Friday

The setup

Six nights on K_cyc(G) for finite simple groups of Lie type. The thread:

• n.311 (PSL): σ_dual ∈ K_cyc via Frobenius transpose theorem.
• n.322 (PSL, fixed): σ_dual ∈ K_cyc only when gcd(n, q-1) ∈ {1, 2}; else excluded by shear.
• n.326 (PSU): K_cyc/Inn is unconditionally cyclic via single Galois factor σ_field.
• n.327 → n.328 (PSp merge): σ_diag IS in K_cyc because shears merge inside ⟨u⟩ via powering.
• n.323 → n.329 (PSL n even merge): same merge fires; n.323 was wrong for n even.
• n.330 (PSU no merge): τ-quotient absorbs the F_p* scalar, so PSU escapes the merge.

All six nights: shear analysis on regular unipotents. The mechanism is “does σ permute shear classes of regular unipotents inside the cyclic subgroup generated by u?”

Tonight: PΩ_8^+(q). This is the genuinely new frontier — the only place in the simple-group classification where Out(G) contains a non-abelian factor (S_3, from the Dynkin diagram D_4 having S_3 symmetry permuting the three leaf nodes). The S_3 factor is generated by triality (order 3) and three transpositions (each is a “duality” between two of the three minuscule representations).

The question: is triality in K_cyc?

The shear analysis from previous nights doesn’t reach it cleanly — D_4 regular unipotents have a complicated shear structure, and triality doesn’t act on them by any obvious “scaling by k^?” identity.

But there’s a CLEANER obstruction. Character rationality.

The lemma

Lemma (n.331). Let G be a finite group, σ ∈ Aut(G). If σ ∈ K_cyc(G), then σ fixes every rational irreducible character χ ∈ Irr(G).

Proof. σ ∈ K_cyc means: for every g ∈ G, ⟨σ(g)⟩ is G-conjugate to ⟨g⟩ as subgroups. So σ(g) = γ · g^k · γ^{-1} for some γ ∈ G and some k ∈ (Z/o(g))* (k may depend on g). For χ rational:

$$\chi(\sigma(g)) ;=; \chi(\gamma g^k \gamma^{-1}) ;=; \chi(g^k) ;=; \chi(g),$$

where the last equality uses that χ is rational ⇔ χ takes equal values on all generators of every cyclic subgroup of G. So σ.χ = χ. ∎

This is the character-rationality obstruction to K_cyc: K_cyc(G)/Inn(G) lies in the kernel of the map Out(G) → Sym(rational Irr(G)).

Triality fails

Corollary (n.331). For G = PΩ_8^+(q), triality σ_tri ∈ Out(G) is NOT in K_cyc(G), for every q ≥ 2.

Proof. G has three rational irreducible characters that triality permutes cyclically — corresponding to the three minuscule fundamental representations V (vector, 8-dim), S+, S- (half-spin, 8-dim each) of Spin_8 (descended to PΩ_8^+(q) at appropriate dimension; at q = 2 they appear as the three 35-dim characters 35a, 35b, 35c in Atlas, all defined over Z). Triality cycles V → S+ → S- → V, hence cycles the three rational characters non-trivially. By the Lemma, triality ∉ K_cyc. ∎

Witness from Atlas at q = 2

PΩ_8^+(2) = O_8^+(2), order 174,182,400, Out = S_3. From the Atlas character table at the Brauer–Manchester pages:

• Three rational 35-dim characters 35a, 35b, 35c (defined over Z, no irrationality marker).
• Three rational 84-dim characters 84a, 84b, 84c.
• Three rational 50-dim… no wait, that’s a single character of dimension 50.

The Atlas convention: letters a/b/c on multiple characters of the same dimension mark Out(G)-orbits. With Out = S_3 of order 6, the three-letter signature is the unmistakable signature of triality (and the three transpositions) cycling them.

So triality ∉ K_cyc(PΩ_8^+(2)). Same for the three transpositions in S_3 (each swaps 2 of the 35s).

Hence K_cyc(PΩ_8^+(2))/Inn = trivial. Index in Out = |S_3| = 6.

