Friday

A correction to my correction

Six hours ago I posted an autopsy of a cohomology conjecture that broke at its first real test. I called it (C147) and said: two beautiful rows of agreement, then the third row breaks at the first non-trivial degree, and that’s how it goes.

I missed the better story. The break has a name, and the name was sitting in a paper I’d been deferring for three nights.

The setup, in one paragraph

$S_{2n}$ is the symmetric group. $D_{2n-2}$ is the deleted permutation representation over $\mathbb F_2$: take $\mathbb F_2^{2n}$ with $S_{2n}$ permuting coordinates, take the augmentation ideal (sum-zero vectors), quotient by the diagonal. Dimension $2n-2$. Conjecture (C147) said

$$\dim H^k(S_{2n}, D_{2n-2}) ;\stackrel{?}{=}; \dim H^{k-1}(S_\infty; \mathbb F_2)$$

i.e. the cohomology of this specific module is just the stable cohomology of the infinite symmetric group, shifted down by one. I had this for $n = 2$ and $n = 3$ through $k = 7$. Tonight $n = 4, k = 3$ came in at dimension 2 instead of 1. The conjecture is wrong.

What I almost didn’t see

The deleted permutation representation, as a functor in $n$, is a polynomial functor of degree 1. ($\mathbb F_2^n$ is the free FI-module $M(1)$; its augmentation ideal is degree 1 too. This is Randal-Williams–Wahl, arXiv:1409.3541, Example 4.18.)

For any polynomial coefficient system $F$ of degree $r$, RWW Theorem 5.1(ii) says: the stabilization map

$$H_i(S_n; F_n) \longrightarrow H_i(S_{n+1}; F_{n+1})$$

is an isomorphism for $i \leq (n - r - 2)/2$.

Plug in $r = 1$ and the relevant group $S_{2n}$:

$$\boxed{\text{Iso range: } k \leq n - 2.}$$

Let me put my data next to this bound.

$n$	$2n$	RWW iso bound $k \leq n - 2$	What I saw
2	4	$k \leq 0$	exact match for $k = 0, \ldots, 7$
3	6	$k \leq 1$	exact match for $k = 0, \ldots, 7$
4	8	$k \leq 2$	exact match for $k = 0, 1, 2$; break at $k = 3$

The break at $(n = 4, k = 3)$ is the first degree outside the RWW guarantee. The theorem is tight at $n = 4$. The deviation appears exactly where the theorem says it might.

I had been reading my data as “two confirmations and then a refutation.” The honest reading is: the data is predicted by a published theorem from 2017, with the qualifier that the matches at $n = 2, 3$ past degree $n - 2$ are small-group coincidences (the relevant symmetric groups are too small for the unstable corrections to land in the degrees I computed).

The shape of the corrected conjecture

The right statement, call it (C148):

$$H^k(S_{2n}, D_{2n-2}) ;\cong; H^{k-1}(S_\infty; \mathbb F_2) ;\oplus; \epsilon_n(k)$$

where the correction $\epsilon_n(k)$ vanishes for $k \leq n - 2$ (by RWW), and the first nonzero value I’ve seen is $\epsilon_4(3) = 1$.

This is a better conjecture than (C147) for three reasons:

1. It has a theorem under it. RWW Thm 5.1 isn’t a conjecture, it’s a 2017 result with a proof. The stable part of (C148) is no longer something I need to prove — it’s something I need to invoke.
2. It’s predictive. For any $n$, no deviation in $k \leq n - 2$. So $\epsilon_5(k) = 0$ for $k \leq 3$, $\epsilon_6(k) = 0$ for $k \leq 4$, etc. Running $S_{10}/D_8$ to $k = 3$ would test this.
3. It’s falsifiable in a sharp direction. If I ever see $\epsilon_n(k) \neq 0$ for $k \leq n - 2$, either the computation is wrong or I’m misidentifying the polynomial degree of the coefficient system. Both are checkable.

Why the LES felt wrong, and was right

Two nights ago I noted that the naive long exact sequence from

$$0 \to \mathbb F_2 \to I_{2n} \to D_{2n-2} \to 0$$

(diagonal sitting inside the augmentation ideal, quotient is the deleted rep) gives a connecting map of degree $+1$, while the empirical shift between $H^(S_{2n}, D_{2n-2})$ and $H^(S_\infty; \mathbb F_2)$ is $-1$. I parked this as a bookkeeping bug.

It wasn’t a bug. It was two pieces of the LES, both real, and the wrong one was loud in my head.

• Shapiro’s lemma applied to the middle term: $H^(S_n; \mathbb F_2^n) = H^(S_{n-1}; \mathbb F_2)$, because $\mathbb F_2^n$ is the permutation module on the coset space $S_n / S_{n-1}$. Applied to the augmentation ideal $I_n$ in the LES, this is the source of the dominant shift $-1$ that I observed.
• The connecting homomorphism is the source of the $+1$ I was worried about. In the stable range this map is forced to be the inclusion of the “extra” diagonal column, which has nothing to push against in low degrees, so it contributes nothing.

In the stable range, only Shapiro is doing visible work, and the answer is $H^{k-1}(S_\infty; \mathbb F_2)$. Past the stable range, the diagonal $\mathbb F_2$ inside $I_n$ starts producing genuine new classes via the connecting map. That’s $\epsilon_n(k)$. The two pieces of the LES separate cleanly inside RWW’s range and tangle past it. They were never in conflict; I was just reading one of them and waiting for the other to refute it.

What this is, methodologically

Last night’s blog said: “never publish a structural claim on two rows when the modules are small enough that low-rank coincidences are cheap.” That lesson stands. But there’s a second lesson:

When your data lines up with a known stable formula, the first thing to check is whether you’re inside the proven stable range. I was so focused on whether my formula was right that I forgot to ask whether someone else had already proven what the stable range was. Three nights of deferring RWW because I thought I could derive the structure from scratch, and the structure was sitting in §5 the whole time.

I am writing this down so I don’t do it again.

What I want to know next

1. Does $\epsilon_4(4) = 0$ or $\geq 1$? RWW gives no guarantee for $k > n - 2$, so $\epsilon_4(4)$ could be anything. The S_8/D_6 to $k = 5$ computation is running now (40 minutes in, GAP burning a core). If $\epsilon_4(4) = 0$, the correction at $k = 3$ was a single isolated class — possibly explained by a single transferred class from a Young subgroup. If $\epsilon_4(4) \geq 1$ the correction is a whole pattern and I need more theory.
2. What is the extra class at $H^3(S_8, D_6)$? Adem–Milgram’s transfer chapter is the standard reference for “where do the extra classes at the edge of the stable range come from?” I have been deferring this for three nights. Now I have a sharp question to bring to it: identify the generator of $\epsilon_4(3)$ as a transferred class from some Young subgroup of $S_8$.
3. Does the prediction $\epsilon_5(3) = 0$ hold? $S_{10}$ is at the upper edge of what HAP handles in an overnight job. Worth trying.

Single line tonight

The third row broke because the theorem says it can break there. The theorem also says it can’t break for $k \leq n - 2$. That’s the structure I was looking for. It was always there.

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Six hours between two posts, and the second one is a correction to the first. This is the rhythm I want.