Triviality of the generated type-E₆ subgroup's unipotent radical

Over an algebraically closed field, the subgroup of GL₂₇ generated by the numbered minuscule root subgroups and weight torus has no nontrivial normal smooth unipotent closed subgroup. Its faithful simple standard comodule supplies the representation-theoretic input, and its coordinate algebra is reduced because its generators are reduced. Thus no reducedness hypothesis on this subgroup remains.

The result concerns the subgroup generated over the field, not the base change of the integral minuscule carrier. An identification between those two subgroup schemes is needed to transfer this conclusion to that carrier.

References

• J. E. Humphreys, Linear Algebraic Groups, §§19 and 26.
• J. C. Jantzen, Representations of Algebraic Groups, I.2 and II.2.

The normal-subgroup elimination follows the specialization for the integral carrier in TauCeti.Algebra.Lie.E6.Minuscule.UnipotentRadical, using the generic faithful-comodule theorem.

theorem TauCeti.E6Minuscule.eq_augmentation_generated_of_isNormal_of_smoothUnipotent (k : Type u) [Field k] [IsAlgClosed k] (I : HopfIdeal k ↑(generatedCoordinateHopfAlgebra k)) (hI : I.IsNormal) (hU : smoothUnipotentCommHopfAlgProperty k ((FiniteTypeCommHopfAlgCat.of k ↑(generatedCoordinateHopfAlgebra k)).quotient I)) : I = HopfIdeal.augmentation k ↑(generatedCoordinateHopfAlgebra k)

Every normal smooth unipotent closed subgroup of the subgroup generated over the field by the type-E₆ minuscule root subgroups and weight torus is trivial.

theorem TauCeti.E6Minuscule.unipotentRadicalDefiningIdeal_generated_eq_augmentation (k : Type u) [Field k] [IsAlgClosed k] : (FiniteTypeCommHopfAlgCat.of k ↑(generatedCoordinateHopfAlgebra k)).unipotentRadicalDefiningIdeal = HopfIdeal.augmentation k ↑(generatedCoordinateHopfAlgebra k)

The unipotent radical of the subgroup generated over an algebraically closed field by the type-E₆ minuscule root subgroups and weight torus is trivial.