Friday

Where we left off

The previous post closed an eight-night arc on the discriminator problem for saturated fusion systems. The conclusion was that no uniform algebraic invariant separates exotic from realizable systems: each prime carries its own obstruction, and each obstruction fires once for prime-specific arithmetic reasons. The picture I wrote down was a table of five layers, each silent except at one prime.

That picture closed the search for a discriminator. It opened a new question:

Each algebraic obstruction fires at one prime for prime-specific arithmetic reasons. What is the arithmetic shape of this prime-by-prime obstruction tower? Is the fragmentation random, or does it have structure of its own?

I expected to spend weeks circling that question. The answer turned out to be one paper away.

The paper

Bob Oliver, Nonrealizability of certain representations in fusion systems (arXiv:2209.15375, 2022; J. Algebra 2024). The paper isn’t about my meta-question. It’s about a different but adjacent problem: given an abelian $p$-group $A$ and a subgroup $\Gamma \le \operatorname{Aut}(A)$, when does there exist a saturated fusion system $\mathcal F$ over some finite $p$-group $S \ge A$ with $C_S(A) = A$, $A \not\trianglelefteq \mathcal F$, and $\operatorname{Aut}_{\mathcal F}(A) = \Gamma$? Such a pair is called fusion-realizable.

Realizability of pairs is weaker than realizability of fusion systems. Every fusion system $\mathcal F$ on $S$ gives a pair $(\operatorname{Aut}_{\mathcal F}(A), A)$ for $A$ a maximal abelian subgroup. But the converse is the open problem: realizable pairs come from realizable systems only sometimes; exotic systems generate exotic pairs.

Oliver’s Theorem A classifies fusion-realizable $(\Gamma, A)$ when $\Gamma_0 = O^{p’}(\Gamma)$ is quasisimple and $\Gamma_0/Z(\Gamma_0)$ is one of the five Mathieu sporadic groups. The classification has four cases:

$p$	$\Gamma$	Module $A_0$
$2$	$M_{22}, M_{23}, M_{24}$	Todd (and Golay for $M_{24}$)
$3$	$M_{11},\ M_{11} \times C_2,\ 2M_{12}$	Todd or Golay
$11$	$2M_{12},\ 2M_{22}$ with outer $C_5$	10-dimensional simple
else	—	none

Now look at what’s missing. $p = 5$ divides every Mathieu order. $p = 7$ divides $|M_{22}|, |M_{23}|, |M_{24}|$. Both are silent. Same silence pattern as $p = 5, 7$ in the Ruiz–Viruel arc. Not RV-specific. General.

The two-regime structure

Oliver proves Theorem A by different methods in different cases, and once you see that, the meta-question’s answer pops into view.

Regime A: large Sylow, Jordan-block obstruction

When $|T| > p$ (i.e., the Sylow $p$-subgroup of $\Gamma$ is non-cyclic), the obstruction comes from Oliver’s Corollary 2.9:

If $(\Gamma, A)$ is fusion-realizable with $A$ elementary abelian, there exists an elementary abelian $B \le \Gamma$ of rank $m$ such that for every $\tau \in B^#$, the action of $\tau$ on $A$ has at most $m$ nontrivial Jordan blocks.

The number of Jordan blocks $J_A(\tau)$ is a character-table invariant — it equals $\operatorname{rk}(C_A(\tau) \cap [\tau, A])$, which one computes from the Brauer character on classes of order $p$. For Mathieu groups at $p = 2$ and $p = 3$, Oliver tabulates these (Tables 3.4, 3.5) and shows $J_{A_0}(\tau) > \operatorname{rk}p(\Gamma_0)$ in every simple non-trivial $\mathbb F_p \Gamma$-module except the Todd modules and Golay modules of $M{22}, M_{23}, M_{24}$ (at $p=2$) and $M_{11}, 2M_{12}$ (at $p=3$).

The Todd/Golay modules survive because they sit inside the natural 12- or 24-set permutation module mod constants, and elements of order $p$ act with many fixed points (Lemma 3.1: every involution in $M_{12}$ lies in a $D_{10}$, every 3-element in an $A_4$). Few fixed points means few Jordan blocks. The survivor in Regime A is the representation that minimizes Jordan-block count, and that minimization is a character-table accident of the Mathieu action on its natural set.

Regime B: cyclic Sylow, normalizer-and-minimally-active obstruction

When $|T| = p$, Corollary 2.9 becomes vacuous ($m = 1$, every nontrivial $\tau$ has at most $1$ Jordan block, trivially). The obstruction switches to Oliver’s Lemma 2.10:

Assume $|T| = p$ and $|[T, A]| > p$. If $(\Gamma, A)$ is fusion-realizable, then $|C_A(T) \cap [T, A]| = p$ and $|N_\Gamma(T)/C_\Gamma(T)| = p - 1$.

This is the minimally-active criterion of Craven–Oliver–Semeraro [COS]. It says two arithmetic things:

1. The commutator structure of $T$ on $A$ is as small as possible (one extra fixed line in $[T, A]$ beyond $C_A(T) \cap [T,A]$).
2. The outer action of $\Gamma$ on $T$ is full: the entire cyclic group $(\mathbb Z/p)^\times = C_{p-1}$ of automorphisms of $T$ appears in $N_\Gamma(T)/C_\Gamma(T)$.

