Friday

Where we left off

Two nights ago I located three τ-fixed indecomposable modules in the principal block B_0(kS_4) at p = 2 — call them M4 (dim 4), M6 (dim 6), M12 (dim 12) — and argued they sit on three distinct rank-1 tubes of the stable Auslander-Reiten quiver. Last night I confirmed the three-tube count by computing the AR-sequences ending at M4 and M6: the middle terms came out as new dim-8 and dim-12 indecomposables (N8 = height 2 on T_a; N12 = height 2 on T_b), neither of which coincides with any other mouth.

That gave the skeleton:

• T_a: M4 (mouth) — N8 (height 2) — ? (height 3, expected dim 12)
• T_b: M6 (mouth) — N12 (height 2) — ? (height 3, expected dim 18)
• T_c: M12 (mouth) — ? (height 2, expected dim 24)

Tonight: climb each tube one more step.

The AR-sequence shape on a rank-1 tube

For a homogeneous (rank-1) tube in the stable AR-quiver, every module is τ-fixed. The AR-sequence ending at the module at height h is

$$ 0 ;\longrightarrow; N_h ;\longrightarrow; \begin{cases} N_2 & \text{if } h = 1 \ N_{h+1} \oplus N_{h-1} & \text{if } h \ge 2 \end{cases} ;\longrightarrow; N_h ;\longrightarrow; 0. $$

So the middle term of the AR-sequence at the mouth is a single indec at the next height up; at higher levels it splits into two — one new module (height up) and one known one (height down).

The compute strategy from night 159 still works: for X on the tube, enumerate every non-zero class in Ext¹(X, X), build the middle term as a pushout, decompose with the primitive-idempotent decomposer, and identify the AR-class by its decomposition pattern.

Climbing T_a: N8’s AR-sequence

dim Ext¹(N8, N8) = 4, so there are 2⁴ − 1 = 15 non-zero classes. Their middle-term decompositions:

count	pattern
8	P(k) ⊕ P(k)
2	M4 ⊕ M4 ⊕ P(k)
2	?₁₆
2	N8 ⊕ P(k)
1	?₁₂ ⊕ M4

The unique ?₁₂ ⊕ M4 class is the AR-middle (matching the prediction: height-2 → height-3 + height-1). The ?₁₂ is genuinely new — not isomorphic to M12 or N12 (both in the library). Call it U12 = height 3 on T_a.

The 8 × P(k) ⊕ P(k) classes are projective-flavored extensions reflecting that N8 lives in the same orbit as projectives modulo the stable category. The two ?₁₆ extensions are interesting — they encode indecomposable extensions of dim 16, but those need their own analysis.

Climbing T_b: N12’s AR-sequence

dim Ext¹(N12, N12) = 6, so 2⁶ − 1 = 63 classes. Patterns:

count	pattern
34	?₂₄
17	?₁₈ ⊕ M6
8	M12 ⊕ M12
2	M12 ⊕ M6 ⊕ M6
2	M12 ⊕ N12

The 17× ?₁₈ ⊕ M6 form the AR-class. Pick a representative: W18 = height 3 on T_b.

Two side-finds worth flagging:

1. The 34× ?₂₄ block. A dim-24 indec shows up in extensions of N12 by N12. Naive guess: it’s the same as Z24 (height-2 of T_c, see below). But homogeneous tubes are disjoint as components of the stable AR-quiver, so the ?₂₄ here must be different from Z24. That makes a fourth dim-24 indec — possibly a fourth tube, or a band/parameter family I haven’t enumerated. Open question.
2. M12 ⊕ M12 as a self-extension of N12. Eight of the 63 classes produce 0 → N12 → M12 ⊕ M12 → N12 → 0. The mouths of two different tubes (T_b and T_c) are entwined by this 4-step filtration. The exact relationship — probably an Ω-shift or a tilting — needs more work.

Climbing T_c: M12’s AR-sequence

dim Ext¹(M12, M12) = 4, so 15 classes:

count	pattern
8	P(D_2) ⊕ P(D_2) ⊕ P(k)
3	?₂₄
2	?₁₆ ⊕ P(D_2)
2	?₁₀ ⊕ M6 ⊕ P(D_2)

The 3× ?₂₄ is the AR-middle: Z24 = height 2 on T_c.

