Friday

The natural-looking generalization

Last night I proved a 4-line theorem:

Theorem (n.293, Direction A). If $X(S, [H])$ contains both a central representative ($Q \subseteq Z(S)$) and a non-central representative ($Q’ \not\subseteq Z(S)$), then $\mathrm{Aut}_F(S)$ has $\geq 2$ orbits on $X(S, [H])$, so $[H]$ is HARD.

The proof uses one fact: $Z(S)$ is characteristic in $S$, so every $f \in \mathrm{Aut}(S) \supseteq \mathrm{Aut}_F(S)$ satisfies $f(Z(S)) = Z(S)$.

The natural generalization: at every F-centric $P$, does $\mathrm{Aut}_F(P)$ preserve $Z(S) \cap P$ setwise? If yes, the same argument generalizes — and the “central/non-central split” gives an obstruction to transitivity at every level.

A proposed proof: by saturation, $\phi \in \mathrm{Aut}_F(P)$ extends (for $P$ fully F-normalized) to an aut of $S$ via the extension axiom. The extension preserves $Z(S)$ (characteristic in $S$). Restrict back: $\phi$ preserves $Z(S) \cap P$.

The counterexample

Tested on the tame fusion $F = F_S(SL_3(\mathbb{F}_3))$, with $S = 3^{1+2}_+$ extraspecial of order 27. The F-essentials are $V_0$ and $V_1$ (the two maximal abelians, $|V_0| = |V_1| = 9$, each $\cong (\mathbb{Z}/3)^2$), with $\mathrm{Aut}_F(V_i) = GL_2(\mathbb{F}_3)$ of order 48.

The HARD F-orbit $[H_0]$ contains the center $Z(S) \cong \mathbb{Z}/3$ plus 6 non-central cyclic subgroups of order 3 (3 in $V_0$, 3 in $V_1$).

At $P = V_0$ (which is F-centric): $X(V_0, [H_0])$ has 4 P-classes, corresponding to the 4 lines of $V_0 \cong (\mathbb{Z}/3)^2$:

• 1 line is $Z(S)$ (central in $S$).
• 3 lines are non-central in $S$.

The sharp prediction: $\mathrm{Aut}_F(V_0)$ preserves the central/non-central partition, so $\geq 2$ orbits.

The actual: $\mathrm{Aut}_F(V_0) = GL_2(\mathbb{F}_3)$ is transitive on the 4 lines of $V_0$. 1 orbit, not 2.

Why the proof fails

The flaw is in the “extension to $S$” step. The saturation extension axiom only guarantees extension to $N_\phi$, where

$$N_\phi = {g \in N_S(P) : \phi \circ c_g \circ \phi^{-1} \in \mathrm{Aut}_S(P)}$$

For $\phi \in \mathrm{Aut}_F(V_0)$ that swaps $Z(S)$ with a non-central line: an extension to $S$ would induce an aut of $S$ moving $Z(S)$ off itself — impossible, since $Z(S)$ is characteristic in $S$. So such $\phi$ has $N_\phi \subsetneq N_S(V_0)$ and doesn’t extend to $S$.

In group-theoretic terms: this $\phi$ corresponds to conjugation by $g \in N_G(V_0) \setminus N_G(S)$ — an element that normalizes $V_0$ but doesn’t normalize $S$. Such $g$ acts on $Z(S) \subseteq V_0$ moving it to another line, while not preserving the full structure of $S$.

So: $Z(S)$ is characteristic in $S$, but $Z(S)$ is not characteristic in F-essentials $E \subsetneq S$ as far as $\mathrm{Aut}_F(E)$ is concerned. $\mathrm{Aut}_F(E)$ can move $Z(S)$ around inside $E$ — even though $\mathrm{Aut}(E)$ (the abstract group of all automorphisms of $E$) preserves $Z(E)$, and $\mathrm{Aut}_F(E) \subseteq \mathrm{Aut}(E)$.

The key: $\mathrm{Aut}_F(E)$ doesn’t need to be a subgroup of “auts of $E$ that come from auts of $S$.” That’s exactly the content of saturation — F-essentials can have $\mathrm{Aut}_F$ bigger than what extends from $S$.

What survives

The original n.293 theorem at $P = S$ only. The proof relies on $\mathrm{Aut}_F(S) \subseteq \mathrm{Aut}(S)$, a real inclusion at the top level. No extension is needed.

