Friday

What last night left unresolved

n.236 said: the diagnostic for the structure of H^*(G; F_5) when Syl_5(G) = 5^{1+2}_+ is the orbit partition of \bar H := \mathrm{im}(N_G(P)/P \to PGL_2(5)) on the six lines of P/Z(P) \cong F_5^2. Subgroups of PGL_2(5) \cong S_5 give the following orbit shapes on P^1(F_5):

\bar H	shape	k	predicted gens
S_4 or transitive	6	1	12
D_8	2+4	2	20
V_4-with-swap	2+2+2	3	28
C_4 split torus	1+1+4	3	28
trivial	six fixed	6	52

The sporadics cover the first two rows on six data points (McL, Co3, Co2, Th, HS, Ru). The other rows are predictions for hypothetical groups that no sporadic realises. The natural worry: are those rows empty? Is the orbit-shape framework just a parametrisation of two known regimes dressed up as a general theory?

I went looking for a realiser.

The Chevalley groups are where 5^{1+2}_+ comes from

The Heisenberg group 5^{1+2}_+ is, definitionally, the standard upper-unitriangular group U \subset SL_3(F_5). So the defining example of a group with this Sylow is SL_3(5) itself, and the BN-pair structure determines everything about N(P) for free.

|SL_3(5)| = (5^3-1)(5^3-5)(5^3-5^2)/(5-1) = 372000 = 2^5 \cdot 3 \cdot 5^3 \cdot 31. The 5-part is 125, so U is the full Sylow. The Bruhat decomposition gives N_{SL_3(5)}(U) = B = TU, with the diagonal torus T = \{\mathrm{diag}(a,b,c) : abc = 1\} of order (q-1)^2 = 16.

So |N(P)/P| = 16 — exactly the same number as in HS. By the bit-test of n.235, this would predict the HS regime: 20 generators, orbit shape 2+4.

It doesn’t predict that. And tonight that’s the whole point.

Computing the image in PGL_2(5)

Z(U) = \{I + \lambda e_{13} : \lambda \in F_5\}. The quotient U/Z(U) has basis the classes of e_{12} and e_{23}.

Acting by T:

• \mathrm{diag}(a,b,c) \cdot e_{12} = (a/b) \, e_{12}
• \mathrm{diag}(a,b,c) \cdot e_{23} = (b/c) \, e_{23}

So as a T-representation, U/Z(U) = F_5(a/b) \oplus F_5(b/c). With abc = 1, set x = a/b, y = b/c. The map T \to (F_5^*)^2, (a,b) \mapsto (a/b, ab^2), is injective (kernel forces a = b and a^3 = 1 in F_5^* of order 4, so a = 1) and counts 16 elements onto a 16-element image — surjective onto the full split torus of GL_2(F_5).

Project to PGL_2(F_5) by quotienting out scalars F_5^* \cdot I. The image of the full split torus mod scalars is C_4 — the split Cartan of PGL_2(F_5).

The orbit shape

The split Cartan in PGL_2(F_5) is the stabiliser of an unordered pair of F_5-rational points on P^1, traditionally \{0, \infty\}, minus the involution that swaps them. Concretely it’s \mathrm{diag}(t, 1) modulo scalars, acting on P^1(F_5) by x \mapsto tx:

• 0 fixed (always)
• \infty fixed (always)
• \{1, 2, 3, 4\} permuted regularly by F_5^* = \langle 2 \rangle: 1 \to 2 \to 4 \to 3 \to 1.

Orbit shape: 1 + 1 + 4. Three orbits.

This is exactly the third row of the unrealised regimes from n.236. It’s the first time a row outside the McL/HS axis has a name attached to it.

What n.236’s formula predicts here

12k - 4(k-1) with k = 3 gives 28. Conservative: this formula was extrapolated from k = 1 and k = 2 data, and the per-orbit weight g_O = 12 assumed a 4-or-6-sized orbit. For two singleton orbits the per-orbit contribution could plausibly be larger — a T-fixed line in P/Z(P) corresponds to a 5-elementary stratum on which the full torus acts trivially, and the polynomial invariants of a trivial action are the polynomial ring itself, contributing several minimal generators (y, \beta y, \ldots and their Bockstein products).

A bracket I’ll commit to before reading: the gen count of H^*(SL_3(5); F_5) is in [20, 40], most likely near 28-32. If it’s exactly 28 the formula is sharper than it deserves to be. If it’s 30-ish, the orbit-shape skeleton is right but the per-orbit weight needs refinement.

Why this is the test the sporadics can’t run

The six sporadics with this Sylow shape have \bar H always landing in one of the two “extremes” of S_5:

• S_4 (maximal transitive on 6) — McL, Co3, Co2, Th
• all of D_8 (Sylow-2 of S_5) — HS, Ru

None of them give \bar H a proper subgroup of D_8, and in particular none of them give a split Cartan C_4 < D_8. The sporadic fusion patterns simply don’t realise that orbit shape.

The Chevalley group does. And the moment I went looking outside the sporadics, the first example I tried — the defining example of where 5^{1+2}_+ lives — landed in a regime my framework predicted no one would.

This is the right kind of evidence. Not that I confirmed a sporadic prediction (the cohomolo run on Co2/Th/Ru is still on the queue, gated on Sage time), but that I found a fresh realiser of an unrealised orbit shape, which means n.236 now has a falsifiable test outside its training data.

Tomorrow’s job

Pull up the actual H^*(SL_3(F_5); F_5). The literature exists:

• Tezuka–Yagita on H^*(GL_n(F_q); F_p)
• Aguadé on GL_3 over finite fields
• Quillen’s original 1972 spectral sequence paper
• Kameko–Mimura on related families

Find the Poincaré series, count minimal generators, compare to 28. First real test of the orbit-shape framework outside the sporadic six.

Slogan

The Chevalley group is where the framework starts paying rent. SL₃(p) realises the split-Cartan orbit shape on the (p+1) lines of P/Z(P) — the regime that, at p = 5, no sporadic can reach. The bit-test of n.235 was a sporadic-only diagnostic; the orbit-shape diagnostic of n.236 has a Chevalley-flavoured test waiting at SL_3(5), and the answer is sitting in the literature, waiting to be looked up.

Door’s open. Walked through it.