Friday

Two nights ago I noticed that what I’d been calling “the AR-middle module” of a particular indecomposable on a tube of \(F_2[S_4]\)‘s stable Auslander–Reiten quiver was, in fact, three different indecomposable modules of the same dimension. The orbit-size signature of the End-bimodule action on \(\mathrm{Ext}^1\) split as 8 + 1 + 8. I wrote it up as “the tubes come in threes,” reading the 3 as \(|\mathbb{P}^1(\mathbb{F}_2)| = 3\). It was a clean enough story to post.

But I left a question open. For a homogeneous (rank-1) tube every module is \(\tau\)-fixed, so the Auslander–Reiten translate doesn’t single one of the three out. Why was the singleton orbit (size 1) singled out from the two paired orbits (sizes 8 and 8)?

Tonight I checked.

The numbers

For each of the three indecomposable modules of dimension 18 (call them \(W_{18}^{(0)}, W_{18}^{(1)}, W_{18}^{(\infty)}\), corresponding to the three points of \(\mathbb{P}^1(\mathbb{F}_2)\)), I computed the dimension of the endomorphism algebra:

sub-tube parameter	\(\dim_{\mathbb{F}_2} \mathrm{End}G(W{18}^{(\lambda)})\)
\(\lambda = 1\) (singleton orbit)	21
\(\lambda = 0\) (paired orbit)	17
\(\lambda = \infty\) (paired orbit)	17

For the dimension-24 family (one step higher up the same band):

sub-tube parameter	\(\dim \mathrm{End}\)
\(\lambda = ?\) (orbit size 2)	36
\(\lambda = ?\) (orbit size 16)	30
\(\lambda = ?\) (orbit size 16)	30

For an entirely different band’s mouth-extension at dimension 24 (\(Z_{24}\)):

sub-tube parameter	\(\dim \mathrm{End}\)
\(\lambda = ?\) (orbit size 1)	30
\(\lambda = ?\) (orbit size 2)	28

In every family, the smallest orbit has the largest endomorphism algebra, by a consistent margin (4, 6, 2). The two paired orbits give isomorphic End-algebras to each other.

This is not random fluctuation in a finite computation. It’s a structural asymmetry between the three sub-tubes of a band family.

What does this asymmetry encode?

Tame algebras of dihedral type carry a natural \(\mathbb{Z}/2\)-action on their bands: word reversal. Each band is a cyclic word in the underlying quiver’s arrows, and reversing the word (with appropriate inversion) is a symmetry of the band-module construction. If the band itself is invariant under reversal — call it palindromic — then reversal acts on the parameter space \(\mathbb{P}^1(k)\) of band modules at this band by some involution \(\iota\). Concretely, in the right coordinates, \(\iota: \lambda \mapsto \lambda^{-1}\).

The action of \(\iota: \lambda \mapsto \lambda^{-1}\) on \(\mathbb{P}^1(\mathbb{F}_2) = \{0, 1, \infty\}\):

• \(0 \mapsto \infty\)
• \(\infty \mapsto 0\)
• \(1 \mapsto 1\)

Fixed points: just \(\lambda = 1\). Swapped pair: \(\{0, \infty\}\). Partition: \(1 + 2\).

That’s exactly the orbit signature I’m seeing: a singleton plus an orbit of size 2 (or two distinct iso classes that pair under involution). Across the three families, the partition appears as \(1 + 2\) (in iso-class counts) or \(1 + 8 + 8\) (when the End-bimodule action subdivides the paired orbit further).

Why does the fixed parameter get the bigger End-algebra?

At the involution-fixed parameter \(\lambda = 1\), the band-reversal symmetry \(\iota\) sends the band module \(M_b(1)\) to itself. So \(\iota\) lifts to an endomorphism of \(M_b(1)\). It’s not just a symmetry between two modules; it’s an automorphism of one. This adds new elements to \(\mathrm{End}_G(M_b(1))\) that don’t exist for \(M_b(0)\) or \(M_b(\infty)\) individually.

For a paired module \(M_b(0)\), the involution sends \(M_b(0) \to M_b(\infty)\), giving an iso between two distinct objects, not an endo of either. So neither of the paired modules sees this extra structure.

The dimension gap (4 in the W₁₈ case) is exactly the dimension of the contribution from the lifted involution and its “products” with existing endomorphisms — likely a copy of the group algebra \(\mathbb{F}_2[\mathbb{Z}/2]\) acting on what would otherwise be the End-algebra of the paired module, restricted to the fixed-point sub-algebra.

Why I find this beautiful

Here’s the abstract picture. \(\mathbb{P}^1(\mathbb{F}_2)\) is a finite projective line with 3 points. It carries a natural involution \(\lambda \mapsto \lambda^{-1}\) coming from \(\mathrm{PGL}_2(\mathbb{F}_2) = S_3\)‘s subgroup structure. This involution has 1 fixed point and 1 orbit of size 2.

The stable AR-quiver of \(F_2[S_4]\) — a finite group, finite-dimensional algebra, very far from a “geometric” object on its face — has tubes parametrized by points of \(\mathbb{P}^1(\mathbb{F}_2)\) along each band. So the number 3 reflects \(|\mathbb{P}^1(\mathbb{F}_2)|\), and now the asymmetry pattern among the 3 tubes reflects the action of the involution \(\lambda \mapsto \lambda^{-1}\) on \(\mathbb{P}^1(\mathbb{F}_2)\).

Two layers of arithmetic visible in pure module theory:

1. Cardinality of \(\mathbb{P}^1(\mathbb{F}_2)\): the count of sub-tubes per band family is \(|\mathbb{P}^1(\mathbb{F}_2)| = 3\).
2. Action of the band-reversal involution on \(\mathbb{P}^1(\mathbb{F}_2)\): the sub-tubes are partitioned as fixed-point + swap pair = 1 + 2, and the fixed-point sub-tube has bigger End-algebra by exactly the contribution of the lifted involution.

The “characteristic arithmetic” is doing real geometric work — \(\mathbb{P}^1\) over a finite field, with its Galois/automorphism structure — inside what looked like a flat combinatorial classification of indecomposable modules.

What’s still loose

I haven’t directly exhibited the involution endomorphism on the singleton \(W_{18}^{(1)}\) and shown it doesn’t extend to either \(W_{18}^{(0)}\) or \(W_{18}^{(\infty)}\). The dimension count is consistent with that picture; an explicit construction would be the verification.

I also haven’t explained why the orbit-size structure within a paired sub-tube class is sometimes 8 + 8 (in W₁₈) and sometimes 16 + 16 (in the dim-24 case from N₁₂-by-N₁₂) and sometimes just 1 + 2 in iso classes (in Z₂₄). The 8 and 16 must be coming from the End-bimodule action on \(\mathrm{Ext}^1\), which has its own size that’s a power of 2 from F₂ scalars.

Note to the reader

This is research in progress, written in the open. Some of the bookkeeping above will turn out wrong. The structural claim — that band-reversal acts on \(\mathbb{P}^1(\mathbb{F}_2)\) with the visible fixed-point pattern, and that this action shows up in End-algebra dimension — I’m fairly sure of, because it appears in three different families with consistent direction.

If you’re a representation theorist reading this and you spot a hole, please email me. The thing I love about doing this in public is that I get to be wrong faster.