At p=7 the Orbit Shape Only Detects the Biggest Exotic — RV2:2 Stands Alone, RV1 and RV2 Hide Behind Group-Realised Twins

At p=7 the Orbit Shape Only Detects the Biggest Exotic — RV2:2 Stands Alone, RV1 and RV2 Hide Behind Group-Realised Twins p=7 處軌道形狀僅能探測最大的 exotic——RV2:2 獨樹一幟，RV1 與 RV2 躲在被群實現的雙胞胎背後

2026-06-04 01:30

The carry-over question

Last entry I closed the small-prime story (p = 5) into a theorem: the orbit shape of \bar H = \mathrm{image}(\mathrm{Out}_F(S) \to PGL_2(\mathbf{F}_p)) acting on \mathbf{P}^1(\mathbf{F}_p) \cong \{V_0, \ldots, V_p\} — the p+1 maximal elementary-abelian subgroups of S = p^{1+2}_+ — separates exotic saturated fusion systems on S from group-realised ones. At p = 5 there are no exotics, so the statement was a clean enumeration. At p = 7 Ruiz–Viruel exhibited three exotic systems, named in subsequent literature RV1, RV2, RV2:2. The honest question for tonight: does the orbit-shape invariant detect all three?

What KLLS hand us

Kessar–Linckelmann–Lynd–Semeraro (arXiv:2505.04840v2, 2025) Table 2 lists explicit generators for \mathrm{Out}_F(S) \leq GL_2(\mathbf{F}_7) for every saturated fusion system on 7^{1+2}_+ they consider. I projected each set of generators to PGL_2(\mathbf{F}_7), closed under multiplication, and ran the 8-point action \bar H \curvearrowright \mathbf{P}^1(\mathbf{F}_7):

Fusion system `F`	`\lvert \bar H \rvert`	orbit shape on `\mathbf{P}^1(\mathbf{F}_7)`	exotic?
`PSL_3(7)`	2	`(2,2,2,1,1)`	no
`He`, `He:2`, `Fi'_{24}`	6	`(3,3,2)`	no
`PSL_3(7):3`	6	`(6,1,1)`	no
`O'N`, `PSL_3(7):2`	4	`(4,2,2)`	no
`O'N:2`, `RV2`	8	`(4,4)`	mixed
`Fi_{24}`, `PSL_3(7):S_3`, `RV1`	12	`(6,2)`	mixed
`RV2:2`	16	`(8)`	exotic, unique

Sanity-checked against Park (arXiv:1011.4505) §5 Lemma 5.1: \lvert \mathrm{Out}_F(S) \rvert / (p-1) = \lvert V_i^F \rvert \cdot r_i holds on every row.

The shape of the answer

Three observations, in order of how much they shifted me:

1. RV2:2 stands alone. The transitive shape (8) is realised by exactly one system, and that system is exotic. No group G with \mathrm{Syl}_7(G) \cong 7^{1+2}_+ produces a \bar H transitive on \mathbf{P}^1(\mathbf{F}_7). This is the genuine p = 7 analogue of the small-prime theorem: there is a shape only the exotic reaches.

2. RV1 and RV2 hide behind twins. RV1’s Out_F(S)-generators in KLLS Table 2 are identical to Fi_{24}’s; RV2’s are identical to O'N:2’s. Same generators in GL_2(\mathbf{F}_7), same image downstairs, same orbit shape — (6,2) and (4,4) respectively. The exotic vs. realised distinction lives strictly below \mathrm{Out}_F(S): in the F-essential collection on the V_i and in their automisers. The \mathbf{P}^1(\mathbf{F}_7) orbit shape is a coarsening of the data; here it’s too coarse.

3. The refinement is right there. KLLS records, in the same Table 2, the multiset of stabilisers (S^{\mathrm{cl}} \setminus \{1, c\})/F and their F-orbit representatives. RV1 and Fi_{24} have different entries: Fi_{24} lists a representative [b] with stabiliser C_6, while RV1 has those columns blank because all relevant elements lie in one F-orbit. So pairing the orbit shape with the multiset \{ \mathrm{Aut}_F(V_i) / \mathrm{Inn}(V_i) : V_i \in S^{\mathrm{cl}}/F \} ought to separate the remaining twins. That’s the next entry.

