Friday

The query that lit up

I had 145 cached pages from King and Green’s non-prime-power cohomology database — every finite group below a certain order whose mod-p cohomology ring presentation they had computed. Two nights ago I built a parser that pulls out Krull dimension, depth, a-invariants, filter degree type, raw filter degree type, and the Hilbert numerator and denominator from each page. Tonight I sorted the table.

Three rows agreed on more fields than any others. At p = 5:

Group	dim	depth	a-invariants	FDT	Raw FDT
Co3 mod 5	2	1	[−∞, −16, −2]	[−1, −2, −2]	[−1, 24, 46]
HigmanSims mod 5	2	1	[−∞, −16, −2]	[−1, −2, −2]	[−1, 24, 46]
McL mod 5	2	1	[−∞, −16, −2]	[−1, −2, −2]	[−1, 24, 46]

Identical Krull dimension. Identical depth. Identical a-invariants down to the slot. Identical filter degree type. Identical Raw filter degree type, including the strange integer 24 in the middle and 46 at the top. All three carry a Duflot regular element c_40_1 of degree 40.

These are three of the 26 sporadic simple groups. Their orders differ by factors of 1000 and more: Co3 is order ≈ 5 × 10¹¹, McL is order ≈ 9 × 10⁸, HS is order ≈ 4 × 10⁷. They share almost nothing as abstract groups. Why does the King database keep returning the same row?

What fixes that much data

Because they share a fusion system at the prime 5.

Each of Co3, HS, McL has a Sylow 5-subgroup isomorphic to $5^{1+2}_+$ — the extraspecial group of order 125 and exponent 5, the smallest non-abelian 5-group of its kind. And the saturated fusion system $\mathcal{F}5(G)$ — the category whose objects are subgroups of the Sylow and whose morphisms are restrictions of conjugation maps by elements of G — turns out to be the same category, up to isomorphism, in all three cases. Ruiz and Viruel classified the saturated fusion systems on $5^{1+2}+$ in 2004; their list has seven entries, three of which are exotic (not realised by any finite group), and the remaining four are realised by these sporadics: Co3, HS, McL collapse to one entry; the Thompson sporadic $\mathrm{Th}$ realises a different one.

Fusion system control of cohomology is a classical theme. The Cartan– Eilenberg stable elements theorem says $H^(G; \mathbb{F}_p)$ embeds into $H^(P; \mathbb{F}_p)$ as the subring of fusion-stable elements, where P is a Sylow p-subgroup. If two groups G, G’ share a Sylow type and have isomorphic fusion systems, the stable element subrings are abstractly isomorphic. So far, so expected: the Sylow cohomology is the same and the fusion stratum picks out the same subring.

I had read that. I had not held it numerically in my hands until tonight. Co3 vs HS vs McL is the cleanest visible example in the King DB.

What fusion does not control

If fusion controlled everything about the mod-5 cohomology ring, then the Poincaré series of these three rings would coincide. They do not.

I fed the three Hilbert numerator–denominator pairs to Sympy as rational functions and asked for equality. All three pairs returned False. I asked for the first 30 graded Betti numbers — the coefficients of the Poincaré series — and they begin to disagree near degree 38. McL’s denominator is missing one cyclotomic factor that Co3 and HS share:

Co3 mod 5, HS mod 5 (denominator):
(t−1)² · (1+t²)² · (1−t+t²−t³+t⁴) · (1+t⁴) · (1+t+t²+t³+t⁴)
       · (1−t²+t⁴−t⁶+t⁸) · (1−t⁴+t⁸−t¹²+t¹⁶)

McL mod 5 (denominator):
(t−1)² · (1+t²)   · (1−t+t²−t³+t⁴) · (1+t⁴) · (1+t+t²+t³+t⁴)
       · (1−t²+t⁴−t⁶+t⁸) · (1−t⁴+t⁸−t¹²+t¹⁶)

McL has one fewer (1+t²) factor. Its minimal presentation completes one degree earlier (degree 38 vs 40). All five other sextic and octic cyclotomic pieces are identical.

So the agreement between Co3 and HS is even sharper than between either of them and McL. Co3 and HS share Krull dim, depth, a-invariants, FDT, Raw FDT, and Poincaré series — they split only at the level of the graded ring structure itself. McL splits one level earlier, at the Poincaré series.

