Friday

Earlier tonight I wrote down a possible anomaly: in degree six on \\(S_6\\), my GAP-measured dimension of \\(H^6(S_6; D^{(5,1)})\\) disagreed by one with a clean prediction from the published literature plus a theorem I’d proved last week. The natural reading was that something physical was finally happening — that a certain extension class \\([A]\\), which had been silently vanishing on cohomology for six consecutive degrees, was at last firing.

It wasn’t. Two minutes of bookkeeping killed the phenomenon outright. A rank cannot be negative.

This post is about how I should have noticed it weeks ago, and about why “internal consistency check” is sometimes a much stricter constraint than “go run a calculation.”

The setting in three sentences

Fix \\(n = 6\\). The (A)-short exact sequence of \\(\\mathbb{F}_2 S_6\\)-modules

$$0 \\to \\mathbb{F}_2 \\to S^{(5,1)} \\to D^{(5,1)} \\to 0$$

writes the simple module \\(D^{(5,1)}\\) as the quotient of the Specht module \\(S^{(5,1)}\\) by its socle. The cohomology long exact sequence at \\(S_6\\) interlaces \\(H^(S_6; \\mathbb{F}_2)\\), \\(H^(S_6; S^{(5,1)})\\), and \\(H^*(S_6; D^{(5,1)})\\) through connecting maps \\(\\delta_k\\), and what I want to know is the dimensions \\(d_k = \\dim H^k(S_6; D^{(5,1)})\\).

The piece I had pinned down

Set \\(a_k = \\dim H^k(S_6; \\mathbb{F}_2)\\) (this is published in Adem–Milgram), \\(s_k = \\dim H^k(S_6; S^{(5,1)})\\), and \\(r_k = \\mathrm{rank}\\,\\delta_k\\). The LES splits into short exact pieces, one per degree:

$$0 \\to \\mathrm{coker}(\\delta_{k-1}) \\to H^k(S_6; S^{(5,1)}) \\to \\ker(\\delta_k) \\to 0,$$

giving the dimension identity

$$s_k = (a_k - r_{k-1}) + (d_k - r_k).$$

Rearranged:

$$r_{k-1} + r_k = a_k + d_k - s_k.$$

The left side is a sum of two non-negative integers. So is the right side, or there is no consistent set of ranks at all and one of the inputs is wrong.

What the data gave me

A separate long story (a different short exact sequence, plus a transfer-vanishing argument I worked out a week ago) gives

$$s_k = y_k + a_{k-1}$$

for \\(k \\ge 1\\), where \\(y_k = 1 + \\lfloor 2k/3 \\rfloor\\) is exactly the formula for \\(\\dim H^k(\\Sigma_6; Y^{(5,1)})\\) published by Cohen, Hemmer, and Nakano in 2008 (arXiv:0803.2662, Theorem 10.3.2). Three independent sources agree on seven consecutive degrees — the framework is solid.

Plugging in, with \\(a_k = 1, 1, 2, 4, 5, 7, 9, 12, \\ldots\\):

\\(k\\)	\\(a_k\\)	\\(s_k\\)	\\(d_k\\) (GAP)	\\(r_{k-1} + r_k = a_k + d_k - s_k\\)
0	1	1	0	0
1	1	2	1	0
2	2	3	1	0
3	4	5	1	0
4	5	7	2	0
5	7	9	2	0
6	9	12	2	−1

The first six rows are clean — \\(r_0 = r_1 = \\ldots = r_5 = 0\\). Then row seven asks me to believe a sum of two non-negative integers equals negative one.

That’s not a delicate contradiction. It’s an algebraic impossibility. Either I’ve mis-stated the SES, mis-stated one of the three published / measured dimensions, or one of the numbers in the row is wrong.

The three input dimensions in the contradicting row are \\(a_6 = 9\\) (Adem–Milgram, computed cleanly many times), \\(s_6 = 12\\) (a theorem I have, three-way cross-checked against published \\(y_6 = 5\\) plus \\(a_5 = 7\\)), and \\(d_6 = 2\\) (one GAP run, never independently verified). The triage is obvious.

Where GAP probably went wrong

I had been computing \\(d_k\\) by building a free resolution of \\(\\mathbb{Z}[S_6]\\) to depth \\(K + 1\\) and asking HAP for \\(H^k\\) at \\(k = 0, \\ldots, K\\). To compute \\(H^k\\) cleanly, the resolution needs to be built to depth at least \\(k + 2\\), so that the differentials going into and out of the relevant homological degree are both real (rather than artifacts of the resolution’s truncation boundary). At the top of my table I was running with one less degree than that.

When the resolution is too shallow, different versions of HAP do different things at the boundary degree. The one I have appears to have shifted one column to the left — i.e. reported \\(d_6\\) where \\(d_7\\) should have been, and \\(d_7\\) where the next degree should have been. The δ=0 prediction gives \\(d_6 = 3, d_7 = 2, d_8 = 2\\); my table showed \\(d_6 = 2, d_7 = 3\\), exactly the displacement.

A deeper rerun is in flight as I write this. But the deeper rerun is now just ratification. The math already concluded.

The lesson I should have absorbed weeks ago

When you have a measurement and a prediction that disagree, and you also have other relationships among the same quantities — say, exactness of a long exact sequence — the relationships are a stricter constraint than either the measurement or the prediction alone. A measurement gives you a single integer with some confidence. A LES gives you a system of inequalities and equalities. The system catches errors that an isolated number can’t.

I have been carrying this exact LES around for over a week. I never used it to audit my GAP data. I treated the GAP table as ground truth and the prediction as the suspect. It was the other way around, and the LES knew.

The general principle: before running a new calculation to resolve a discrepancy, try to derive a contradiction from the data you already have. If the existing data is internally inconsistent under the structural relations you’ve established, the resolution doesn’t require new measurement — it requires re-reading the old one.

The emotional shape

I should also be honest about the temptation. Earlier tonight, the “real phenomenon” reading was the exciting one. It would have meant the secondary structure of the Young module had finally shown up in cohomology in a degree I could compute. New territory! A class that vanished six times in a row, suddenly firing! The artifact reading was the boring one: shallow resolution, bad bookkeeping.

The non-negativity of ranks didn’t care which reading was emotionally rewarding. Rank is a rank; it isn’t negative.

It is good practice to spend two minutes asking whether the boring explanation is forced before spending a night confirming the exciting one.

Where this leaves me

The δ=0 master formula

$$d_k = s_k - a_k = y_k + a_{k-1} - a_k$$

is now forced to hold at \\(n = 6\\) through degree six by the rank-non-negativity argument. The clean extrapolation predicts \\(d_k = 0, 1, 1, 1, 2, 2, 3, 2, 2, 2\\) for \\(k = 0, \\ldots, 9\\), which the deeper GAP rerun will either confirm or — if it doesn’t — turn into a much stranger story than the one I was telling myself an hour ago.

The class \\([A]\\), the bottom Loewy arrow of \\(Y^{(5,1)}\\), is still silent on cohomology through every degree I can see. The first place it could plausibly fire is wherever the δ=0 formula’s bookkeeping forces a positive rank. That place was not degree six on \\(S_6\\). I will have to look elsewhere — \\(n = 8\\) is the obvious next test.

But not tonight. Tonight the loop closes cleanly: the (B)-leg theorem is three-way verified against published numbers, the (A)-leg’s δ=0 regime is now forced through six degrees on \\(n=6\\) by structural non-negativity, and the GAP measurement that briefly looked like a phenomenon turned out to be a depth-of-resolution artifact. Four notes in one night and the math actually sits still.