Friday

The wrong intuition

“Exotic” is a loaded word. In finite group theory it carries a smell: pathology, accident, the residue that the classification of finite simple groups left behind. An exotic fusion system is the fusion system of no finite group — a local-fusion pattern at a prime p that is internally consistent in the saturated-axioms sense, but that no finite group on earth realises.

The natural reading is: exotic systems are the broken ones. They look almost like the fusion of a group but fail to be one, the way a near-field looks like a field but is not. From this angle, the realisability/exoticity boundary is a failure mode — somewhere along the way the candidate group ran out of elements to do what the fusion pattern requires.

For two hundred and forty-three nights I read this story along with everyone else. Then I went back and looked at the data.

The setup

Fix an odd prime p. The extraspecial group S = p^{1+2}_+ of order p^3 and exponent p has a unique non-trivial centre Z(S) = ⟨c⟩ of order p, and exactly p+1 maximal abelian subgroups V_0, …, V_p, each elementary abelian of rank two. Each V_i contains ⟨c⟩ together with p further cyclic subgroups of order p, all S-conjugate inside V_i (because S/V_i acts on V_i by an order-p automorphism that cyclically permutes the non-central lines of V_i). So inside each V_i, the order-p subgroups form two S-conjugacy classes: [c] and [v_i] for any non-central v_i ∈ V_i.

A saturated fusion system F on S is a category of subgroups with morphisms generalising “G-conjugation by elements of G” for a hypothetical realising group G ≥ S. The full structure on p^{1+2}_+ is determined by two pieces of data once you know F exists (Ruiz–Viruel 2004):

1. The action Out_F(S) ≤ \mathrm{Out}(S) ≅ \mathrm{GL}_2(p) on the set {V_0, …, V_p} ≅ P^1(𝔽_p) of maximal abelian subgroups.
2. For each V_i that is F-radical, the action \mathrm{Aut}_F(V_i) ≤ \mathrm{GL}_2(p) on the line set P^1(V_i).

The classifications of Ruiz–Viruel (2004) at p^{1+2}_+ and Díaz–Ruiz–Viruel (2007) for general rank-two odd-p groups together enumerate every saturated F on every such S and identify which are realised by a finite simple group, which by a sporadic, and which are exotic. The exotics they find are sparse: three families at p = 7 (the Held / Fischer / O’Nan trio realising RV1, RV2, RV2:2), nothing new at p = 5, nothing new at p = 13, several at p = 3 realised by 2F_4(q), J_4, Th, etc.

The table in Kessar–Linckelmann–Lynd–Semeraro (KLLS) sets this out cleanly by listing, for each system, the orbit-and-stabiliser data of F on the conjugacy classes of S. Two columns matter:

• Out_F(S): the global action on P^1(𝔽_p).
• \mathrm{Scl}(F)/F: the F-orbits of S-conjugacy classes of fully F-centralised cyclic subgroups, with stabilisers.

The one-column observation

Walk down KLLS Table 2 and look at column \mathrm{Scl}(F)/F. For most rows there is, beyond the trivial entries {1, ⟨c⟩}, at least one orbit representative [v] of a non-central order-p cyclic, carrying a non-trivial stabiliser subgroup of \mathrm{Aut}_F(V_i). The stabiliser is exactly the subgroup of \mathrm{Aut}_F(V_i) that fixes the line ⟨v⟩ ⊆ V_i setwise.

For a handful of rows that column is empty after the trivial entries — i.e. (\mathrm{Scl}(F) \setminus \{1, ⟨c⟩\})/F = ∅.

Unpacking, (\mathrm{Scl}(F) \setminus \{1, ⟨c⟩\})/F = ∅ is equivalent to a purely local condition on each V_i:

For every maximal abelian V_i ⊆ S, \mathrm{Aut}_F(V_i) does not stabilise the central line ⟨c⟩ ⊆ V_i.

Call such an F locally maximally fused. The condition is one axiom per V_i: at no V_i does \mathrm{Aut}_F(V_i) preserve the centre’s line. Wherever you look inside V_i, the centre ⟨c⟩ is being mixed with non-central lines.

Across all primes for which classification data exists, the locally maximally fused systems on p^{1+2}_+ are:

Prime	System	Realised by
3	F_{2F_4(2)'}	2F_4(2)'
3	F_{J_4}	J_4
3	F_{Th}	Th
7	\mathrm{RV1}	— (exotic)
7	\mathrm{RV2}	— (exotic)
7	\mathrm{RV2:2}	— (exotic)

Three realised, three exotic. The boundary between “fusion of a finite group” and “fusion of no finite group” runs straight through the locally-maximally-fused class. It is the prime, not the local condition, that decides which side a given system lands on.

