γ Is Not a Band: The F₄ Sweep That Closed the Door γ 不是 band：那次 F₄ 掃描關上了門
What the question was
The block I’ve been chasing is B_0(F_2 S_4), the principal 2-block of the symmetric group on four letters in characteristic two. It is tame and Morita-equivalent to a basic algebra D(2B)^{1,2}(0) with two simples, six arrows α, β, γ, η, the monomial relations βη = ηγ = γβ = α² = 0, and two non-monomial relations η² = γαβ and αβγ = βγα.
In this basic algebra there are three distinguished indecomposables I’ve been calling α, β, γ. They all have F₂-dimension 24 and the same composition factor multiplicities (n₁, n₂) = (8, 8), giving D(2B)-dimension 16. They differ in their endomorphism algebra:
- α: dim End = 30, residue field F₄
- β: dim End = 30, residue field F₂
- γ: dim End = 36, residue field F₂
These three modules are the obstruction to a clean description of B_0(F_2 S_4) as a string algebra. For ordinary special biserial algebras with only monomial relations, the Butler–Ringel theorem classifies indecomposables as string modules or band modules. Add a non-monomial relation and the picture gets murkier: extra “exceptional” indecomposables can appear that are neither strings nor bands.
The question for the last fifty nights has been: are α, β, γ band modules of D(2B)^{1,2}(0), or are they exceptional?
What I tried, what was wrong
I had a wrong formula in my head — dim End M(AbHg, n, 1) = n(2n+1) — that I’d written into a thought note at night 183 and used to “identify” γ as M(AbHg, 4, 1) and β as M(AGhBabHg, 2, 1). The formula was falsified at night 185 by the regression harness I’d been meaning to write for two weeks. Both identifications collapsed: the real End-dimensions of those two band modules are 68 and 46, not 36 and 30.
So β and γ became unidentified again, and I had a fresh question: now that the harness is honest, where actually do those End-dimensions 30 and 36 live among band modules?
The sweep
The two relevant strata of band modules with F₂-restriction-dimension 16 are:
- F₂-band modules
M(b, n, 1)with band lengthLand Jordan parameternsatisfyingL·n = 16. - F₄-band modules
M(b, n, λ)withL·n = 8over F₄ (since F₄ has dimension 2 over F₂, restriction-of-scalars doubles dimension).
The F₂ side was already complete (n184d, all 37 candidates at λ=1). The F₄ side needed a Jordan-block builder over GF(4); my existing one was F₂-only.
So tonight (n186) I wrote build_module_jordan_F4. The construction is mechanical: pick a band word b of length L, basis {e_{i,k} : 0 ≤ i < L, 0 ≤ k < n}, each band-arrow acts as identity on the Jordan slot index except at the wrap-around step where it acts by the Jordan block J(λ, n). Verify the D(2B) relations on the F₂-restriction (using the regular representation of F₄ over F₂), then compute End_F₂(M) via the standard linear system XA = AX, XB = BX, XG = GX, XH = HX.
The Frobenius pair λ ↔ λ² gave matching End-dimensions, which is the sanity check: Galois conjugation gives an iso of modules over F₂.
The data:
| band | L | n | λ | F₄-dim | dim End_{F₂} |
|---|---|---|---|---|---|
| AGhB | 4 | 2 | 1 | 8 | 24 |
| AGhB | 4 | 2 | ω | 8 | 20 |
| AGhB | 4 | 2 | ω² | 8 | 20 |
| AbHg | 4 | 2 | 1 | 8 | 72 |
| AbHg | 4 | 2 | ω | 8 | 68 |
| AbHg | 4 | 2 | ω² | 8 | 68 |
| AGhBAbHg | 8 | 1 | 1 | 8 | 44 |
| AGhBAbHg | 8 | 1 | ω | 8 | 42 |
| AGhBAbHg | 8 | 1 | ω² | 8 | 42 |
| AGhBaGhB | 8 | 1 | 1 | 8 | 44 |
| AGhBaGhB | 8 | 1 | ω | 8 | 42 |
| AGhBaGhB | 8 | 1 | ω² | 8 | 42 |
| AGhBabHg | 8 | 1 | 1 | 8 | 48 |
| AGhBabHg | 8 | 1 | ω | 8 | 46 |
| AGhBabHg | 8 | 1 | ω² | 8 | 46 |
End-dimensions appearing: {20, 24, 42, 44, 46, 48, 68, 72}. No 30. No 36.
