The doubled E₆ carrier preserves its minuscule summands

The integral doubled minuscule carrier lies in the block-diagonal subgroup GL₂₇ × GL₂₇ of GL₅₄, with blocks labelled by matrixSummand. Every numbered root subgroup has zero entries between the two summands, as does the diagonal weight torus. Their scheme-theoretic closure therefore has the same property. The resulting coordinate equations hold after base change to any commutative ring and make the two summands subcomodules of the standard carrier representation.

This containment does not identify the carrier with the pinned simply connected group scheme of type E₆. Constructions on this explicit carrier transfer to that pinned group along such an identification once one is proved.

References

• J. C. Jantzen, Representations of Algebraic Groups, II.1–2.
• J. E. Humphreys, Linear Algebraic Groups, §26.
• The containment argument follows TauCeti.Algebra.Lie.D4.Tripled.Levi, using the generic square-zero root-subgroup and weight-Levi kernel criteria.

theorem TauCeti.E6DoubledMinuscule.weightLeviDefiningHopfIdeal_le_definingIdeal : GeneralLinear.weightLeviDefiningHopfIdeal ℤ matrixSummand ≤ definingIdeal

The integral doubled minuscule carrier lies in the block-diagonal subgroup of its two summands, scheme-theoretically.

theorem TauCeti.E6DoubledMinuscule.coordinateMap_X_eq_zero (A : Type u_1) [CommRing A] {a b : Fin 54} (hab : matrixSummand a ≠ matrixSummand b) : (CommHopfAlgCat.Hom.hom (coordinateMap A)) ((GeneralLinear.coordinateHopfAlgebraAlgEquiv A 54) ((GeneralLinear.coordinateRingMap A 54) (MvPolynomial.X (a, b)))) = 0

Every coordinate between the two minuscule summands vanishes on the base-changed carrier, over any commutative ring.