The tripled minuscule weight table of type D4 #
The three eight-dimensional representations of type D₄, the natural representation V(ϖ₁) and
the two half-spin representations V(ϖ₃) and V(ϖ₄), are minuscule: every pairing of one of
their weights with a simple coroot is -1, 0 or 1. This file enumerates their twenty-four
weights in the fundamental-weight basis Fin 4 → ℤ, eight per summand, beginning each block at
its fundamental weight, and records the structure a Chevalley carrier built on this weight family
needs.
The table is closed under the four Bourbaki-numbered simple reflections through explicit
permutations of Fin 24, with the reflection equation s_i μ = μ - ⟨μ, αᵢ∨⟩ αᵢ, and the orbits
of those reflections are exactly the three summands. Its weights
generate the full character lattice of D₄: the three blocks represent the three nonzero cosets
of the root lattice in the weight lattice, and their weights together generate the whole of it.
What makes all three blocks necessary is not that generation but stability under triality, which
cycles the blocks, so that no one block and no pair of blocks is stable.
Triality, the order-three symmetry TauCeti.trialityPermD4 of the D₄ diagram, fixes the central
node and cycles the three outer nodes, so it cycles the three fundamental weights ϖ₁, ϖ₃, ϖ₄
and with them the three summands. The table is stable under it: d4TripledTrialityPerm is the
permutation of Fin 24 carrying each weight μ to μ ∘ σ⁻¹, which is the equivariance
wt (π a) (σ k) = wt a k under which a numbered permutation of the coordinates of a Kostant
toral-closure carrier extends to an automorphism of the carrier. Its action on this table has order
three.
No representation or group scheme is constructed here. This is the weight-diagram input for the
tripled type-D₄ Chevalley carrier, the carrier on which triality acts.
Main declarations #
TauCeti.DynkinType.d4TripledWeight: the twenty-four weights in fundamental coordinates.TauCeti.DynkinType.d4TripledReflection: the permutation induced by a simple reflection, withTauCeti.DynkinType.d4TripledWeight_reflectionthe simple-reflection equation.TauCeti.DynkinType.d4TripledSummand: the summand containing a weight, withTauCeti.DynkinType.exists_foldl_d4TripledReflection_eq_iffidentifying the three summands with the orbits of the simple reflections.TauCeti.DynkinType.span_range_d4TripledWeight_eq_top: the weights span the character lattice.TauCeti.DynkinType.d4TripledTrialityPerm: the permutation of the table realizing triality, withTauCeti.DynkinType.d4TripledWeight_d4TripledTrialityPerm_applyits equivariance andTauCeti.DynkinType.d4TripledTrialityPerm_pow_threeits order relation, whose pointwise forms for the permutation and its inverse areTauCeti.DynkinType.d4TripledTrialityPerm_apply_apply_applyandTauCeti.DynkinType.d4TripledTrialityPerm_symm_apply_symm_apply_symm_apply.
References #
The node numbering and the identification of the three minuscule weights follow Bourbaki, Lie
Groups and Lie Algebras, Chapters 4--6, Plate IV. The minuscule-orbit description of the
representations follows J. E. Humphreys, Introduction to Lie Algebras and Representation Theory,
§13.4. That triality permutes the three eight-dimensional representations, and the conventions
for ³D₄(q) that make this relevant, are R. W. Carter, Simple Groups of Lie Type, §12.2.
The formal organization follows the type-E₆ minuscule orbit in
TauCeti.LinearAlgebra.RootSystem.SimplyConnectedRootDatum.E6.MinusculeWeight.
The weight table #
The twenty-four weights of the type-D₄ representation V(ϖ₁) ⊕ V(ϖ₃) ⊕ V(ϖ₄).
Coordinates are pairings with the four Bourbaki-numbered simple coroots. Indices 0 to 7 carry
the weights of the natural representation, beginning at ϖ₁ = (1, 0, 0, 0); indices 8 to 15
those of the half-spin representation V(ϖ₃), beginning at ϖ₃; and indices 16 to 23 those
of V(ϖ₄), beginning at ϖ₄. Within each block every weight after the first is a simple
reflection of an earlier weight of the block; no mathematical structure depends on the ordering.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The twenty-four tripled weights are pairwise distinct.
The first weight of the natural block is the first fundamental weight ϖ₁.
The first weight of the second block is the fundamental weight ϖ₃.
The first weight of the third block is the fundamental weight ϖ₄.
Every pairing of a tripled weight with a simple coroot is -1, 0, or 1: the three
summands are minuscule.
Every simple-coroot coordinate takes the value -1 on some tripled weight. Equivalently,
every positive simple-root operator has a nonzero step on the tripled weight graph.
Simple reflections #
The permutation of the twenty-four tripled weights induced by the i-th simple
reflection. It preserves each of the three blocks.
Equations
Instances For
Applying the same simple reflection twice fixes every index in the weight table.
The coordinate equation for a simple reflection on the tripled weights. Reflection in the
i-th simple root subtracts the pairing with the i-th simple coroot times that root, the root
being the i-th row of the type-D₄ Cartan matrix.
The coordinate change under a simple reflection, entry by entry.
The three summands #
The summand containing a tripled weight, numbered 0, 1 and 2 for V(ϖ₁), V(ϖ₃)
and V(ϖ₄): the table lists the eight weights of each summand consecutively. The label is an
integer so that it can serve directly as a block labelling of the coordinates of GL₂₄.
Equations
- TauCeti.DynkinType.d4TripledSummand a = ↑(↑a / 8)
Instances For
Simple reflections preserve each of the three summands.
The summands are the orbits of the simple reflections on the tripled table. One index is carried to another by a word in the simple reflections exactly when the two lie in the same summand.
Generation of the character lattice #
Every type-D₄ spin weight occurs in one of the two half-spin blocks of the tripled table.
The tripled weights span the full type-D₄ character lattice. The last sixteen entries
are the two half-spin blocks, hence contain the full type-D₄ spin-weight family, which already
spans the simply connected character lattice.
Triality on the weight table #
The permutation of the twenty-four tripled weights realizing triality. It carries the
weight μ to μ ∘ σ⁻¹, where σ = TauCeti.trialityPermD4, so it cycles the three blocks
V(ϖ₁) → V(ϖ₃) → V(ϖ₄) → V(ϖ₁). Its inverse is its own square.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Triality carries the highest weight ϖ₁ of the natural block to the highest weight ϖ₃ of
the second block.
Triality carries the highest weight ϖ₃ of the second block to the highest weight ϖ₄ of
the third block.
Triality carries the highest weight ϖ₄ of the third block back to ϖ₁.
The triality permutation of the weight table has order dividing three.
The inverse triality permutation is its square.
A single inverse triality step is two forward triality steps.
The triality permutation of the tripled table has order exactly three.
Applying the triality permutation of the weight table three times is the identity.
Applying the inverse of the triality permutation of the weight table three times is the identity, the inverse having order three with the permutation itself.
The tripled weight family is equivariant for triality: the weight at the image index,
read at the image node, is the weight at the original index read at the original node. This is
the hypothesis wt (π a) (σ k) = wt a k under which a numbered permutation of the coordinates of a
Kostant toral-closure carrier extends to an automorphism of the carrier.
The functional form of d4TripledWeight_d4TripledTrialityPerm_apply: triality carries the
weight μ to μ ∘ σ⁻¹.
Triality intertwines the simple reflections of the weight table with the diagram
permutation: π ∘ s_i = s_{σ i} ∘ π.