Exact low-degree Chevalley action in type F₄ #
This file transports the pinned forty-eight-root indexing to the rational Killing root system and records the exact degree-one and degree-two adjoint actions needed for the characteristic-two short-root submodule.
References #
- R. W. Carter, Simple Groups of Lie Type, §12.3.
- R. Steinberg, Endomorphisms of linear algebraic groups, Memoirs AMS 80 (1968), §11.
Identify the fixed forty-eight F₄ root labels with the root index of DynkinType.F4.
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The Killing-root label belonging to a pinned integral root index.
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The Killing weight indexed by the corresponding root of the pinned F₄ root datum.
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A chosen F₄ root-vector system compatible with the pinned Chevalley involution.
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The chosen F₄ root vectors form a Chevalley system for the pinned involution.
The rational root-system equivalence carries a pinned root to its Killing root.
Integral root relations are equivalent to the corresponding relations among Killing weights.
The rational Killing-root identification preserves the root-string coefficient inherited from the integral pinned F₄ datum.
The rational Killing-root identification preserves the ascending root-string coefficient inherited from the integral pinned F₄ datum.
The two root-string lengths agree for the Killing weights and the pinned integral roots.
Every Killing coroot expands in the simple Killing coroots with the coordinates of the corresponding pinned integral F₄ coroot.
The Cartan integer of a pinned Killing root against a simple Killing coroot is the corresponding integral pairing in the pinned F₄ datum.
The bracket along any nondegenerate F₄ root edge has the integral Chevalley coefficient prescribed by the descending root string.
A root edge whose descending string has length zero carries a unit Chevalley coefficient.
Along a two-step F₄ string from a long root in a short-root direction, the square of the
adjoint root-vector action has coefficient of absolute value two. Dividing by 2! therefore has
unit coefficient, which becomes exactly one after reduction modulo two.
The second divided adjoint power along a long--short--long F₄ string has unit coefficient. It sends the initial root vector to plus or minus the final root vector and therefore preserves the integral lattice on this root string.
The pinned opposite-root index has the negative rational Killing-root label.
Distinct pinned indices define distinct rational Killing roots.
The Killing root at the opposite pinned index is the negative root.
Two indexed rational Killing roots do not sum to zero unless their pinned labels are opposite.
Every rational Killing root label is one of the pinned forty-eight indices.
An edge labelled by a nonzero root changes the pinned root label.
A nonzero weight whose root space is present has a pinned index, including when it is presented as an endpoint of a root string.
A nonzero, present sum of two Killing roots has the corresponding pinned integral root label.
A missing endpoint in the pinned root string forces the corresponding adjoint power to vanish. The only excluded case is the string through the opposite root.
A missing second root-string endpoint makes the second adjoint divided power vanish.
The second divided adjoint power sends the opposite root vector to the negative root vector.
Away from the opposite-root string, the second divided adjoint power annihilates every short-root vector.
Away from the opposite string, the second divided adjoint power annihilates a long root vector when both roots have long length.
The second divided adjoint power annihilates every Cartan element.
Three adjoint applications annihilate every F4 root vector, regardless of root length.
The third adjoint power annihilates every Cartan element.