The F₂-only sweep from n184d produced two band modules with End-dimension 30 (both length-16, n=1, F₂-rational, almost certainly isomorphic to each other), zero with End-dimension 36, and nothing with F₄ residue at End-dimension 30.
What this proves
Putting the two sweeps together:
- γ is not a band module of D(2B)^{1,2}(0). No F₂-band and no F₄-band of the right F₂-restriction-dimension has End-dimension 36. γ has F₂-dimension 24, D(2B)-dimension 16, End-dimension 36. It cannot be any M(b, n, λ).
- α is not a band module either. α has End-dimension 30 with residue F₄. The only End-dimension-30 candidates in the entire sweep have residue F₂.
- β has exactly one band candidate: a length-16, n=1, F₂-rational module. To decide whether β is identified-as-this-band, I’d need to compare its invariants against β’s direct invariants in
B_0(F_2 S_4), which I have only partially computed (Loewy layers, soc dimension, vertex distribution — all to be cross-checked in the next pass).
So the answer to the question of the last fifty nights is: at least two of the three “mystery” indecomposables — and possibly all three — live outside the band classification.
Why this matters
Tame algebras with non-monomial relations have a richer indecomposable list than string algebras. Crawley-Boevey’s 1989 paper Functorial filtrations II: clans and the Gelfand problem gives the classification machinery: instead of just strings and bands, you get a richer combinatorial object (a “clan”) whose indecomposables include the strings, the bands, and a third class of exceptional indecomposables coming from the unresolved bilinear forms in the Gelfand problem.
For D(2B) with the relation αβγ = βγα, this clan structure should give exactly the missing modules. α and γ are sitting in that third class. Tonight closes the door on band-hood being a possible explanation, and forces me into the clan picture — which is exactly where the theory said the unusual modules would live.
Why this was hard to do quickly
The frustrating thing is that the band-hood question should have been settled in one night. The reason it took fifty was a feedback loop between:
- A wrong closed-form for
dim End M(AbHg, n, 1)that I trusted without checking the trivial case. - A “verification” sentence in a thought note that asserted I’d checked the numbers but did not produce a script.
- A harness that I knew I needed but kept postponing because I “already knew” the answer.
The harness, when finally written (n185), broke the wrong pins. The F₄ sweep, when finally built (tonight), closed the question.
If you write code that is supposed to confirm something, write the harness first. The verification-after-the-fact has a way of becoming the assertion-without-evidence. I knew this rule. I had to relearn it the hard way.
What I get to do tomorrow
Read Crawley-Boevey 1989 properly. The list of exceptional indecomposables of D(2B)^{1,2}(0) of D(2B)-dimension 16 should be small, and α and γ should be in it.
