The special length-exchanging map of the pinned type G₂ root datum #
In characteristic three, the Ree construction uses a special isogeny of the simply connected group
of type G₂. At the root-datum level its character-lattice map exchanges the two Bourbaki nodes
and multiplies in the long-root direction. In the fundamental-weight coordinates of
TauCeti.DynkinType.g2Root, that map is
A = !![0, 3; 1, 0].
This file computes the action of A on every root and of Aᵀ on every coroot. The induced root
permutation exchanges the two simple roots, exchanges the two remaining pairs of positive roots,
and commutes with root negation. The rescaling exponent is the existing squared-length table
TauCeti.DynkinType.g2Length: it is 1 on short roots and 3 on long roots. Applying the data
twice multiplies both lattices by 3, and the two exponents along each orbit multiply to 3.
These equations are the explicit G₂ input for the root-datum special-isogeny construction, the
odd characteristic beside the type B₂ case of
TauCeti/LinearAlgebra/RootSystem/SimplyConnectedRootDatum/B/SpecialMap.lean. They do not yet
construct a group-scheme morphism; Layer 9 of the reductive-groups roadmap requires that later
lift, together with its action on root subgroups, before the Suzuki--Ree lane can use it.
Main definitions #
TauCeti.DynkinType.g2SpecialIsogenyMatrix: the character-lattice matrix.TauCeti.DynkinType.g2SpecialIsogenyIndex: the induced permutation of the twelve root indices, andTauCeti.DynkinType.g2SpecialIsogenyIndexEquivthe same permutation as anEquiv.Perm.
The rescaling exponent needs no definition of its own here. The pinned G₂ datum is tabulated on
its own root indices, so TauCeti.DynkinType.g2Length already is that exponent; the rank-two type
B datum is indexed uniformly in the rank instead, which is why its exponent carries a name of its
own.
Main results #
TauCeti.DynkinType.g2SpecialIsogenyMatrix_mulVec_rootandTauCeti.DynkinType.g2SpecialIsogenyMatrix_transpose_mulVec_coroot: the equations on the pinned datum.TauCeti.DynkinType.g2SpecialIsogenyMatrix_mul_self: the square of the lattice map is multiplication by3.TauCeti.DynkinType.det_g2SpecialIsogenyMatrix: the map has determinant-3, so it is an isogeny and not a lattice automorphism.TauCeti.DynkinType.g2Length_mul_g2Length_g2SpecialIsogenyIndex: the two rescaling exponents on an orbit multiply to the defining characteristic.TauCeti.DynkinType.g2SpecialIsogenyIndex_castLE: on the two simple roots the index permutation isTauCeti.lengthPermRankTwo, the pinned length-exchanging permutation of the diagram.TauCeti.DynkinType.g2Length_mul_pairing_g2SpecialIsogenyIndexEquiv: the Cartan integers transform by the rule a special isogeny forces.
References #
The node numbering and coordinates follow Bourbaki, Lie Groups and Lie Algebras, Chapters 4--6,
Plate IX. The special-isogeny equations follow R. Steinberg, Endomorphisms of Linear Algebraic
Groups, §11; the Ree convention is also described in R. W. Carter, Simple Groups of Lie Type,
§§12.3--12.4. The exponent is 1 on long root subgroups and the characteristic on short root
subgroups when the group-scheme map is read contravariantly on its root datum. Here the lattice map
consequently rescales a root by its own squared length before exchanging its index.
The lattice map and the root permutation #
The character-lattice matrix of the special length-exchanging map of the pinned G₂ root
datum, in the fundamental-weight basis.
Equations
- TauCeti.DynkinType.g2SpecialIsogenyMatrix = !![0, 3; 1, 0]
Instances For
The explicit entries of the character-lattice special-isogeny matrix.
The permutation of the twelve G₂ roots induced by
TauCeti.DynkinType.g2SpecialIsogenyMatrix. It exchanges long roots with short roots and commutes
with root negation.
Instances For
The special root permutation is an involution.
Applying the special root permutation twice fixes every root index.
The special permutation of the root indices of the pinned type G₂ datum.
Equations
Instances For
The special root permutation commutes with passing to the negative root.
The special root permutation preserves positivity: it maps the six positive roots, which are the first six indices, among themselves.
On the two simple roots the special permutation is the pinned length-exchanging permutation of the diagram.
Action on the pinned root datum #
The special matrix carries every tabulated root to its indexed image with the prescribed exponent.
The transposed special matrix satisfies the contragredient equation on every tabulated coroot.
The special matrix carries every root of the pinned datum to its indexed image with the prescribed exponent.
The transposed special matrix satisfies the contragredient equation on every coroot of the pinned datum.
The square of the character-lattice special matrix is three times the identity matrix.
The special map is an isogeny and not an automorphism of the character lattice: its
determinant is -3, of absolute value the characteristic.
Length and exponent conventions #
The exponents at a root index and its image multiply to the characteristic 3.
The special permutation exchanges long roots with short ones.
The Cartan integers transform by the rule a special isogeny forces. Writing α' for the
image of a root α under the special permutation, pairing the root equation against the coroot
equation gives ℓ(α) ⟨α', β'∨⟩ = ℓ(β) ⟨α, β∨⟩. No diagram automorphism satisfies this, since the
two lengths differ.