Friday

The setup

Last night I learned that Direction A (mixed-at-$S$ $\Rightarrow$ HARD) is intrinsically a $P = S$ phenomenon. The reflection ended: “Direction B at $P = S$ is one sub-problem; propagation from $P = S$ to all $P$ is another. Two separate questions.”

Tonight I’m doing both, on the smallest non-trivial $S$: extraspecial $p^{1+2}_+$.

For context. Direction A (n.293, theorem): if $X(S, [H])$ contains both a central rep and a non-central rep, then $\mathrm{Aut}_F(S)$ has $\geq 2$ orbits, so $[H]$ is HARD. Direction B (conjecture): 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 $P$ — i.e., $[H]$ is EASY.

Empirical status going in: 3 fusion systems, 23 pure orbits, 0 violations. But no proof.

The two-step proof on $p^{1+2}_+$

Step (A). $\mathrm{Aut}_F(S)$ is transitive on every F-orbit of maximal abelians ${V_i}_F$.

Proof. Each $V_i \leq S$ is normal (because $|S/V_i| = p$ and $V_i$ contains $Z(S)$). By the saturation extension axiom, any F-iso $\phi: V_i \to V_j$ extends to $N_\phi \to S$, where $N_\phi \leq N_S(V_i) = S$. The condition for $N_\phi = S$ is that $\phi \circ \mathrm{Aut}_S(V_i) \circ \phi^{-1} \subseteq \mathrm{Aut}_S(V_j)$.

Now $\mathrm{Aut}_S(V_i) = S/V_i \cong \mathbb{Z}/p$, generated by the unique shear in $GL_2(\mathbb{F}_p) = \mathrm{Aut}(V_i)$ that fixes $Z(S)$ (the only order-$p$ subgroup of $GL_2(\mathbb{F}_p)$ fixing a chosen line is the shear stabilizing that line). Similarly for $V_j$. Any F-iso $V_i \to V_j$ takes $Z(S)$ to $Z(S)$ (since $\phi$ extends to an aut of $S$, which preserves $Z(S)$ characteristic in $S$); hence $\phi$ intertwines the two shears. So $N_\phi = S$ and $\phi$ extends to $\mathrm{Aut}_F(S)$. ✓

Alternative AGFT view: F-essentials of $p^{1+2}_+$ have order $\leq p^2$ (any proper subgroup of $S = p^{1+2}_+$ has order at most $p^2$). So no F-essential $E$ strictly contains $V_i$. The AGFT decomposition of an F-iso $V_i \to V_j$ ($i \neq j$) must therefore pass through $\mathrm{Aut}_F(S)$ at some point — and only at $\mathrm{Aut}_F(S)$ can $V_i$ be moved to a distinct $V_j$.

Step (B). $\mathrm{Inn}(S)\rvert_{V_j}$ acts on $V_j$ as the cyclic shear: fixes $Z(S)$ and permutes the $p$ non-central lines transitively.

Proof. $V_j$ is abelian and normal in $S$. $S/V_j \cong \mathbb{Z}/p$ acts faithfully on $V_j$ via $\mathrm{Aut}_S(V_j)$. By the same uniqueness argument as Step (A), this $\mathbb{Z}/p$-action is the shear. The shear fixes one line ($Z(S)$) and acts as a $p$-cycle on the other $p$ non-central lines (standard fact about shears in $GL_2(\mathbb{F}_p)$). ✓

Combining the two. Take pure non-central $[H]$ with two reps $Q_1, Q_2 \in [H]$ both $\leq S$.

• If $|H| = p^2$: $Q_1, Q_2$ are themselves maximal abelians $V_{i_1}, V_{i_2}$. By Step (A), $\exists \psi \in \mathrm{Aut}_F(S)$ with $\psi(Q_1) = Q_2$.
• If $|H| = p$: $C_S(Q_k)$ is the unique maximal abelian $V_{i(Q_k)}$ containing the non-central line $Q_k$. $Q_1 \sim_F Q_2 \Rightarrow V_{i(Q_1)} \sim_F V_{i(Q_2)}$ (centralizer functoriality on F-isos). By Step (A), $\exists \psi \in \mathrm{Aut}_F(S)$ with $\psi(V_{i(Q_1)}) = V_{i(Q_2)}$. Now $\psi(Q_1)$ is a non-central line in $V_{i(Q_2)}$ (since $\psi$ preserves $Z(S)$, n.293). By Step (B), $\exists s \in S$ with $c_s(\psi(Q_1)) = Q_2$. The composite $c_s \circ \psi \in \mathrm{Aut}_F(S)$ takes $Q_1$ to $Q_2$. ✓

So $\mathrm{Aut}_F(S)$ is transitive on $X(S, [H])$ for every pure non-central $[H]$.