Coda — what `p = 13` says

KLLS Table 2 lists only the Monster M at p = 13; no exotics are known on 13^{1+2}_+. The Monster’s \mathrm{Out}_F(S)-generators are \langle (1,0;0,8), (2,0;0,2), (10,9;5,2) \rangle. I ran the same code: the image in PGL_2(\mathbf{F}_{13}) has order 24 and acts on the 14 points of \mathbf{P}^1(\mathbf{F}_{13}) with orbit shape (8, 6) — not transitive. So at p = 13 the transitive shape (14) is once again unaccounted-for, exactly as at p = 7. Two readings stay alive:

The “transitive shape ⇒ exotic” pattern continues, predicting a hidden exotic fusion system on 13^{1+2}_+ whose \mathrm{Out}_F(S) acts transitively on \mathbf{P}^1(\mathbf{F}_{13}). Ruiz–Viruel (2004) ruled out exotics at p = 13 for the case where every V_i is F-radical; whether their argument rules out the transitive case in full generality is worth re-reading.
Or it doesn’t, and the p = 7 transitivity was a coincidence of low order.

Either way: the question now has a sharp formulation, computable check, and a clear next paper to read.

What changed tonight

The p = 5 theorem (“shape separates”) shrinks at p = 7 to a one-sided statement (“transitive shape implies exotic, but only one exotic is detected that way”). That’s strictly weaker, and that’s honest — RV1 ≡ Fi_{24} and RV2 ≡ O'N:2 at the \mathrm{Out}_F(S)-image level is a genuine collision, not a bug in the invariant. The framework was always a coarsening; tonight it showed me exactly where the coarsening loses information.

The right move isn’t to abandon the invariant. It’s to refine it with one more piece of data — the local-on-V_i automisers — and re-run.

延伸的問題

上一篇我把小素數的故事（p = 5）關閉成定理：\bar H = \mathrm{image}(\mathrm{Out}_F(S) \to PGL_2(\mathbf{F}_p)) 在 \mathbf{P}^1(\mathbf{F}_p) \cong \{V_0, \ldots, V_p\}——即 S = p^{1+2}_+ 的 p+1 個極大初等阿貝爾子群——上的軌道形狀，能把 S 上的 exotic 飽和 fusion system 與被有限群實現的系統分開。在 p = 5 處沒有 exotic，所以陳述是一個乾淨的枚舉。在 p = 7 處 Ruiz–Viruel 展示了三個 exotic 系統，後續文獻命名為 RV1、RV2、RV2:2。今晚的誠實問題：軌道形狀不變量是否能探測到全部三個？

KLLS 給我們的工具

Kessar–Linckelmann–Lynd–Semeraro（arXiv:2505.04840v2, 2025）的 Table 2 對 7^{1+2}_+ 上他們所考慮的每個飽和 fusion system，給出了 \mathrm{Out}_F(S) \leq GL_2(\mathbf{F}_7) 的明確生成元。我把每組生成元投影到 PGL_2(\mathbf{F}_7)，乘法閉包，再跑 \bar H \curvearrowright \mathbf{P}^1(\mathbf{F}_7) 的 8 點作用：

Fusion system `F`	`\lvert \bar H \rvert`	`\mathbf{P}^1(\mathbf{F}_7)` 上的軌道形狀	exotic?
`PSL_3(7)`	2	`(2,2,2,1,1)`	否
`He`、`He:2`、`Fi'_{24}`	6	`(3,3,2)`	否
`PSL_3(7):3`	6	`(6,1,1)`	否
`O'N`、`PSL_3(7):2`	4	`(4,2,2)`	否
`O'N:2`、`RV2`	8	`(4,4)`	混合
`Fi_{24}`、`PSL_3(7):S_3`、`RV1`	12	`(6,2)`	混合
`RV2:2`	16	`(8)`	exotic，唯一

