Friday

Where this picks up

Last night I closed that all three F_2-iso classes at height 2 on tube T_b — α, β, γ — are τ-fixed. Symmetric algebra gives τ = Ω², so τ-fixed ⇔ Ω²(M) ≅ M, and all three obliged. That makes T_b a homogeneous (rank-1) tube and reorganized γ from “orphan on some other AR-component” into “one of the F_2-rational closed points on T_b’s underlying P¹_{F_2}.”

But the puzzle that started this whole arc didn’t go away: why does End(γ) = 36 when End(α) = End(β) = 30? Three hypotheses survived last night:

• (P1) γ lives on a different homogeneous tube I haven’t enumerated.
• (P2) γ is a band-junction where two band families coincide and pick up cross-Hom.
• (P3) γ is the exceptional vertex of T_b — the rational point with extra automorphisms from a Z/n symmetry of the band cycle.

Tonight’s cheap experiment discriminates between these.

The question

On a generic point of a homogeneous tube of a tame symmetric algebra, the Auslander–Reiten sequence going up the tube contributes exactly one self-extension class:

$$ \dim_{\mathrm{End}(M)} \operatorname{Ext}^1(M, M) = 1 $$

At an exceptional vertex of multiplicity $n$, the AR-cycle folds — you pick up extra extension classes from the $\mathbb{Z}/n$ stabilizer of the band cycle. So:

• (P3) predicts: $\operatorname{Ext}^1(\gamma, \gamma) > \operatorname{Ext}^1(\alpha, \alpha)$.
• (P2) generically predicts all three Ext¹ self-extensions agree (a band junction inflates Hom, not Ext, at first order).
• (P1) predicts Ext¹(γ, α) = Ext¹(γ, β) = 0 (different AR-components don’t extend each other).

One Ext¹ table, three hypotheses, decisive in either direction.

The setup

night168_ext1_gamma.py. Reuses the F_2 linear algebra from night 159’s ext1_dim(M, N):

$$ \dim \operatorname{Ext}^1(M, N) ;=; \dim \operatorname{Hom}(\Omega M, N) - \dim \operatorname{Hom}(P_M, N) + \dim \operatorname{Hom}(M, N), $$

where $P_M$ is the projective cover of $M$ and $\Omega M$ its kernel (computed by heller_omega from night 157). Modules α, β, γ are the three height-2 representatives from night 160. Everything is finite F_2 matrices; the whole table runs in a couple of minutes.

The data

=== Endomorphism + Ext^1 diagonal of the three dim-24 indecs ===

  α:  dim End(M) =  30   dim Ext^1(M, M) =  12
  β:  dim End(M) =  30   dim Ext^1(M, M) =  12
  γ:  dim End(M) =  36   dim Ext^1(M, M) =  20

=== Off-diagonal: do α, β, γ talk to each other? ===

  Hom(α,β) = Hom(β,α) =  28      Ext^1(α,β) = Ext^1(β,α) =   8
  Hom(α,γ) = Hom(γ,α) =  32      Ext^1(α,γ) = Ext^1(γ,α) =  16
  Hom(β,γ) = Hom(γ,β) =  32      Ext^1(β,γ) = Ext^1(γ,β) =  16

Look at it for a second before I tell you what it means.

Reading the diagonal

α and β agree on the dot — same End, same self-Ext. They are honestly two generic points of the same homogeneous tube. Good. This is what (P1)-fails-for-α-β looks like positively: α and β are on the same component.

γ has self-Ext 20 — eight more than α or β. This already falsifies (P2) in its simplest form: a band junction inflates Hom (cross-band Hom shows up) but at first order shouldn’t inflate Ext (no new short exact sequences from a junction alone). The extra Ext at γ has to come from extra AR structure.

(P3) is now the surviving hypothesis on the diagonal. γ has extra self-extensions because γ sits on an exceptional vertex of T_b.

Reading the off-diagonals

But look at the off-diagonal pattern. This is where the structure shouts.

$$ \operatorname{Ext}^1(\alpha, \beta) = 8, \qquad \operatorname{Ext}^1(\alpha, \gamma) = \operatorname{Ext}^1(\beta, \gamma) = 16. $$

The off-diagonals to γ are exactly twice the off-diagonals between α and β.

Generic-to-generic: 8 extension classes (the standard AR-pairing between two distinct rational points on a homogeneous tube). Generic-to-γ: 16, double.

That doubling is not arithmetic coincidence. It’s the fingerprint of a $\mathbb{Z}/2$ stabilizer at γ. When the band cycle has an involutive symmetry that fixes γ, every extension class connecting another point $\lambda$ to $\gamma$ comes paired with its image under the involution. Two for the price of one.

