The Steinberg endomorphism and candidate group of the Suzuki family #
The Steinberg endomorphism of ²B₂(2^(2m+1)) is not a Frobenius but an odd power of a
half-Frobenius: the exceptional isogeny τ of the ambient group, which squares to the prime-field
Frobenius, raised to the odd exponent 2m+1. This file forms that map on the ambient group of a
Suzuki index, proves the required square relation,
steinberg (m) ^ 2 = Frob_(2 ^ (2m+1)),
and names the family's candidate group: the derived subgroup of the Steinberg fixed points modulo its centre.
The half-Frobenius is available because the ambient group of a Suzuki index is the rank-two
type-C carrier over an algebraically closed field of characteristic two, and that carrier already
carries the special isogeny. So nothing is constructed here: the half-Frobenius is that isogeny,
and the work is the odd power and its square.
The exponent is not a new parameter. TauCeti.ValidLieTypeIndex.fieldExponent already writes the
field order of an index as a power of its characteristic, and on a Suzuki index it is the odd
number 2m+1, so the Steinberg map is the fieldExponent-th power throughout and squaring it
lands on the q-power Frobenius TauCeti.RankTwoBLieIndex.frobenius that the index records rather
than on a separately tabulated field order.
The simple-root-subgroup action #
What is recorded of τ is its action on the numbered simple root subgroups:
τ (x_{α i}(t)) = x_{α (σ i)}(t ^ e i),
for σ the permutation exchanging the long and short simple roots and e the exponent that is
1 on a long simple root and the defining characteristic on a short one. Those are
TauCeti.SuzukiReeIndex.lengthPerm and TauCeti.SuzukiReeIndex.exponent. On the B₂ diagram the
long simple root is Bourbaki node zero, which TauCeti.RankTwoBLieIndex.carrierNode carries to the
final carrier node, so the indexed equation is proved from the two equations at the carrier nodes
and that numbering correspondence.
Main definitions #
TauCeti.SuzukiLieIndex.halfFrobenius: the special isogeny of the ambient group.TauCeti.SuzukiLieIndex.steinberg: its odd powerτ ^ (2m+1).TauCeti.SuzukiLieIndex.FixedPoints: the fixed subgroup ofsteinberg.TauCeti.SuzukiLieIndex.Group: the candidate group,FixedPointCandidate steinberg.
Main results #
TauCeti.SuzukiLieIndex.steinberg_simpleRootSubgroup: the Steinberg map's own action formula at every numbered simple root, exchanging the two roots and raising the parameter top ^ m * exponent i.TauCeti.SuzukiLieIndex.halfFrobenius_simpleRootSubgroup: the action formula at every numbered simple root, against the index's own length permutation and exponent.TauCeti.SuzukiLieIndex.halfFrobenius_halfFrobenius: the square of the half-Frobenius is the prime-field Frobenius.TauCeti.SuzukiLieIndex.steinberg_steinberg: the square of the Steinberg endomorphism is theq-power Frobenius.
What is not here #
Nothing is proved finite, perfect or simple of Group, and Mathlib's separate suzukiGroup is not
mentioned, so no comparison with it is claimed. The fixed points of an odd half-Frobenius power are
not the 𝔽_q points of the carrier, which is why this family is not an instance of the Frobenius
machinery the untwisted ones use.
The ambient group is the explicit rank-two type-C carrier, and it is not identified with the
pinned simply connected group scheme of type B₂: no pinning datum is constructed for the carrier
here or in the files it imports, and the constructions below transfer to that pinned group only
along such an identification, once one is proved. The identification with the B₂ diagram that
is available is the one on numbered root characters,
TauCeti.RankTwoBLieIndex.rootGeneratorWeight_carrierNode_eq_root_simpleIndex, and the
simple-root-subgroup action equations below are stated against it.
References #
- R. W. Carter, Simple Groups of Lie Type, §13.
- R. Steinberg, Endomorphisms of linear algebraic groups, Memoirs AMS 80 (1968), §11.
- On the cohomology of the Ree groups and kernels of exceptional isogenies,
arXiv:2108.06291, for the formulation
τ ^ 2 = Frob_pand its odd powers.
The half-Frobenius of a Suzuki index: the special isogeny of its ambient group.
Equations
Instances For
The half-Frobenius is the carrier's special isogeny.
The square of the half-Frobenius is the prime-field Frobenius, that is τ ^ 2 = Frob_p at
the defining characteristic p = 2. No uniqueness is claimed: nothing here shows that this
relation, or the action on the simple root subgroups, determines an endomorphism of the ambient
group.
The Steinberg endomorphism of a Suzuki index: the odd power τ ^ (2m+1) of the
half-Frobenius, for 2m+1 the field exponent the index records.
Equations
- d.steinberg = d.halfFrobenius ^ (↑d).fieldExponent
Instances For
The Steinberg endomorphism is the fieldExponent-th power of the half-Frobenius.
The square of the Steinberg endomorphism is the q-power Frobenius: squaring the odd
power τ ^ (2m+1) doubles the exponent, and τ ^ 2 is the prime-field Frobenius.
Applying the half-Frobenius after the Suzuki Steinberg map gives the
2^(m+1)-power Frobenius.
The simple-root-subgroup action formula for the half-Frobenius at every numbered simple root, stated against the index's own length permutation and exponent rather than against the two carrier nodes:
τ (x_{α i}(t)) = x_{α (lengthPerm i)}(t ^ exponent i).
The permutation exchanges the long and short simple roots and the exponent is 1 on the long one
and the defining characteristic 2 on the short one, so this is the two equations above read
through the Bourbaki numbering the index carries.
The simple-root-subgroup action formula for the Steinberg endomorphism at every numbered simple root. It exchanges the two simple roots exactly as the half-Frobenius does, its odd power acting on the parameter by the remaining even power of the characteristic:
steinberg (x_{α i}(t)) = x_{α (lengthPerm i)}(t ^ (p ^ m * exponent i)).
The finite-group candidate #
The fixed subgroup of the Steinberg endomorphism attached to a Suzuki index.
Equations
Instances For
The finite-simple-group candidate attached to a Suzuki index: the derived subgroup of the
Steinberg fixed points, modulo the centre of that derived subgroup. No finiteness or simplicity
assertion is part of this definition, and no comparison with Mathlib's suzukiGroup is made.