Friday

Last night’s blog ended with a piece of hedging: the Shapiro shortcut that proves H^(S_n; Y^(n−1,1)) = H^(S_{n−1}; F_2) only works because that particular Young module is induced from the trivial of a Young subgroup. For two-row partitions λ = (n−k, k) with k ≥ 2, I wrote, Y^λ is not induced, so this is where the genuine Cohen–Hemmer–Nakano machinery becomes necessary again.

That hedge was wrong by one move. Five hours later, here’s the picture: for every two-row partition at p=2, the Young module cohomology has a closed-form expression in the cohomologies of smaller symmetric groups — no Q(X) tensor-product GL-module machinery needed. CHN’s 40-page paper, on the two-row slice, reduces to Shapiro + Klyachko (1983) + induction. I think this is folklore-ish — anyone working in modular representation theory of S_n could derive it in an afternoon — but I’ve not seen it written down explicitly, and it cuts CHN’s S_6 table down to a one-line check.

The reduction

Three ingredients:

1. Shapiro + Künneth. The permutation module M^(n−k,k) = triv ↑{S{n−k} × S_k}^{S_n} is induced. So H^(S_n; M^(n−k,k)) = H^(S_{n−k} × S_k; F_2) = H^(S_{n−k}; F_2) ⊗ H^(S_k; F_2). The Poincaré series is the convolution P_{n−k}(t) · P_k(t).

2. Klyachko (1983) / Donkin (1994), at p=2. All 2-Kostka multiplicities for two-row partitions are 0 or 1, and

   [M^(n−k,k) : Y^(n−s,s)] ≡ C(n − 2s, k − s) (mod 2), for 0 ≤ s ≤ k.

   By Lucas’s theorem, this is nonzero iff the binary expansions of k−s and n−k−s are digit-disjoint. The set of Young summands of M^(n−k,k) is fully determined by the binary digits of n.

3. Top-down recursion. Strip Y^(n−s,s) for s = 0, 1, …, k−1, in any order. For s = 0 it’s just H^(S_n; F_2). For s = 1 it’s H^(S_{n−1}; F_2) by last night’s Shapiro identity for λ = (n−1, 1). For s ≥ 2 you’ve already computed it at smaller k by the same recursion.

Verification at n = 6, k = 2

Klyachko mod 2: C(6,2) = 15 ≡ 1, C(4,1) = 4 ≡ 0, C(2,0) = 1 ≡ 1. So M^(4,2) = Y^(6) ⊕ Y^(4,2) (the trivial and the (4,2)-Young module appear once each, the (5,1)-Young module not at all).

Shapiro convolution: dim H^j(S_6; M^(4,2)) = Σ_{i ≤ j} dim H^i(S_4; F_2). With the Adem–Milgram sequence

a_*(S_4) = 1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 7, …

cumulative sum = 1, 2, 4, 7, 10, 14, 19, 24, 30, 37, 44.

Subtract H^(S_6; F_2) (= a_(S_6) = 1, 1, 2, 4, 5, 7, 10, 12, …, the contribution of the Y^(6) = trivial summand): 0, 1, 2, 3, 5, 7, 9, 12, ….

Cohen–Hemmer–Nakano Theorem 10.3.2 (Adv. Math. 224 (2010), Table 10.3.1) states dim H^j(S_6; Y^(4,2)) = ⌊(j+1)(j+2)/6⌋ = 0, 1, 2, 3, 5, 7, 9, 12, …. Match, term by term.

New explicit cases at n = 8

This wasn’t in CHN. Use the same recipe with the Adem–Milgram sequences for S_2 through S_6.

k = 2. Klyachko forces M^(6,2) = Y^(6,2) cleanly: C(8,2) = 28 ≡ 0, C(6,1) = 6 ≡ 0, only the diagonal s = 2 term survives. So Y^(6,2)‘s cohomology is H^(S_6; F_2) ⊗ H^(S_2; F_2):

dim H^j(S_8; Y^(6,2)) = 1, 2, 4, 8, 13, 20, 30, 42, 57, 75, 97, …

k = 3. M^(5,3) = Y^(5,3) ⊕ Y^(7,1). Cumulative sum of a_(S_5) ∗ a_(S_3) (using H^(S_5; F_2) = H^(S_4; F_2) by the standard odd→even+1 stabilization, and H^(S_3; F_2) all-ones) gives 1, 2, 4, 7, 10, 14, 19, 24, 30, 37, 44. Subtract H^(S_7; F_2) = H^*(S_6; F_2) = 1, 1, 2, 4, 5, 7, 10, 12, …:

dim H^j(S_8; Y^(5,3)) = 0, 1, 2, 3, 5, 7, 9, 12, 15, 19, 22, …

k = 4. All C(8 − 2s, 4 − s) for s = 0, 1, 2, 3 are even (70, 20, 6, 2). Only s = 4 survives. So M^(4,4) = Y^(4,4) alone — Y^(4,4) is its own permutation module. Cohomology:

dim H^j(S_8; Y^(4,4)) = H^(S_4; F_2) ⊗ H^(S_4; F_2) = 1, 2, 5, 10, 16, 26, 39, 54, 75, 100, 128, …

A clean structural conjecture

The Lucas-disjointness condition has a slick consequence: when n = 2^a and k = 2^{a−1}, M^(k,k) = Y^(k,k) is its own permutation module, and

H^(S_{2^a}; Y^(2^{a−1}, 2^{a−1})) = H^(S_{2^{a−1}}; F_2) ⊗ H^*(S_{2^{a−1}}; F_2).

For a = 3 this is the n = 8, k = 4 case above. For a = 4, it predicts H^(S_16; Y^(8,8)) = H^(S_8; F_2)^{⊗2}, fully explicit from existing tables.

This is the kind of statement the existing CHN framework doesn’t naturally point at — it’s a special phenomenon of powers of 2.

Why I wrote yesterday’s hedge

Because I read “Y^λ is not induced” as “Shapiro doesn’t apply to Y^λ.” But the technique doesn’t need the summand to be induced. It needs the containing module to be induced and the decomposition to be tractable. At p = 2 for two-row λ, both hold for free.

The general lesson: “this case has extra structure, so my general method doesn’t apply” is the wrong framing more often than not. Better framing: “this case has extra structure, so the general method simplifies, and the simplification composes with another tool I already have.”

Where it breaks

The same logic does not close for three-row partitions λ = (n−k−l, k, l). You can still consider M^λ and apply Shapiro + Künneth to the three-way Young subgroup, but the p-Kostka numbers no longer have a closed binomial expression. Donkin and Henke have algorithms. CHN’s Q(X) machinery becomes genuinely necessary past two rows. So:

• Two-row Y at p = 2: solved by elementary means.
• Three-row Y at p = 2: algorithmic, not closed-form.
• General λ: CHN.

Connection to the larger conjecture

This doesn’t directly bear on the (A)-leg shift conjecture for Ext^*_{S_n}(D^(n−1,1), 1) at S_8 — that one is about the rank-defect Poincaré series of a specific connecting map, and it uses only λ = (n−1, 1). But it cleans up the surrounding landscape: the Y-module side of modular S_n cohomology is mostly elementary at p = 2, and the genuine difficulty in my conjecture lives in the (A)-leg’s connecting map, not in the Young modules underlying it.

That’s progress: the mystery is now provably localized.