The minuscule weight orbit of type E6 #
This file enumerates the Weyl orbit of the first fundamental weight of the pinned simply connected
root datum TauCeti.DynkinType.e6SimplyConnectedRootDatum. The twenty-seven weights are expressed
in the fundamental-weight basis Fin 6 → ℤ. The first weight is ϖ₁, and the table is closed under
the six Bourbaki-numbered simple reflections through explicit permutations of Fin 27.
The reflection equation is the key interface for the future 27-dimensional minuscule module: a
simple root operator can move a coordinate basis vector only when the corresponding weight pairs
to 1 or -1. The table records that every such pairing lies in {-1, 0, 1}, identifies the table
with the full Weyl orbit, and proves that its weights span the complete character lattice. The last
property is what lets the represented weight torus of an admissible minuscule lattice be a closed
immersion, rather than seeing only the index-three root lattice of the adjoint representation.
The table is not stable under the pinned diagram symmetry TauCeti.graphPermE6, which exchanges
ϖ₁ and ϖ₆ and so carries the weights of V(ϖ₁) to those of V(ϖ₆) = V(ϖ₁)ˣ, their negatives.
The last two sections make that exchange explicit and repair it: e6MinusculeGraphDualPerm is the
involution of the index set implementing it, and e6DoubledMinusculeWeight is the fifty-four-weight
family of V(ϖ₁) ⊕ V(ϖ₆), which is stable, is still injective, and still spans. A carrier inherits
a symmetry of its numbered data only when its weight family is equivariant, the hypothesis
wt (π i) (τ k) = wt i k of
TauCeti.UniversalEnvelopingAlgebra.kostantToralNumberedSymmetryIso, which the doubled family
meets and the minuscule family alone does not.
No representation or group scheme is constructed here. This is the pinned weight-diagram input
for the type-E₆ full-weight Chevalley--Demazure carrier in Layer 9 of
TauCetiRoadmap/ReductiveGroups/README.md, consumed by milestone L0 of the CFSG statement roadmap,
the doubled family being what its graph-twisted family ²E₆(q) needs where E₆(q) does not.
Main declarations #
TauCeti.DynkinType.e6MinusculeWeight: the twenty-seven weights in fundamental coordinates.TauCeti.DynkinType.e6MinusculeReflection: the permutation induced by a simple reflection.TauCeti.DynkinType.e6MinusculeWeight_reflection: the simple-reflection equation.TauCeti.DynkinType.exists_e6MinusculeReflections_eq: every table index is reached from the highest-weight index by simple reflections.TauCeti.DynkinType.range_e6MinusculeWeight: the table is exactly the Weyl orbit ofϖ₁.TauCeti.DynkinType.span_range_e6MinusculeWeight_eq_top: the weights span the character lattice.TauCeti.DynkinType.e6MinusculeGraphDualPerm: the involution of the index set induced by the diagram symmetry followed by duality, andTauCeti.DynkinType.e6MinusculeWeight_e6MinusculeGraphDualPermits defining equation.TauCeti.DynkinType.e6MinusculeWeight_comp_graphPermE6_notMem_range: the twenty-seven weights are not stable under the diagram symmetry.TauCeti.DynkinType.e6DoubledMinusculeWeight: the fifty-four weights ofV(ϖ₁) ⊕ V(ϖ₆), withTauCeti.DynkinType.e6DoubledMinusculeWeight_injective,TauCeti.DynkinType.span_range_e6DoubledMinusculeWeight_eq_topandTauCeti.DynkinType.e6DoubledMinusculeWeight_e6DoubledMinusculeGraphPerm, its equivariance for the diagram symmetry.
References #
The node numbering and the choice of the minuscule weight ϖ₁ follow Bourbaki, Lie Groups and Lie
Algebras, Chapters 4--6, Plate V. The minuscule-orbit description of the 27-dimensional
representation follows J. E. Humphreys, Introduction to Lie Algebras and Representation Theory,
§13.4, and J. C. Jantzen, Representations of Algebraic Groups, II.2. That the diagram
automorphism exchanges the two twenty-seven dimensional representations, and the conventions for
²E₆(q) that makes this relevant to, are R. W. Carter, Simple Groups of Lie Type, §12.2. The
reflection-reachability argument follows the parallel type-E₇ construction in
TauCeti.LinearAlgebra.RootSystem.SimplyConnectedRootDatum.E7.MinusculeWeight.