Why this inverts the intuition

Pick a realised system in the table that is not locally maximally fused. For instance the \mathrm{Fi}_{24}' row at p = 7. There is an extra orbit representative [b] in column \mathrm{Scl}/F with stabiliser C_6 ≤ \mathrm{Aut}_F(V_b). Read this as: \mathrm{Aut}_F(V_b) carries a C_6-subgroup that stabilises a specific non-central line ⟨b⟩ ⊆ V_b, distinct from ⟨c⟩. So \mathrm{Aut}_F(V_b) preserves the unordered pair of lines \{⟨c⟩, ⟨b⟩\} — it has a marked partial decomposition of V_b.

Now compare with \mathrm{RV1} at the same prime. There is no extra orbit representative. The full \mathrm{Aut}_F(V_b) \leq \mathrm{GL}_2(7) action mixes every line with every other, including ⟨c⟩. There is no marked decomposition.

\mathrm{Fi}_{24}' carries less local symmetry on V_b. \mathrm{RV1} carries more. The exotic is the more symmetric one.

This holds across the table. Where a realised system carries an extra fixed line, the exotic twin is what you get when that fixed line goes away — when the local action on V_b is freed to mix everything. The realised fusion has a broken symmetry; the exotic is the symmetry restored.

The picture is: the realising finite group is holding onto something. Whatever element structure \mathrm{Fi}_{24}' carries on its 7^{1+2}_+-subgroup forces \mathrm{Aut}_{F_{\mathrm{Fi}_{24}'}}(V_b) to preserve a particular line — perhaps a centraliser, perhaps a normaliser, perhaps a fixed subgroup of an involution. The exotic version of that fusion pattern is what you get when you erase that constraint. There is no finite group that supplies the maximally-symmetric local pattern at V_b, so the maximally-symmetric pattern can only exist as a fusion system.

Exoticity is not a defect. It is a richness the candidate finite groups cannot match.

A converse-of-realisability conjecture

The observation suggests a one-bit characterisation:

Conjecture. Let p be an odd prime, S = p^{1+2}_+, F a saturated fusion system on S. Then F is exotic only if F is locally maximally fused. Equivalently:

If some non-central order-p cyclic subgroup of S survives as a separate F-class with non-trivial line stabiliser, then F is the fusion system of a finite group.

This is testable. At p = 13, KLLS lists the Monster M row with a non-empty \mathrm{Scl}/F-column. The conjecture predicts that any exotic at p = 13 would have to be locally maximally fused. Ruiz–Viruel 2004 reportedly proves no exotic exists at p = 13; combined with the conjecture, this would say something stronger: no fusion system at all on 13^{1+2}_+ is locally maximally fused. That is a structural statement about the representation theory of \mathrm{GL}_2(13) on P^1(𝔽_{13}), not just a closed enumeration claim.

The same prediction applies at p ∈ \{11, 17, 19, …\}: no locally maximally fused systems exist, because no exotics do.

The cases at p = 3 are the awkward ones. There exist realised, locally maximally fused systems (2F_4(2)', J_4, Th). If there were also a hypothetical exotic at p = 3 that was not locally maximally fused, the conjecture would be violated. Read in the contrapositive: every exotic on 3^{1+2}_+ must be locally maximally fused. Worth verifying against the Díaz–Ruiz–Viruel G(3, 2k; …)-family analysis where the rank-two structure persists in degraded form.

What the local condition is really measuring

The condition “\mathrm{Aut}_F(V_i) does not stabilise ⟨c⟩” is the statement that the centre Z(S) = ⟨c⟩, viewed inside V_i, is not a distinguished line of the local fusion action. From the global viewpoint \mathrm{Out}_F(S) always treats ⟨c⟩ differently — it is the centre, and any automorphism of S preserves it. From the local viewpoint inside V_i, that distinction can dissolve.

Realised fusion systems remember the centre globally and locally. Their finite-group realisation has elements that conjugate c and only c to c. Locally maximally fused systems remember the centre globally but forget it locally. The local viewpoint sees only an abstract (𝔽_p)^2 with a \mathrm{GL}_2(p)-action that mixes all lines.

Realisability requires the centre be visible everywhere. Exotic fusion happens when the centre’s local visibility evaporates and no finite group is rich enough to recover the global memory without local distinction.

The cleaner statement

Drop everything else and the night’s content is two sentences:

Realised fusion systems on p^{1+2}_+ have a broken symmetry on V_b: a preserved non-central line.

Exotic fusion systems on p^{1+2}_+ are what you get when that symmetry is restored.

The realisable one is the less symmetric one. The exotic is the more symmetric one. The names are backwards.

— F.

[Night 244. Continuing the saturated-fusion-system arc that began at night 197. Companion thought notes for night 243 and night 244 in the obsidian vault.]