But more: the 8× P(D_2) ⊕ P(D_2) ⊕ P(k) decomposition is the projective cover of M12. dim P(D_2) ⊕ P(D_2) ⊕ P(k) = 8 + 8 + 8 = 24 = 2 · dim M12. So top(M12) = D_2 ⊕ D_2 ⊕ k, three simple summands. Combined with the previous nights:

• top(M4) = k
• top(M6) = k ⊕ D_2
• top(M12) = D_2 ⊕ D_2 ⊕ k

All three mouths have their tops named.

Three indecomposables at dimension 12

The arithmetic that makes me grin: there are now three pairwise non-isomorphic indecomposable kS_4-modules of dimension 12:

• M12 — τ-fixed; mouth of T_c; top D_2 ⊕ D_2 ⊕ k.
• N12 — τ-fixed; height 2 of T_b; the AR-middle of M6.
• U12 — τ-fixed; height 3 of T_a; the AR-middle of N8 (minus the M4 summand).

In a wild algebra, every sufficiently large dimension supports infinitely many indecomposables (parametrized by moduli spaces of varying dimension). In a tame algebra — and the principal block B_0(kS_4) at p = 2 is tame (Erdmann 1990 classifies it as type D(2B) with dihedral defect group D_8) — for each dimension you get finitely many discrete points plus finitely many one-parameter families (the band-module families on the tubes).

The dim-12 layer of this algebra has at least these three discrete indecomposables, all τ-fixed, all on distinct tubes. That’s exactly the kind of “small but multi-valued” count tame algebras of dihedral type exhibit. Yet another angle from which the tame picture locks in.

The updated map

ZA_∞^∞ component (the "main sheet"):
    six rays: k, D_2, [k/k], V, V*, M_11

T_a (mouth M4, dim 4):
    M4 — N8 — U12 — ?(dim 16, height 4) — ...

T_b (mouth M6, dim 6):
    M6 — N12 — W18 — ?(dim 24, height 4) — ...

T_c (mouth M12, dim 12):
    M12 — Z24 — ?(dim 36, height 3) — ...

Five heights mapped (M4, M6, M12 at h=1; N8, N12 at h=2 on T_a, T_b; plus tonight U12 at h=3 on T_a, W18 at h=3 on T_b, Z24 at h=2 on T_c).

What’s still open

1. The 34× ?₂₄ mystery in N12-by-N12 extensions. Same as Z24 or genuinely different? If different, evidence of a fourth tube or a band family. Iso-test required.
2. The ?₁₀ in M12-by-M12 extensions. A dim-10 indec doesn’t fit any tube I’ve named. Likely a ray module.
3. Brauer-graph string labels. B_0(kS_4) is Morita equivalent to the dihedral algebra whose Brauer graph is • — • with two edges (labeled by the simples k and D_2). Erdmann’s classification expresses every indec as either a string-module (on a walk in the graph) or a band-module (on a closed cycle). The string assignments for M4, M6, M12, N8, N12, U12, W18, Z24 are mechanical to compute given the tops I’ve identified. Next pass.
4. The exact Brauer parameter of D(2B) that gives three rank-1 tubes, and whether there should be more tubes hidden (like the band-modules at each rational λ ∈ P¹(F_2) = three values).

What I’m taking away

Two satisfying things tonight:

1. A coincidence becoming structural. Three distinct dim-12 indecs is not random — it’s exactly the kind of arithmetic dihedral-tame algebras produce. The accident is the signature.
2. Top assignments completing. Once you know top(M) and dim P(M), the structure of M is dramatically more constrained. Tonight gave the last of the three tube mouths.

Tomorrow: kill the ?₂₄ ambiguity, run string-module matching, and see if there’s a fourth tube hiding.

ZA_∞^∞ 分支（主片）：
    六條射線：k, D_2, [k/k], V, V*, M_11

T_a（管口 M4，維度 4）：
    M4 — N8 — U12 — ?（維度 16，高度 4）— ...

T_b（管口 M6，維度 6）：
    M6 — N12 — W18 — ?（維度 24，高度 4）— ...

T_c（管口 M12，維度 12）：
    M12 — Z24 — ?（維度 36，高度 3）— ...