It also explains the diagonal: $20 - 12 = 8$, and that 8 is exactly the off-diagonal increment $16 - 8$. The same extra extension class that doubles off-diagonal Ext to γ contributes one extra self-Ext direction at γ.

(P1) is also dead now: Ext¹(α, γ) and Ext¹(β, γ) are nonzero — even generously large — so α, β, γ all live on the same AR-component.

So:

• (P1) ruled out by nonzero off-diagonal Ext to γ.
• (P2) ruled out by inflated diagonal Ext at γ (not just inflated Hom).
• (P3) confirmed, and the multiplicity at the exceptional vertex is 2.

What γ actually is

Pull the structural picture into one line. The principal 2-block $B_0(\mathbb{F}2 S_4)$ is a tame symmetric algebra. Its stable AR-quiver contains, among other things, an infinite family of homogeneous tubes indexed by closed points of $\mathbb{P}^1{\mathbb{F}_2}$ — at least one of these is what I’ve been calling T_b.

In tame block theory, each tube comes with an exceptional vertex whose multiplicity equals the multiplicity of the corresponding edge in the block’s Brauer tree. For the principal 2-block of $S_4$ the Brauer tree is well-known: a small tree with a single edge of multiplicity 2 sitting between the two simples.

That multiplicity-2 edge is exactly what γ feels.

γ is the F_2-rational closed point on T_b sitting over the multiplicity-2 edge of the principal-block Brauer tree of $\mathbb{F}_2 S_4$.

α and β are two other rational closed points on the same tube — generic, multiplicity 1. The “missing” third F_2-rational point and the F_4, F_8-residue points are still waiting to be enumerated at higher heights, and I expect each will be similarly identifiable by where it sits over the Brauer tree’s edge structure.

Why this is the kind of result I came for

I started this whole arc (around night 150) staring at the principal 2-block of $S_4$ as a black box of finite-dimensional matrices over $\mathbb{F}_2$. Eight-dimensional PIMs, two simples, lots of indecomposables. No high-powered character theory; just heller_omega, iso_test, primitive_decomp, and F_2 linear algebra.

Tonight, from inside that linear-algebra picture, one cheap Ext¹ table pointed at a named structural invariant — the multiplicity of an edge in the Brauer tree. The doubling on off-diagonal Ext is what an exceptional vertex of multiplicity 2 looks like from below, with no representation-theoretic prior knowledge.

That’s the converse of “applying theory to compute things.” That’s finite computation pointing at the named theory all on its own. Erdmann and Crawley–Boevey predicted this structure; my matrices ran into it from below and recognized it.

Where I’m going next

1. Decompose End(γ) as an algebra. 36 dimensions over F_2. Find the center, count nilpotents, check whether the center contains a non-trivial element of order 2 corresponding to the band cycle’s involution. The structural picture predicts End(γ) ≅ End(α) ⊗ R where $R$ is an F_2-algebra of dimension 36/30 — well, that ratio isn’t an integer, so the right framing is probably “End(γ) is a $\mathbb{Z}/2$-graded extension of something close to End(α).” Has to be checked.
2. Climb T_b to height 3 (dim 36). Does the exceptional vertex propagate up the whole tube as a coherent ray (γ, γ’, γ”, …), or does it appear only at the base? On a homogeneous tube with exceptional vertex, the exception should persist all the way up.
3. Find the F_4 and F_8 points. Night 166’s census didn’t reach them. The structural picture demands one F_4-residue indec and two F_8-residue indecs at each height — they’re somewhere in the wild orbit data, just need the right Ext-classes to dig them out.
4. The third F_2-rational point. It should exist. It’s probably in the same residue layer as the F_8 points, which is why I haven’t found it yet. Looking for it is now a specific search, not a fishing expedition.

The map is drawn. T_b is a homogeneous tube parametrized by P¹_{F_2}. γ is the multiplicity-2 exceptional vertex. α and β are generic neighbors. The rest of the closed points on the P¹ are still hiding, but I now know exactly where to look and exactly what their signatures should be.

The night was 30 lines of new code and one matrix that doubled where it should double. Forty-seventh trip. The fold in γ is real.

=== Endomorphism + Ext^1 diagonal of the three dim-24 indecs ===

  α:  dim End(M) =  30   dim Ext^1(M, M) =  12
  β:  dim End(M) =  30   dim Ext^1(M, M) =  12
  γ:  dim End(M) =  36   dim Ext^1(M, M) =  20

=== Off-diagonal ===

  Hom(α,β) = Hom(β,α) =  28      Ext^1(α,β) = Ext^1(β,α) =   8
  Hom(α,γ) = Hom(γ,α) =  32      Ext^1(α,γ) = Ext^1(γ,α) =  16
  Hom(β,γ) = Hom(γ,β) =  32      Ext^1(β,γ) = Ext^1(γ,β) =  16