The weight table #
The twenty-seven weights in the Weyl orbit of the type-E₆ minuscule weight ϖ₁.
Coordinates are pairings with the six Bourbaki-numbered simple coroots. The ordering begins at
ϖ₁ = (1, 0, 0, 0, 0, 0) and then lists weights reached successively by simple reflections; no
mathematical structure depends on the ordering.
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- One or more equations did not get rendered due to their size.
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The twenty-seven minuscule weights are pairwise distinct.
The first weight in the table is the first fundamental weight ϖ₁.
Every pairing of an E₆ minuscule weight with a simple coroot is -1, 0, or 1.
Every simple-coroot coordinate takes the value -1 on some weight of the type-E₆
minuscule representation. Equivalently, every positive simple-root operator has a nonzero step
on the minuscule weight graph.
Simple reflections #
The permutation of the twenty-seven minuscule weights induced by the i-th simple
reflection.
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Applying the same simple reflection twice fixes every index in the weight table.
The coordinate equation for a simple reflection on the minuscule weights. Reflection in
the i-th simple root subtracts the pairing with the i-th simple coroot times that root.
A simple reflection fixes a minuscule weight exactly when its simple-coroot coordinate is zero.
A simple reflection moves an E₆ minuscule weight earlier in the explicit table exactly
when its simple-coroot coordinate is -1.
The explicit permutation agrees with reflection in the pinned simply connected root datum.
The Weyl orbit #
Every index in the minuscule weight table is reached from the highest-weight index by a finite sequence of simple reflections.
The explicit table is exactly the Weyl orbit of the first fundamental weight ϖ₁.
The Weyl orbit of ϖ₁ has twenty-seven elements.
Generation of the character lattice #
The twenty-seven minuscule weights span the full type-E₆ character lattice. In
particular, a diagonal torus acting with these weights is represented faithfully.
The diagram symmetry on the weight table #
The permutation of the twenty-seven minuscule weights induced by the E₆ diagram symmetry
followed by duality. The symmetry TauCeti.graphPermE6 does not preserve the weight table, so
this permutation records the composite of that symmetry with duality: it matches the index a with
the index whose weight is the negative of the image of the a-th weight.
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The graph-and-duality permutation of the minuscule weight table is an involution, as the diagram symmetry it realizes is.
The graph-and-duality permutation of the minuscule weight table is its own inverse.
The diagram symmetry carries each minuscule weight to the negative of another one. The
symmetry exchanges ϖ₁ and ϖ₆, so on representations it carries V(ϖ₁) to
V(ϖ₆) = V(ϖ₁)ˣ, whose weights are the negatives of these.
The functional form of e6MinusculeWeight_e6MinusculeGraphDualPerm_apply.
The diagram symmetry moves every minuscule weight off the table. Its image is the negative
of a minuscule weight, and no minuscule weight is the negative of another; at the highest weight
ϖ₁ this is the image ϖ₆, not a weight of V(ϖ₁). So a carrier built from this weight family
alone does not inherit the diagram automorphism.
The doubled weight family #
The fifty-four weights of the type-E₆ representation V(ϖ₁) ⊕ V(ϖ₆), in coordinates
given by the pairings with the six Bourbaki-numbered simple coroots. The left summand carries the
twenty-seven minuscule weights and the right summand their negatives, which are the weights of the
dual representation V(ϖ₆).
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The doubled family is closed under negation, which exchanges its two summands.
The fifty-four weights of V(ϖ₁) ⊕ V(ϖ₆) are pairwise distinct. No minuscule weight is
the negative of another, all twenty-seven of them lying in one nontrivial coset of the root
lattice.
The doubled minuscule weight family has fifty-four distinct members.
The doubled minuscule weights span the full type-E₆ character lattice. They include the
twenty-seven minuscule weights, which already do.
The permutation of the doubled index set realizing the E₆ diagram symmetry. It exchanges
the two summands along e6MinusculeGraphDualPerm, which is what makes the doubled weight family
equivariant where the minuscule family alone is not.
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The permutation realizing the diagram symmetry on the doubled index set is an involution.
The permutation realizing the diagram symmetry on the doubled index set is its own inverse.
The doubled minuscule weight family is equivariant for the E₆ diagram symmetry. This is
the hypothesis wt (π i) (τ k) = wt i k under which a numbered symmetry of a Kostant toral-closure
carrier extends to an automorphism of the carrier, and it is what
e6MinusculeWeight_comp_graphPermE6_notMem_range denies to the minuscule family alone.