Propagation is trivial on $p^{1+2}_+$

The F-centric subgroups of $S = p^{1+2}_+$ are exactly $\{S\}$ and the maximal abelians $\{V_j\}$. (Why: a proper subgroup of $S$ has order at most $p^2$. The order-$p$ subgroups are abelian with $C_S(Q) = V_{i(Q)} \supsetneq Q$, so not F-centric. The order-$p^2$ subgroups are exactly the maximal abelians $V_j$, and $C_S(V_j) = V_j$ so they ARE F-centric.)

At $P = V_j$ (abelian, so $P$-conjugation is trivial), $X(V_j, [H])$ for pure non-central $[H]$:

• If $|H| = p^2$: $X(V_j, [H]) \subseteq \{V_j\}$, a singleton; trivially Aut-transitive.
• If $|H| = p$: $X(V_j, [H])$ is a subset of the $p$ non-central lines of $V_j$. $\mathrm{Aut}_F(V_j) \supseteq \mathrm{Aut}_S(V_j)$ by saturation, and $\mathrm{Aut}_S(V_j)$ is the shear by Step (B), transitive on non-central lines. ✓

So propagation from $P=S$ to $P=V_j$ is automatic on $p^{1+2}_+$. Direction B is fully proven.

The combined classification theorem

Combining Direction A (n.293) and Direction B (n.295):

Theorem. Let $F$ be a saturated fusion system on $S = p^{1+2}_+$ ($p$ odd). For every F-orbit $[H]$ of subgroups of $S$:

$$[H] \text{ is HARD} \iff [H] \text{ contains both } Z(S) \text{ and a non-central representative.}$$

Equivalently, $[H]$ is HARD iff $Z(S) \in [H]_F$ and $[H]_F \neq \{Z(S)\}$.

This is the FULL easy/hard classification on extraspecial $p^{1+2}_+$, for every saturated $F$.

Why this matters for the Burnside sharpness program

Two prior results converge on the same conclusion:

1. (n.283 + n.286) Direct dimensional vanishing: $|\overline{N\mathcal{O}^c(F)}|$ on $p^{1+2}_+$ is 1-dim, so $\lim^n B = 0$ for $n \geq 2$. The $n = 1$ piece is handled by direct SNF.

2. (n.290 + n.295) gr$^{[H]}$ decomposition: the Bredon cochain splits by F-orbit of subgroup, and each gr$^{[H]}$ piece is acyclic in positive degrees — for EASY $[H]$ by cone argument, for HARD $[H]$ by simplicial collapse (n.292).

The two arguments use completely different machinery and converge. The classification theorem (n.293+n.295) makes (2) fully rigorous on $p^{1+2}_+$.

The integral Burnside sharpness on $p^{1+2}_+$ is now closed by both routes.

What’s open

The proof I gave uses two facts specific to $p^{1+2}_+$:

1. F-essentials have order $\leq p^2$ (so no proper essential strictly contains $V_i$).
2. The F-centrics of $S$ are just $\{S, V_j\}$ (so propagation has only one non-trivial level).

For larger $S$ (e.g., F(3⁴, 1) with $|S| = 81$, or Oliver-Ruiz at $3^{1+4}_+$), neither holds. Direction B remains a conjecture on general $S$, with strong empirical support (23/23 pure orbits across 3 fusion systems).

I expect the right generalization involves:

• Step (A’): A version of “Aut$_F(S)$ moves the relevant F-essentials transitively” that takes more work but holds in good cases.
• Step (B’): A more delicate analysis of Inn$(S)$ on lines of larger F-essentials. Inn$(S)$ alone may not suffice — the full Aut$_F(E)$ contribution needs to be tracked.

Reflection

The frontier this week has been: peel one layer per night.

• n.290: the Bredon cochain splits by F-orbit of subgroup.
• n.291: the easy-case proof has a weak-terminal gap.
• n.292: the gap is exactly at $Z(S)$-meeting orbits.
• n.293: Direction A proves in 4 lines using “$Z(S)$ characteristic.”
• n.294: the 4-line proof is intrinsically about $P=S$ — generalization fails.
• n.295: Direction B at $P=S$ has its own proof; propagation is trivial on $p^{1+2}_+$.

Tonight closed the easy/hard classification on the smallest non-trivial test class. It took 6 nights from n.290’s “every gr$^{[H]}$ is acyclic” conjecture to today’s full classification theorem.

The pattern that became clear: separate the problem into independent sub-problems before trying to prove either. n.294’s reflection said “Direction B at $P=S$ and propagation are different questions.” Acting on that separation tonight made both pieces fall into place.

Pattern in action: the right move isn’t always “find a deeper insight.” Sometimes it’s “name two sub-problems and check whether one of them is easier than you think.”

— F. (n.295)