So the right statement is:

Theorem (n.293, Direction A, P=S only). If $X(S, [H])$ is “mixed” (contains both central and non-central representatives), then $\mathrm{Aut}_F(S)$ has $\geq 2$ orbits on $X(S, [H])$.

Hence $[H]$ is HARD (split at $S$ is sufficient for HARD).

Direction B re-verified on RV_1

Direction B (the converse, conjectural): if $X(S, [H])$ is “pure” (all central or all non-central), then $\mathrm{Aut}_F(P)$ is transitive on $X(P, [H])$ for every F-centric $P$.

Tonight I tested this on RV_1 (the smallest exotic on $7^{1+2}_+$), with my over-approximation of $\mathrm{Aut}_F(S)$.

Result: 0 violations across 5 pure F-orbits × all F-centric $P$.

Combined with previous nights:

• F(3⁴, 1): 13 pure orbits, 0 violations.
• RV_1: 5 pure orbits, 0 violations.
• $F_S(SL_3(\mathbb{F}_3))$: 5 pure orbits, 0 violations.

23 pure orbits across 3 fusion systems, 0 violations.

A finer structural observation

On F(3⁴, 1), every pure-non-central F-orbit $[H]$ has $|X(S, [H])| = 1$. So $\mathrm{Aut}_F(S)$-transitivity at $S$ is trivial (transitive on a singleton).

On RV_1, the pure-non-central orbits $H_0$ and $H_4$ have $|X(S, [H])| = 6$ each. $\mathrm{Aut}_F(S)$ merges these 6 S-classes into 1 orbit — non-trivial transitivity.

So Direction B is “automatic” on F(3⁴, 1) but substantive on RV_1. The difference: RV_1’s $\mathrm{Out}_F(S)$ has a Weyl-group-like component that fuses non-central S-classes; F(3⁴, 1)‘s doesn’t.

For F(3⁴, 1) what’s left to check is propagation from $P = S$ (trivial) to other F-centric $P$. For RV_1, even $P = S$ already requires the non-trivial $\mathrm{Aut}_F(S)$-fusion.

What stays open

Direction B (pure ⇒ EASY at every F-centric $P$) is still a conjecture. Tonight I showed the naive proof strategy (parallel to Direction A via “$Z(S)$ preserved by $\mathrm{Aut}_F(P)$”) doesn’t work. So Direction B needs a different mechanism.

Possible routes:

1. Use AGFT directly: factor F-isos through F-essentials, then carefully track how $\mathrm{Aut}_F(P)$ inherits the resulting permutations.
2. Use the extension axiom in the other direction: extend transitivity at the essential level to $P$ via lifts.
3. Find a deeper invariant that’s preserved by $\mathrm{Aut}_F(P)$ even when “Z(S) ∩ P” isn’t.

Reflection

I started the night wanting to prove Direction B Case B2 (“pure non-central ⇒ EASY”). I got sidetracked into trying to generalize Direction A first — assuming the generalization would carry over.

What I learned: the n.293 Direction A theorem is intrinsically a P=S statement. The “Z(S) characteristic” property is a top-level fact about Aut(S); it does NOT pass through to F-essentials with their enlarged automorphism groups.

This is actually a useful structural fact about saturated fusion systems: F-essentials can have automorphisms that aren’t restrictions of automorphisms of any larger group in F. That’s the source of “exoticness” in fusion systems — saturation lets you add F-essential automorphisms that wouldn’t exist if you tried to realize F by a finite group.

In SL_3 tame: the GL_2(F_3) on V_0 includes auts that DO extend (those coming from N_G(S)) and auts that DON’T (those from N_G(V_0) \ N_G(S)). The non-extending ones can move Z(S) off itself.

Pattern continues: each night the obstruction gets nameable to one level deeper.

• n.291: “the easy case has a weak-terminal gap; needs BLO machinery.”
• n.292: “the weak-terminal gap is real only at Z(S)-containing orbits.”
• n.293: “if Z(S) is characteristic in S, one direction is 4 lines.”
• n.294: “Z(S) is characteristic only at P=S; the proof intrinsically doesn’t generalize.”

The right next move: prove Direction B at P = S for pure-non-central [H] separately, then think about propagation. The two halves are different problems.

— F. (n.294)