Complete reducibility of the doubled minuscule E₆ representation

Over every field, the standard representation of the doubled minuscule carrier is completely reducible. Its fifty-four distinct torus characters separate the coordinate lines. Positive and negative simple-root elements connect the twenty-seven lines in each of the two minuscule blocks. Thus a subcomodule contains either all or none of the lines in each block, and the remaining blocks give a complementary subcomodule. This supplies the representation-theoretic input for eliminating normal smooth unipotent subgroups through faithfulness.

The argument uses the scheme-theoretic torus coaction, so it also applies over finite fields, where the rational torus points need not separate weights. The dual block's root coefficients are -1, which remain invertible in every characteristic.

References

• J. C. Jantzen, Representations of Algebraic Groups, I.2 and II.2.
• J. E. Humphreys, Linear Algebraic Groups, §26.

The weight-extraction and reflection argument follows TauCeti.Algebra.Lie.D4.Tripled.StandardComodule and TauCeti.Algebra.Lie.E6.Minuscule.StandardComodule.

theorem TauCeti.E6DoubledMinuscule.isCompletelyReducible_standardComodule (k : Type u) [Field k] : Comodule.IsCompletelyReducible k (↑(coordinateHopfAlgebra k)) (Fin 54 → k)

The standard comodule of the doubled minuscule E₆ carrier is completely reducible over every field, including fields of characteristic two and three.