Geometric connectedness of the symplectic flag stabilizer #
The standard complete isotropic flag stabilizer in Sp₂ₘ is geometrically connected
over every field. Its matrices have the form [A, A S; 0, (A⁻¹)ᵀ], where A is
invertible upper triangular and S is symmetric. Polynomial coordinates for A and
S, localized at det A, give a domain containing the flag coordinate algebra as
a retract. This works in characteristic two because symmetric matrices are
parametrized by their upper triangle, without dividing by two.
The coordinate-ring retraction argument follows
TauCeti.Algebra.AlgebraicGroup.SpecialLinear.UpperTriangular.Connected.
References #
- J. S. Milne, Algebraic Groups (2017), §24.6.
- T. A. Springer, Linear Algebraic Groups, §6.2.
theorem
TauCeti.Symplectic.IsotropicFlag.geometricallyConnectedCommHopfAlgProperty_coordinateHopfAlgebra
(m : ℕ)
(k : Type u)
[Field k]
:
The standard complete isotropic flag stabilizer of Sp₂ₘ is geometrically connected
over every field, including characteristic two and rank zero.