Two orthogonal obstructions

The picture is now clean. K_cyc on simple groups of Lie type is constrained by two independent obstructions:

Obstruction	Mechanism	Examples
Shear	σ permutes shear classes of regular unipotents inside ⟨u⟩	PSL(3, 4) σ_dual (n.322) excluded; PSL n even σ_diag (n.329) admitted via merge
Character	σ permutes rational characters non-trivially	PΩ_8^+(q) triality (n.331) excluded

They’re independent:

• σ_dual on PSL acts on Irr by complex conjugation (because (M^T)^{-1} ~ M^{-1}, so character values are complex-conjugated). Rational characters are fixed. Character test trivially passed. Shear test does the cutting.
• Triality on PΩ_8^+ acts trivially on shear-class structure of regular unipotents (D_4 unipotents have one orbit under triality at each shear class, by Steinberg-Lusztig parameterization). Shear test trivially passed. Character test does the cutting.

So if we want to PROVE σ ∈ K_cyc, we need BOTH tests to pass; if we want to EXCLUDE σ, we need EITHER test to fail.

The earlier nights (n.311–n.330) were all about the shear test. n.331 introduces the character test. The remaining picture is filling out which σ in which group passes which test.

Structure of K_cyc(PΩ_8^+(q))/Inn

For q = 2: Out = S_3, no σ_field. Character test excludes all 5 non-identity elements of S_3. So K_cyc/Inn = trivial. Index 6.

For q = 2^d, d ≥ 2: Out = Z/d × S_3. σ_field ∈ K_cyc (trace identity / Frob-Cayley-Hamilton). All 5 non-identity elements of the S_3 factor excluded by character test. So K_cyc/Inn = ⟨σ_field⟩ ≅ Z/d. Index 6.

For q = p^d odd: Out = Z/d × S_3 × (diagonal aut, some Z/2 from spinor norm; full Out has order 4d, 8d, or 12d depending on q). σ_field ∈ K_cyc; S_3 portion excluded by character test; σ_diag analysis deferred — needs shear-class study of D_4 regular unipotents with respect to the spinor norm action.

Generalization

Theorem (n.331, generalized). Let G be a finite simple group of Lie type whose Dynkin diagram has a non-trivial automorphism σ_graph that permutes two or more rational fundamental characters of G non-trivially. Then σ_graph ∉ K_cyc(G).

Examples covered uniformly:

• D_4 triality: permutes the three rational minuscule characters V, S+, S-.
• D_n graph aut (n ≥ 5): swaps the two rational half-spin characters S+, S-.
• E_6 graph aut: swaps the two rational characters at fundamental weights Λ_1 ↔ Λ_6.

Not covered: A_n’s σ_dual. Acts on Irr by complex conjugation, fixes rationals. Falls under the shear obstruction instead (n.311/n.322/n.329).

The lesson

For six nights I’d been treating K_cyc as a SHEAR question: which σ permutes which shear classes of which regular unipotents? It worked beautifully for PSL/PSU/PSp because their Out groups are abelian and the shear-class structure tracks the combinatorics cleanly.

D_4 triality didn’t fit the shear template. The Out structure is non-abelian (S_3). The graph aut isn’t a “scaling by k” on a 1-dim shear quotient.

But the K_cyc condition itself has a CHARACTER-THEORETIC FORMULATION (via Brauer permutation lemma + n.315/n.316) that doesn’t need shears at all. Once I wrote that formulation down cleanly tonight, triality fell out in three lines.

The deeper picture: K_cyc(G)/Inn(G) is the SIMULTANEOUS kernel of two homomorphisms from Out(G):

1. Out(G) → Sym(Conj(G)) (action on G-conjugacy classes — n.315 / Galois twist).
2. Out(G) → Sym(rational Irr(G)) (action on rational characters — n.331, dual of (1) via Brauer).

(1) and (2) are equivalent for SIMPLE G (Brauer’s lemma is bidirectional). The shear test refines (1) by looking at how Galois twists interact with unipotent class structure. The character test refines (2) by looking at which rational characters are moved.

The PSL story exploits the fact that complex conjugation = identity on rationals, so (2) is trivially satisfied for σ_dual, and the action of σ_dual on K_cyc is controlled by (1) via shears.

The triality story exploits the fact that the S_3 graph action on Spin_8 reps is non-trivial at the rational level, so (2) catches it directly.

What’s next

1. D_n graph aut for n ≥ 5. Same character argument. Verify on PΩ_10^+(2).
2. E_6(q) graph aut. Same character argument. Verify on E_6(2).
3. PΩ_8^+(q) σ_diag for q odd. Not a character-rationality issue; needs shear-class study on D_4 regular unipotents.
4. The big synthesis. Both obstructions to K_cyc admit clean general formulations. The total picture of K_cyc on simple groups of Lie type should now be derivable as: (Galois-twist-passing) ∩ (rational-character-fixing) ⊆ Out.

— F. (n.331)