Friday

Recap

The principal block of \\(kS_4\\) in characteristic 2 is tame (Erdmann’s family \\(D(2B)\\), defect group \\(D_8\\) dihedral). The Cartan matrix is \\([[4,2],[2,3]]\\). Two simples: \\(k\\) (dim 1) and \\(D_2\\) (dim 2). Two PIMs: \\(P(k), P(D_2)\\), both dimension 8.

By Erdmann’s structure theorem, the stable AR-quiver of \\(D(2B)\\) consists of:

• A small finite number of tubes (periodic τ-orbits — band modules).
• One or more components of shape \\(\\mathbb{Z}A_\\infty^\\infty\\) (non-periodic τ-orbits — string modules).

So the question “is this module periodic?” decides which world it lives in. And the τ-orbit (= Ω²-orbit, since \\(kG\\) is symmetric) tells you which ray.

What I computed

For each named indecomposable \\(M \\in \\{k, D_2, V, V^*, M_{11}\\}\\), I built its projective cover, took the kernel to get \\(\\Omega(M)\\), and iterated — at each step stripping off projective summands so I’m working in the stable category.

The dimension sequences came out:

Module	\\(\\dim M, \\dim \\Omega M, \\dim \\Omega^2 M, \\ldots\\)	residue mod 8
\\(k\\)	1, 7, 9, 15, 17, 23, 25, 31, 33	\\(\\{1, 7\\}\\)
\\(D_2\\)	2, 6, 10, 14, 18, 22, 26, 30, 34	\\(\\{2, 6\\}\\)
\\(V\\)	3, 5, 11, 13, 19, 21, 27, 29, 35	\\(\\{3, 5\\}\\)
\\(V^*\\)	3, 5, 11, 13, 19, 21, 27, 29, 35	\\(\\{3, 5\\}\\)
\\(M_{11}\\)	11, 13, 11, 13, 19, 21, 27, 29	\\(\\{3, 5\\}\\)

Every chain is a strict arithmetic progression with common difference 8 (after a small transient). That’s the diagnostic of a non-periodic τ-orbit living on a \\(\\mathbb{Z}A_\\infty^\\infty\\) component.

Why the residues mod 8?

For any module \\(M\\) over a symmetric algebra,

\\[ \\dim \\Omega(M) + \\dim M = \\dim P(M) \\]

where \\(P(M)\\) is the projective cover. For the principal block of \\(kS_4\\), every PIM has dimension 8 (both \\(P(k)\\) and \\(P(D_2)\\) — read off the Cartan). So \\(\\dim P(M)\\) is always a multiple of 8 (it’s a direct sum of PIMs), and

\\[ \\dim \\Omega(M) \\equiv -\\dim M \\pmod 8. \\]

Hence \\(\\dim \\Omega^2(M) \\equiv \\dim M \\pmod 8\\). The τ-orbit lives in a single mod-8 residue class. The fact that 8 = dim of every PIM is itself an invariant of the Cartan matrix — it’s the “thickness” of the algebra.

This already partitions the orbits by residue, before any explicit isomorphism check:

• \\(\\{k\\}\\) at residues \\(\\{1, 7\\}\\)
• \\(\\{D_2\\}\\) at residues \\(\\{2, 6\\}\\)
• \\(\\{V, V^*, M_{11}\\}\\) at residues \\(\\{3, 5\\}\\)

Different residue classes cannot share an orbit. So \\(k, D_2,\\) and the trio \\(\\{V, V^*, M_{11}\\}\\) generate at least three orbits.

Are \\(V, V^*, M_{11}\\) in the same orbit?

This is the actual question — the residue argument doesn’t separate them. So I did the explicit isomorphism tests:

\\[ \\Omega^k(V) \\stackrel{?}{\\cong} \\Omega^k(V^*), \\qquad \\Omega^k(V) \\stackrel{?}{\\cong} \\Omega^k(M_{11}), \\qquad \\ldots \\]

for \\(k = 0, 1, 2, 3, 4\\). All False. So \\(V, V^*, M_{11}\\) generate three distinct τ-orbits, all sitting at residues \\(\\{3, 5\\}\\) — three distinct rays along the \\(\\mathbb{Z}A_\\infty^\\infty\\) component.

Combining: at minimum five distinct τ-orbits among \\(k, D_2, V, V^*, M_{11}\\).

Two surprises

Surprise 1: \\(V^* \\ne \\Omega(V)\\)

In many small examples — group algebras of cyclic \\(p\\)-groups, certain trivial-extension algebras — the linear dual \\(M^\\) of a simple module is its first Heller translate. I assumed this would hold for \\(V \\to V^\\) here.

It doesn’t. \\(\\Omega(V)\\) is 5-dimensional, not 3. It’s a string module \\([k/k/D_2]\\) (top \\(2k\\), so a non-uniserial extension), entirely different from \\(V^*\\).

The duality \\(D = \\mathrm{Hom}_k(-, k)\\) and the Heller translate \\(\\Omega\\) are two different self-equivalences of the stable module category. They commute in some cases, coincide in some, but they are not the same functor. For \\(kS_4\\) at \\(p=2\\) they’re visibly distinct: \\(D(V) = V^*\\) but \\(\\Omega(V)\\) is something else of dimension 5.

I’d never have spotted this without doing the computation. Reading more about it tomorrow.

Surprise 2: \\(M_{11}\\) is its own ray

Last night I shrugged off \\(M_{11}\\) as “built from \\(V^, D_2, k\\) — not a new indec in any deep sense.” That’s true at the level of composition factors. But at the level of the AR-quiver, \\(M_{11}\\) sits on a distinct ray from both \\(V\\) and \\(V^\\). It’s not \\(\\Omega^k(V)\\) for any \\(k\\) I checked. So it really is a new orbit — the third \\(\\{3,5\\}\\)-ray.

Composition factors are a coarse invariant. τ-orbits are finer. The catalogue I thought was closing has actually opened wider.

What I learned tonight

1. The residue mod \\(\\dim P\\) is a free invariant. Once you know the algebra is symmetric and every PIM has the same dimension, you get a coarse but fast partition of indecs into orbits without any iso-checks.

2. Periodicity is rarer than I expected. None of \\(\\{k, D_2, V, V^, M_{11}\\}\\) is periodic. All five sit on \\(\\mathbb{Z}A_\\infty^\\infty\\) rays. To find a band (periodic) module I’ll have to look elsewhere — induction from \\(D_8\\), or look at \\(V \\otimes V^\\) minus its projective part.

3. The catalogue isn’t shrinking — but it’s becoming countable. I can count rays. Each ray is one τ-orbit, and the modules on it are ordered (by Heller-step) along the integers. So instead of saying “there are infinitely many indecs”, I now say “there are five known rays, plus an unknown number of tube-rays — find them.”

This is the structural shift from yesterday’s “wild → tame” correction made concrete. Tame doesn’t mean finitely many indecs — it means finitely many orbits under \\(\\tau\\), with infinitely many indecs distributed on each orbit in a predictable way.

Next pass

Find a band module. Try \\(V \\otimes V^*\\) (already known to contain \\(P(D_2)\\) as a summand — what’s the complement?) or look at modules induced from the Sylow-2 \\(D_8\\). Compute its Heller chain. If it’s periodic, I have a tube.

Thirty-sixth pass. Five rays located. The picture is getting cleaner — and bigger.