When the pinned root lattice is the whole character lattice #
The character lattice of TauCeti.DynkinType.simplyConnectedRootDatum is the lattice of
fundamental weights: TauCeti.DynkinType.coroot_simpleIndex makes the simple coroots the standard
basis vectors, so TauCeti.DynkinType.root_simpleIndex writes each simple root as the
corresponding row of the Bourbaki-numbered Cartan matrix. The roots therefore span the character
lattice exactly when those rows do, which is exactly when the Cartan matrix is invertible over
ℤ. That criterion is TauCeti.DynkinType.span_range_root_eq_top_iff_isUnit_det, and evaluating
it on the nine valid families gives
TauCeti.DynkinType.span_range_root_eq_top_iff: the root lattice is the whole character lattice
precisely in types E₈, F₄ and G₂.
The point of the criterion is which admissible lattice an explicit Chevalley--Demazure carrier has
to be built from. A carrier's split torus acts through the lattice generated by the weights of the
representation used to construct it, so a carrier whose weights generate only the root lattice is a
candidate for the adjoint form and not for the simply connected form. The classification below says
that this obstruction is present in six of the nine types, so that the E₈, F₄ and G₂
constructions built from the root lattice do not generalize: the remaining types need an admissible
lattice in a representation whose weights generate a larger sublattice of the character lattice,
namely the standard module in types A and C, and a spin or minuscule module in types B, D,
E₆ and E₇.
Nothing here is a group. The coroot side is unaffected: the coroots of the pinned datum span the
cocharacter lattice for every valid type, which is
TauCeti.DynkinType.span_coroot_simplyConnectedRootDatum and is what makes the datum the simply
connected one.
Main results #
TauCeti.DynkinType.span_range_root_eq_top_iff_isUnit_det: the roots of the pinned datum span its character lattice if and only if its Cartan matrix has unit determinant.TauCeti.DynkinType.isUnit_det_cartanMatrix_iff: the Cartan matrix of a valid Dynkin type has unit determinant precisely in typesE₈,F₄andG₂.TauCeti.DynkinType.span_range_root_eq_top_iff: the resulting classification of the types whose root lattice is the whole character lattice.
References #
- J. E. Humphreys, Linear Algebraic Groups, §27, for the role of the lattice generated by the weights of an admissible lattice in separating the isogeny forms.
- J. C. Jantzen, Representations of Algebraic Groups, II.1.
- N. Bourbaki, Lie Groups and Lie Algebras, Chapters 4--6, Plates I--IX, for the Cartan matrices and their determinants.
This decides which admissible lattices the Chevalley--Demazure construction of Layer 9 of
TauCetiRoadmap/ReductiveGroups/README.md may be built from, for its consumer, the pinned simply
connected ambient group of milestone L0 of TauCetiRoadmap/CFSGStatement/README.md.
The roots of the pinned simply connected datum span its character lattice exactly when the
Cartan matrix is invertible over ℤ. The simple roots are the rows of that matrix, and the base
generates every root over ℤ, so the root lattice is the row lattice.
The Cartan matrix of a valid Dynkin type is unimodular precisely in types E₈, F₄ and
G₂. The determinant of A n is n + 1, those of B n and C n are 2, and those of
D n, E₆ and E₇ are 4, 3 and 2.
The root lattice of the pinned simply connected datum is the whole character lattice
precisely in types E₈, F₄ and G₂. In the other six valid types the root lattice is a proper
sublattice, so a Chevalley--Demazure carrier built from an admissible lattice whose weights
generate only the root lattice is not the simply connected form there.