Chevalley relations for symplectic root subgroups #
This file lifts the six multiply-laced rank-two commutator relations from the standard
symplectic matrices to the functor of points of Sp₂ₘ. For distinct i and j, they include
⁅x_{eᵢ-eⱼ}(a), x_{2eⱼ}(b)⁆
= x_{eᵢ+eⱼ}(ab) x_{2eᵢ}(a²b),
⁅x_{eᵢ-eⱼ}(a), x_{eᵢ+eⱼ}(b)⁆
= x_{2eᵢ}(2ab).
The opposite root string gives
⁅x_{eᵢ-eⱼ}(a), x_{-2eᵢ}(b)⁆
= x_{-eᵢ-eⱼ}(-ab) x_{-2eⱼ}(a²b),
⁅x_{eᵢ-eⱼ}(a), x_{-eᵢ-eⱼ}(b)⁆
= x_{-2eⱼ}(-2ab).
The two complementary strings starting from the sum-root families are
⁅x_{eᵢ+eⱼ}(a), x_{-2eⱼ}(b)⁆
= x_{eᵢ-eⱼ}(ab) x_{2eᵢ}(-a²b),
⁅x_{-eᵢ-eⱼ}(a), x_{2eⱼ}(b)⁆
= x_{eⱼ-eᵢ}(-ab) x_{-2eᵢ}(-a²b).
The parameters on the right are expressed using the algebra structure of the value ring. Thus
TauCeti.AdditiveGroup.gaPointParamMul f g has parameter ab; its product with itself in the
additive group has parameter 2ab. These are not statements about convolution multiplication
being the multiplication of the value ring.
The relations supply the rank-two C₂ check for the root-subgroup part of the pinned
Chevalley--Demazure interface. In characteristic two, the second relation becomes commutation,
while the first retains the quadratic term used by the B₂/C₂ special isogeny.
References #
- R. W. Carter, Simple Groups of Lie Type (1972), §5.2 and §11.3.
- J. E. Humphreys, Linear Algebraic Groups (1975), §26.3.
The multiply-laced Chevalley relation on algebra-valued points of Sp₂ₘ. For
distinct i and j, the commutator of the roots eᵢ-eⱼ and 2eⱼ is the product of the
root subgroups for eᵢ+eⱼ and 2eᵢ, with parameters ab and a²b.
The structure-constant-two Chevalley relation on algebra-valued points of Sp₂ₘ.
For distinct i and j, the commutator of the roots eᵢ-eⱼ and eᵢ+eⱼ is the long-root
point x_{2eᵢ}(2ab). The square on the right is convolution multiplication in 𝔾ₐ(A),
which adds its parameter to itself.
The negative multiply-laced Chevalley relation on algebra-valued points of Sp₂ₘ.
For distinct i and j, the commutator of the roots eᵢ-eⱼ and -2eᵢ is the product
of the root subgroups for -eᵢ-eⱼ and -2eⱼ, with parameters -ab and a²b.
The negative structure-constant-two Chevalley relation on algebra-valued points of
Sp₂ₘ. For distinct i and j, the commutator of the roots eᵢ-eⱼ and -eᵢ-eⱼ is
the long-root point x_{-2eⱼ}(-2ab). The inverse and square on the right are the inverse
and multiplication of Sp₂ₘ(A); since negativeLongRootSubgroupPoints j is a group
homomorphism out of 𝔾ₐ(A), they correspond to negating and doubling the parameter.
The complementary positive-sum multiply-laced relation on algebra-valued points of
Sp₂ₘ. The inverse long-root point on the right represents the parameter -a²b.
The complementary negative-sum multiply-laced relation on algebra-valued points of
Sp₂ₘ. Both inverses on the right encode the negative parameters in the additive source
group.