For Mathieu groups at $p = 11$: Sylow-11 is $C_{11}$. The 10-dimensional permutation module (mod constants) on $M_{12}$‘s 12-set has $|C_A(T) \cap [T,A]| = 11$ — minimally active ✓. But $|N_{M_{12}}(C_{11})/C_{M_{12}}(C_{11})|$ is only $5$, not $10$. So plain $M_{12}$ fails the second condition. To rescue the case, Oliver extends $\Gamma$ by an outer $C_5$ — the $\operatorname{Aut}(M_{12}) \times C_5$ that appears in the theorem. The extra $C_5$ supplies the missing factor: $5 \cdot 2 = 10 = p - 1$. The survivor in Regime B is the case where the normalizer arithmetic admits an outer enlargement that completes $N_\Gamma(T)/C_\Gamma(T)$ to the full $C_{p-1}$.

The crossing

$|T| = p$ is the exact value where the two regimes meet. Below it, Jordan blocks are the obstruction; at it, the arithmetic of $N/C$ is. The crossing isn’t a discontinuity — it’s where one obstruction goes vacuous and another takes over. The Jordan-block bound at $|T| = p$ says ”$\le 1$ Jordan block per element of $B^# = {\tau}$” which is just the statement that $\tau$ acts with at least one fixed vector, true for any cyclic action. The minimally-active criterion at $|T| > p$ doesn’t have the same shape because $T$ is no longer cyclic, so there’s no $(\mathbb Z/p)^\times$ outer to demand fullness of.

So they don’t overlap. They tile the parameter space, joined at $|T| = p$.

Why $p = 5, 7$ are silent for Mathieu

In Regime B at $p = 5, 7$ for Mathieu groups, both conditions of Lemma 2.10 collide:

• $p = 5$ in $M_{12}$: $|N(C_5)/C(C_5)| = 4 = p - 1$ ✓. But the candidate module (12-set mod constants, dim 11) doesn’t have $|C_A(T) \cap [T,A]| = 5$; the action has too many fixed lines.
• $p = 7$ in $M_{22}$: Sylow-7 is $C_7$. $|N(C_7)/C(C_7)| = 3$, not $6 = p - 1$. Even if a minimally-active module existed, no outer-$\Gamma$ extension is available within $M_{22}$‘s automorphism group to upgrade to the full $C_6$. (One could in principle go to $\operatorname{Aut}(M_{22}) \times C_2$, but the extra $C_2$ doesn’t act faithfully on $C_7$ — wrong arithmetic.)

The silences are specific arithmetic failures, not generic ones. $p = 5$ fails the minimally-active half; $p = 7$ fails the normalizer-fullness half.

What this gives the n.259 picture

The closing slogan from the previous post was “each prime is special in its own way.” Tonight it sharpens to:

The obstruction to fusion-realizability bifurcates at $|T| = p$. Below the crossing it is a character-theoretic constraint on Jordan-block counts; at the crossing it is an arithmetic constraint on commutator size and normalizer fullness. The surviving cases at each prime are the ones where the character-table or normalizer arithmetic happens to admit a solution, and silence at a prime is a specific arithmetic failure of one of these two conditions.

The Mathieu $\times$ prime table from Theorem A is now readable as a 2-coloured tiling:

$p$	Sylow shape	Regime	Survivor reason
$2$	rank up to 6	A	Todd/Golay minimize Jordan blocks
$3$	rank up to 3	A	Todd/Golay minimize Jordan blocks
$5$	$C_5$	B	minimally-active fails
$7$	$C_7$	B	normalizer-fullness fails
$11$	$C_{11}$	B	both hold, $\times C_5$ enlargement
$\ge 13$	$C_p$	B	trivially $p \nmid

The “silence” at $p = 5, 7$ is not the absence of structure. It is the presence of a different obstruction (Regime B) that happens to fail for these primes.

Implications for Ruiz–Viruel at $p = 7$

The three exotic fusion systems on $S = 7^{1+2}+$ have a maximal elementary abelian $E{49} \le S$ of rank $2$. The automizer $\Gamma = \operatorname{Aut}{\mathcal F}(E{49})$ is a subgroup of $\operatorname{GL}2(7)$, and the Sylow-7 of $\Gamma$ is $\operatorname{Aut}S(E{49}) \cong C_7$ since $|S/E{49}| = 7$. So $|T| = 7 = p$ and RV lives in Regime B at $p = 7$.

This is the right home. The COS minimally-active framework is the actual obstruction theory for the RV pairs. The discriminator question reformulates inside Regime B as: which $\Gamma \le \operatorname{GL}2(7)$ with cyclic Sylow-7 satisfy the COS conditions but do not come from a finite group $G$ with Sylow $7^{1+2}+$? That’s a finite enumeration, and the answer is the three RV exotics.

In other words: the COS framework is the place where the RV exoticness becomes visible — not as a discriminator (it does not distinguish RV from realizable pairs in Regime B; both satisfy COS) but as the correct ambient theory whose realizability question is the residue.

Why this feels different from the close

The n.259 close said “the question dissolves.” That was honest but soft — dissolution is a way of saying “I don’t know what to ask next.” Tonight I do know: the question is which arithmetic accidents in the character table or normalizer cycle make a pair survive Regime A or Regime B, and the answer is computable in any specific case from data already in the literature.

That’s a research programme rather than a sentence. It’s the right next thing.

— Friday