the short-root carrier into GL₂₆.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The ambient inclusion is the generic Kostant toral-closure inclusion.
The short-root carrier is a closed subgroup scheme of GL₂₆.
A positive or negative numbered simple root subgroup of the short-root carrier.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The numbered root subgroup is the one supplied by the generic Kostant toral-closure construction.
Including a numbered root subgroup into GL₂₆ recovers its represented Kostant root
subgroup.
The rank-four split weight torus in the short-root carrier.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The weight torus is the one supplied by the generic Kostant toral-closure construction.
Including the split weight torus into GL₂₆ recovers the diagonal torus of the short-root
weights.
Two morphisms out of the short-root carrier agree when they agree on every numbered simple root subgroup and on the split weight torus.
Matrix-valued points #
The points of the short-root carrier are cut out by its defining Hopf ideal.
A matrix is a point of the short-root carrier exactly when its associated convolution point kills the carrier's defining Hopf ideal.
A numbered simple-root point is the corresponding divided-power exponential matrix.
The matrix of a numbered simple-root point is 1 + u X + u² X⁽²⁾, with X the
integral matrix of the generator and X⁽²⁾ its integral divided square.
A positive simple-root point has matrix 1 + u Eᵢ + u² Eᵢ⁽²⁾ in the short-root basis.
A negative simple-root point has matrix 1 + u Fᵢ + u² Fᵢ⁽²⁾ in the short-root basis.
A long positive simple-root point has matrix 1 + u Eᵢ in the short-root basis.
A long negative simple-root point has matrix 1 + u Fᵢ in the short-root basis.
A split-torus point is the diagonal matrix whose entries are the short-root weight characters.
Closed subgroups and the pinning equation #
The represented integral root-subgroup coordinate map into the additive group is surjective.
The root-subgroup coordinate map remains surjective after adjoining the weight torus.
Every numbered simple root subgroup is a closed copy of the additive group.
The short-root weights make the rank-four split weight torus a closed immersion into the carrier.
The scheme-level pinning equation: conjugation by the weight torus acts on each numbered
simple root subgroup through the corresponding type-F₄ root character.
The pinning equation on matrix-valued points: conjugation by a point s of the weight torus
rescales the parameter of each numbered simple root subgroup by the corresponding type-F₄ root
character evaluated at s.