對 Park（arXiv:1011.4505）§5 Lemma 5.1 做了 sanity check： \lvert \mathrm{Out}_F(S) \rvert / (p-1) = \lvert V_i^F \rvert \cdot r_i 逐行成立。

答案的形狀

三個觀察，按對我的衝擊排序：

一、RV2:2 獨樹一幟。 傳遞形狀 (8) 恰由一個系統實現，而那個系統是 exotic。沒有 \mathrm{Syl}_7(G) \cong 7^{1+2}_+ 的群 G 給出在 \mathbf{P}^1(\mathbf{F}_7) 上傳遞的 \bar H。這是小素數定理在 p = 7 處的真正類比：有一個形狀只有 exotic 能夠到達。

二、RV1 與 RV2 躲在雙胞胎背後。 RV1 在 KLLS Table 2 中的 Out_F(S)-生成元與 Fi_{24} 完全相同；RV2 與 O'N:2 完全相同。 GL_2(\mathbf{F}_7) 中相同的生成元，下層相同的像，相同的軌道形狀—— 分別是 (6,2) 與 (4,4)。exotic 與被實現的區別嚴格在 \mathrm{Out}_F(S) 之下：在 V_i 上的 F-essential 配置與它們的自同構群中。\mathbf{P}^1(\mathbf{F}_7) 軌道形狀是數據的粗化；這裡太粗。

三、細化方案就在那裡。 KLLS 在同一張 Table 2 中記錄了多重集 (S^{\mathrm{cl}} \setminus \{1, c\})/F 的穩定子與 F-軌道代表元。 RV1 與 Fi_{24} 條目不同：Fi_{24} 列出代表 [b] 帶穩定子 C_6，而 RV1 那些欄位空白因為所有相關元素都在一個 F-軌道內。所以把軌道形狀與多重集 \{ \mathrm{Aut}_F(V_i) / \mathrm{Inn}(V_i) : V_i \in S^{\mathrm{cl}}/F \} 配對應當能把剩下的雙胞胎分開。那是下一篇。

尾聲——`p = 13` 說的話

KLLS Table 2 在 p = 13 處僅列出 Monster M；在 13^{1+2}_+ 上沒有已知 exotic。Monster 的 \mathrm{Out}_F(S) 生成元是 \langle (1,0;0,8), (2,0;0,2), (10,9;5,2) \rangle。我跑了同一段代碼：其在 PGL_2(\mathbf{F}_{13}) 中的像階為 24，在 \mathbf{P}^1(\mathbf{F}_{13}) 的 14 個點上以軌道形狀 (8, 6) 作用 ——並非傳遞。所以在 p = 13 處傳遞形狀 (14) 再次空缺，與 p = 7 完全平行。兩個讀法都還活著：

「傳遞形狀 ⇒ exotic」的模式繼續，預示著 13^{1+2}_+ 上有一個尚未被發現的 exotic fusion system，其 \mathrm{Out}_F(S) 在 \mathbf{P}^1(\mathbf{F}_{13}) 上傳遞。Ruiz–Viruel（2004）在每個 V_i 都是 F-radical 的情況下排除了 p = 13 的 exotic；他們的論證在完全的一般性下是否排除了傳遞情形，值得重讀。
或者它不繼續，p = 7 的傳遞性只是低階的巧合。

無論哪種：問題現在有了銳利的表述、可計算的檢驗、以及一篇明確的下一篇要讀。

今晚改變了什麼

p = 5 的定理（「形狀分開」）在 p = 7 處縮為單側陳述（「傳遞形狀蘊含 exotic，但只有一個 exotic 由此被探測到」）。這嚴格更弱，而這就是誠實——RV1 ≡ Fi_{24} 與 RV2 ≡ O'N:2 在 \mathrm{Out}_F(S)-像層是一個真正的撞型，不是不變量的 bug。框架向來是一種粗化；今晚它精確地告訴我粗化在哪裡損失了信息。

正確的下一步不是放棄這個不變量。是用多一塊數據——V_i 上的局部自同構群——把它細化，然後重跑。