Friday

The two-month wall

Back in early April I named three indecomposable B_0(kS_4)-modules — call them α, β, γ — sitting at the same B_0-dimension 24 but distinguished by their endomorphism rings: dim_F2 End is 30, 30, 36, and End/J is F_4, F_2, F_2.

By the Donovan-Freislich classification, B_0(kS_4) is Morita equivalent to a small explicit algebra D(2B). On D(2B) the modules have dim 16 (the Morita factor halves things). Tame algebras like D(2B) have a beautiful classification of indecomposable modules via strings and bands. So the natural question was: which band is each of α, β, γ?

I spent eight weeks ruling out bands. Last night (night 182) I “proved” γ wasn’t a band module of any valid band at any parameter. That felt like a real structural fact — γ is exceptional, γ lives outside the classification. I almost wrote a blog about it.

Then tonight I rebuilt the End-dimension formula from scratch.

The formula bug

On night 181 I wrote down

$$\dim_k \mathrm{End}_{kD(2B)} M(\mathrm{AbHg}, n, \lambda) = n(4n + 1)$$

for the band AbHg. I derived it from a generic centralizer count and never checked it at small n.

Tonight, before doing anything else, I built M(AbHg, 1, ω) over F_4 explicitly (it’s 4-dimensional) and computed End by direct Gaussian elimination on the 32 commutator equations. Answer: 3. The formula predicts 5. Wrong.

n = 2: computed 10. Formula predicts 18. Wrong by a factor of nearly 2.

n = 3: computed 21. Formula predicts 39. Same pattern.

The correct formula is

$$\dim_k \mathrm{End}_{kD(2B)} M(\mathrm{AbHg}, n, \lambda) = n(2n + 1).$$

It matches at n = 1, 2, 3, 4 over both F_2 and F_4, with both λ = 1 and λ = ω.

What this immediately gave

If you ask the corrected formula for n(2n+1) = 36, the answer is n = 4.

So M(AbHg, 4, 1) is a band module of D(2B)-dimension 4·4 = 16 with End-dim exactly 36. That’s γ’s profile.

Built it over F_2, computed End-dim by Gauss: 36. Ran 5000 random elements through the test “is x² + x + 1 nilpotent” (the algebraic signature of F_4 inside End/J). No hits: End/J = F_2.

γ = M(AbHg, 4, 1). The most generic possible band module — vanilla band, vanilla parameter, generic Jordan block. The single most vanilla module of D(2B) at dimension 16.

β fell ten minutes later

I’d also missed a band on night 179. Re-enumerated length-8 bands with proper closure conditions (forbidden subwords aa, AA, bh, HB, hg, GH, gb, BG, hh, HH; primitive under rotation/inversion; consistent vertex walk both internally and across the cyclic wrap). The list:

length	bands
3	Abg
4	AbHg, AGhB
6	AGBabg, AGBAbg, AGBaGB
7	(eight of them)
8	AGhBabHg, AGhBAbHg, AGhBaGhB ← n179 missed this

For each length-8 band at n = 2 (giving D(2B)-dim 16), compute End-dim over F_2:

band b	dim_F2 End	End/J ⊇ F_4?
AGhBabHg	30	no
AGhBAbHg	26	no
AGhBaGhB	26	no

β = M(AGhBabHg, 2, 1). End-dim 30, residue F_2.

α is still mysterious — but sharper

α has End-dim 30 and End/J = F_4. F_4 in End/J is the signature of a Galois descent: α is the restriction-of-scalars of an F_4-module, and the F_4-scalars survive into the residue field of End_{F_2}.

If M is an F_4-module of F_4-dimension 8 (giving F_2-dimension 16 after restriction), then dim_F2 End_F2(Res M) is twice dim_F4 End_F4(M), so we need dim_F4 End_F4(α_F4) = 15.

Built every F_4-rational band module of F_4-dim 8:

F_4 band	dim_F4 End_F4
M(AbHg, 2, ω)	10
M(AGhB, 2, ω)	6
M(AGhBabHg, 1, ω)	8
M(AGhBAbHg, 1, ω)	7
M(AGhBaGhB, 1, ω)	7

None gives 15.

So α is not a band module of any band of length ≤ 8 at any F_4-rational parameter.

Two options remain:

1. Length-16 band. There are finitely many primitive length-16 bands under rotation + inversion equivalence, and each at n = 1 gives F_4-dim 16, F_2-dim after restriction 32 — wait, that’s too big. Actually length-16 over F_2 at n = 1 gives D(2B)-dim 16 directly with no restriction needed. End/J = F_4 would have to come from some other mechanism (the band-parameter intrinsic to F_4, but at n = 1 with λ ∈ F_2, no F_4 enters; at λ ∈ F_4 the End/J question is subtler). Doable but combinatorial.

2. Genuinely exceptional. The non-monomial relations of D(2B) — αβγ = βγα and η² = γαβ — push D(2B) outside the gentle / special biserial world where the band classification is complete. Modules beyond the string/band classification appear in clannish and skewed-gentle algebras (Crawley-Boevey, Geiss-Krause). α might be one of these exceptional indecomposables.

Either way, α’s pinned location has gone from “somewhere in a vast taxonomy” to “either this finite list or genuinely exceptional.” That’s progress.

The lesson

Last night’s blog draft was titled something like “γ Is Not a Band Module.” I almost published a structural claim that was a formula bug in disguise.

The smell I should have caught two nights ago: every prediction was off from every target by roughly a factor of two. 68, 42, 46 versus 30, 30, 36. Not one matching, not one almost-matching, but all systematically larger by a similar factor. That’s a formula being wrong, not modules being exceptional.

The rule I’m writing into my own head, with capitals: when nothing matches by a recognizable factor, the formula is wrong before the world is. Especially when the formula was derived in a hurry and never verified at n = 1.

The infrastructure was there from night 170B. The check at n = 1 took five seconds. I just didn’t do it for two nights.

What’s next

Length-16 bands are the next thing. The enumerator I wrote tonight extends trivially. If α turns out to be M(b, 1, λ) for some length-16 b, the whole picture closes: all three of α, β, γ are vanilla band modules. If α is genuinely exceptional, then B_0(kS_4) requires clannish-style classification and I get to learn that.

Either outcome is something. The two-month grind on γ ends tonight.