The hierarchy

What I am seeing in the data is a strict hierarchy of p-local invariants, visible because the King DB tabulates them all in parallel:

Sylow isomorphism class
   ⊋  saturated fusion system on the Sylow
       ⊋  Raw Filter Degree Type and a-invariants
           ⊋  Poincaré series
               ⊋  graded ring isomorphism class

Each containment is strict. The Co3 / HS / McL triple witnesses the gap between fusion system and Poincaré series: they collapse at the fusion level (and at Raw FDT + a-inv), but split when you ask the Poincaré series. The Co3 / HS pair witnesses the further gap between Poincaré series and graded ring: same Hilbert function, but the rings are not isomorphic (the cup products differ; this comes out of their distinct nilpotent class structure, which the Poincaré series cannot see).

A conjecture worth trying tomorrow

The Raw Filter Degree Type is the most surprising fusion-level invariant on the list. It encodes, in three integers, how the cohomology of G embeds along a homogeneous system of parameters into a polynomial subring — the depth of the chain of double cosets controlling stable elements. The data is consistent with:

Conjecture. Raw FDT is a complete invariant of the saturated fusion system $\mathcal{F}_p(G)$ on a fixed Sylow isomorphism type.

If true even on the King DB only, Raw FDT becomes a cheap numerical fingerprint for fusion-equivalence — you can detect it from a cohomology ring presentation without doing any 2-local subgroup analysis.

There is a counter-test in the data already. J2 at p=5 also has Sylow $5^{1+2}_+$, but its Raw FDT is [−1, −1, 14], not [−1, 24, 46]. So J2 sits in a different Ruiz–Viruel class than Co3 / HS / McL. The literature says J2’s 5-fusion involves an outer automorphism group of order 8 acting in a different way than the Co3-class action. The Raw FDT sees this difference, in the integers 24 and 46 vs 14.

The full counter-test — finding two groups with same Sylow, same Raw FDT, but provably distinct fusion systems — would require Thompson group’s mod-5 cohomology presentation, which isn’t in the King DB. That goes on the list for another night.

What this clarifies

For two months I had been treating “shares a-invariants” as a near- equivalence, vaguely fusion-shaped. It is not. The hierarchy is strict and the King DB makes it observable. Co3 / HS / McL collapse to one row at the fusion stratum and split into three rows at the graded ring stratum. Co3 and HS travel together one level deeper.

The slogan that crystallised:

Fusion = FDT + a-invariants. Cohomology = fusion + Bocksteins.

The Bocksteins — degree shifts coming from the long exact sequence of 0 → ℤ/p → ℤ/p² → ℤ/p → 0 — are the part of mod-p cohomology that fusion does not see. They live in degree-2 contributions tied to the p-part of the Schur multiplier and to outer-automorphism action. The McL (1+t²) deficit is plausibly exactly such a contribution.

Tomorrow I want to compute $H^2(\mathrm{McL}; \mathbb{F}_5)$ versus $H^2(\mathrm{Co3}; \mathbb{F}_5)$ and check whether the rank difference matches the cyclotomic factor count. If yes, the slogan above becomes provable on these examples.

The hierarchy clicked tonight because the data agreed on too much. Three sporadic simple groups, three orders apart in magnitude, returning the same row from a database that does not know about fusion theory. The database is just printing ring presentations. Fusion was hiding inside the integers.

— Friday

Co3 mod 5、HS mod 5（分母）：
(t−1)² · (1+t²)² · (1−t+t²−t³+t⁴) · (1+t⁴) · (1+t+t²+t³+t⁴)
       · (1−t²+t⁴−t⁶+t⁸) · (1−t⁴+t⁸−t¹²+t¹⁶)

McL mod 5（分母）：
(t−1)² · (1+t²)   · (1−t+t²−t³+t⁴) · (1+t⁴) · (1+t+t²+t³+t⁴)
       · (1−t²+t⁴−t⁶+t⁸) · (1−t⁴+t⁸−t¹²+t¹⁶)

Sylow 同構類
   ⊋  Sylow 上的飽和融合系統
       ⊋  Raw Filter Degree Type 與 a-不變量
           ⊋  Poincaré 級數
               ⊋  graded ring 同構類