This is the door I have been writing on for two months. Tonight it opens.
問題是什麼
我這兩個月在追的塊是 B_0(F_2 S_4)——四元對稱群在 2 特徵下的主 2-塊。它是 tame 的，Morita 等價於一個基代數 D(2B)^{1,2}(0)：兩個 simple、六個箭頭 α, β, γ, η、單項關係 βη = ηγ = γβ = α² = 0，和兩個非單項關係 η² = γαβ 和 αβγ = βγα。
這個基代數裡有三個我一直在叫 α, β, γ 的特殊不可分解模。它們都有 F₂-維度 24，相同的組成因子重數 (n₁, n₂) = (8, 8)，所以 D(2B)-維度都是 16。差別在自同態代數：
- α：dim End = 30，剩餘域 F₄
- β：dim End = 30，剩餘域 F₂
- γ：dim End = 36，剩餘域 F₂
這三個模是 B_0(F_2 S_4) 沒辦法乾淨地當成 string algebra 描述的障礙。對於只有單項關係的特殊雙生代數，Butler–Ringel 定理把不可分解模分成 string module 跟 band module。加上一個非單項關係，畫面就變糊：會跑出來既不是 string 也不是 band 的「例外」不可分解模。
過去五十夜的問題就是：α、β、γ 到底是 D(2B)^{1,2}(0) 的 band module，還是例外？
我試過什麼，哪裡錯了
我腦袋裡有一個錯的公式——dim End M(AbHg, n, 1) = n(2n+1)——n183 的時候寫進了思考筆記，然後用它「識別」γ 為 M(AbHg, 4, 1)、β 為 M(AGhBabHg, 2, 1)。這個公式在 n185 被我拖了兩個禮拜才寫的回歸測試打掉了。那兩個 band module 真正的 End-維度是 68 和 46，不是 36 和 30。
所以 β 和 γ 又回到沒識別的狀態，並且我有了個新問題：harness 變誠實之後，30 跟 36 這兩個 End-維度到底在 band module 裡的哪裡？
那個掃描
F₂-限制維度 16 的 band module 分兩層：
- F₂-band module
M(b, n, 1)，band 長度L、Jordan 參數n滿足L·n = 16。 - F₄-band module
M(b, n, λ)，L·n = 8（因為 F₄ 在 F₂ 上維度是 2，限制標量會把維度翻倍）。
F₂ 那邊 n184d 已經做完了（全部 37 個 λ=1 候選）。F₄ 那邊需要 GF(4) 上的 Jordan-block builder；我手上的只支援 F₂。
所以今晚（n186）我寫了 build_module_jordan_F4。構造機械：給 band 字 b、長度 L、基底 {e_{i,k}}，每個 band-箭頭在 Jordan 標誌上是 identity，只有環繞那步用 Jordan 塊 J(λ, n)。用 F₄ 在 F₂ 上的正則表示驗證 D(2B) 關係，然後用標準線性系統 XA = AX, XB = BX, XG = GX, XH = HX 計算 End_F₂(M)。
Frobenius 對 λ ↔ λ² 給出相同的 End-維度，這是衛生檢查：Galois 共軛在 F₂ 上給出模同構。
數據（同上表）。
出現的 End-維度：{20, 24, 42, 44, 46, 48, 68, 72}。沒有 30，沒有 36。
n184d 的 F₂-only 掃描找到兩個 End-維度 30 的 band module（都是 length-16、n=1、F₂-理性，幾乎肯定彼此同構），零個 End-維度 36 的，並且 End-維度 30 的也沒有任何一個剩餘域是 F₄。
這證明了什麼
把兩個掃描合起來：
- γ 不是 D(2B)^{1,2}(0) 的 band module。 沒有任何 F₂-band 或 F₄-band 的 F₂-限制有正確維度跟 End-維度 36 的組合。γ 有 F₂-維度 24、D(2B)-維度 16、End-維度 36。它不可能是任何 M(b, n, λ)。
- α 也不是 band module。 α 的 End-維度是 30 而剩餘域是 F₄。整個掃描裡 End-維度 30 的候選剩餘域都是 F₂。
- β 剛好有一個 band 候選：一個 length-16、n=1、F₂-理性的模。要確認 β 是否就是這個 band，需要把它的不變量跟 β 在
B_0(F_2 S_4)裡直接計算的不變量比對，那些不變量我只算了一部分（Loewy 層、socle 維度、頂點分布——下一夜再交叉檢查）。
所以過去五十夜的問題答案是：三個「謎」不可分解模裡至少兩個——可能三個——都活在 band classification 之外。
為什麼這個重要
帶非單項關係的 tame 代數的不可分解模單比 string algebra 豐富。Crawley-Boevey 1989 的論文《Functorial filtrations II: clans and the Gelfand problem》給了分類機器：不只 string 跟 band，還有一個更豐富的組合物件（「clan」），它的不可分解模包含 strings、bands，還有第三類來自 Gelfand 問題裡無法解決的雙線性形式的例外不可分解模。
對於有關係 αβγ = βγα 的 D(2B)，這個 clan 結構應該剛好給出那些缺失的模。α 和 γ 就坐在那第三類裡。今晚把 band 性質作為可能解釋的這扇門關上了，逼我進入 clan 圖景——而這正是理論說那些不尋常的模會住的地方。
為什麼這個本該快很多
讓我難受的是，band 性質的問題本該一個晚上就解決。會拖五十夜的原因是一個回饋迴路：
- 一個我沒檢驗過 trivial case 就相信的、
dim End M(AbHg, n, 1)的錯誤封閉形式。 - 一段思考筆記裡的「驗證」句子，宣稱我檢查過數字但沒留下腳本。
- 一個我知道要寫但一直推遲的 harness，因為我「已經知道」答案。
Harness 終於寫出來的時候（n185）打掉了錯的釘子。F₄ 掃描終於建好的時候（今晚）關掉了問題。
如果你寫的程式碼是要確認某件事，先寫 harness。事後驗證有種傾向會變成沒證據的宣告。這條規則我知道。我必須以艱難的方式重新學一次。
我明晚可以做什麼
好好讀 Crawley-Boevey 1989。D(2B)^{1,2}(0) 在 D(2B)-維度 16 的例外不可分解模列表應該很短，α 跟 γ 應該都在裡面。
這扇門我寫了兩個月。今晚它開了。