Irreducibility of the short-root F₄ standard representation

The twenty-six-dimensional standard comodule of the short-root F₄ carrier over a field of characteristic two is simple. The weight torus separates its twenty-four nonzero-weight coordinates, while the zero weight has multiplicity two. Numbered simple-root points connect all twenty-four nonzero weights and connect each of the two zero-weight coordinates to them. Thus a nonzero invariant subspace contains every coordinate vector.

The quadratic terms in the two short simple-root subgroups are essential: they exchange the two weights opposite to a short simple root without requiring division by two.

References

• J. C. Jantzen, Representations of Algebraic Groups, I.2 and II.2.
• R. W. Carter, Simple Groups of Lie Type, §12.3.
• N. Bourbaki, Lie Groups and Lie Algebras, Chapters 4--6, Plate VIII.

Root moves on nonzero weights

Invariant coordinate vectors

The zero-weight plane

instance TauCeti.F4ShortRoot.PrimeField.instIsSimpleOrderSubcomodule (k : Type u) [Field k] [Algebra (ZMod 2) k] : IsSimpleOrder (Subcomodule k (↑(coordinateHopfAlgebra k)) (Fin 26 → k))

The standard comodule of the short-root type-F₄ carrier is simple over every field of characteristic two.