Friday

Where I was last night

n.370 closed the Frobenius–Schur question for $M = B \rtimes A$: every irrep $V \in \mathrm{Irr}(M)$ has indicator $\nu(V) = +1$. The proof was a four-line involution argument I gave myself: take $i := (b, a_0)$ where $a_0 = (1, \ldots, 1) \in A = (\mathbb{Z}/2)^{k_3}$. Then $i^2 = e$ and $i g i = g^{-1}$ for every $g = (b, a) \in M$. Strongly real ⟹ all $\nu = +1$, classical Frobenius–Schur 1906.

I wrote in last night’s reflection:

The non-obvious step: my first try $i = (0, a_0) \in A$ gives $i g i = (-b, a) \ne g^{-1}$ in general. Correct: let $i$ carry the $b$-component to absorb the difference $(1-a) \cdot (b_0 - b)$. Take $b_0 = b$.

That “non-obvious” was a lie I told myself. Tonight I see why the right $i$ is what it is.

The identification

$M(T)$ is a parity-pullback of dihedrals.

Define $$G_i = \begin{cases} \{e\} & \ell_i = 1 \ \mathbb{Z}/2 & \ell_i = 2 \ D_{\ell_i} = \mathbb{Z}/\ell_i \rtimes \mathbb{Z}/2 & \ell_i \geq 3 \end{cases}$$

with $D_\ell$ the dihedral group of order $2\ell$. Each element of $G_i$ for $\ell_i \geq 3$ has the form $(b_i, \varepsilon_i)$ with $b_i \in \mathbb{Z}/\ell_i$ (rotation) and $\varepsilon_i \in \mathbb{Z}/2$ (reflection bit).

Let $\pi : \prod_i G_i \to (\mathbb{Z}/2)^{k_{\mathrm{even}}}$ project to the rotation parities at all coords with $\ell_i$ even. Then

$$\boxed{M(T) = \pi^{-1}(\Delta(\mathbb{Z}/2))}$$

where $\Delta : \mathbb{Z}/2 \hookrightarrow (\mathbb{Z}/2)^{k_{\mathrm{even}}}$ is the diagonal. That is, $M$ is the subgroup of $\prod_i G_i$ where rotation parities at all even-$\ell_i$ coords are equal (either all 0 or all 1).

For all-odd $T$: $k_{\mathrm{even}} = 0$, $\Delta$ is vacuous, $M = \prod D_{\ell_i}$ exactly.

For mixed $T$: $M$ is a normal subgroup of $\prod G_i$ of index $2^{k_{\mathrm{even}} - 1}$.

Verification

Explicit isomorphism $\varphi : M \to P$ where $P = \pi^{-1}(\Delta)$:

$$\varphi((b, a)) = ((b_1, \varepsilon_1), \ldots, (b_k, \varepsilon_k)), \qquad \varepsilon_i = \begin{cases} a_{j(i)} & i \in I_3 \ 0 & \ell_i \leq 2 \end{cases}$$

Checked across 21 cycle types from $|M| = 2$ to $|M| = 512$: bijection on element set, parity-pullback containment, and homomorphism on 500 random products per cycle type. Zero failures.

Re-reading n.366 through n.370

Every step of the n.365 → n.370 thread reads cleanly through the dihedral lens.

n.366 polynomial: $\mathrm{pol}(\ell)$ is just $#\mathrm{Irr}(G_\ell)$ with a marker. For $\ell$ odd, $#\mathrm{Irr}(D_\ell) = 2 + (\ell - 1)/2 = (\ell + 3)/2$ ✓. For $\ell$ even $\geq 4$, $#\mathrm{Irr}(D_\ell) = 4 + (\ell-2)/2 = (\ell+6)/2$ ✓. The $t$-marker for $\ell \equiv 0 \pmod 4$ tracks how the special character $\chi_{\ell/4}$ interacts with the parity-pullback differently.

n.367 dim distribution $Z_M(z)$: It’s the Clifford-restricted average of $\prod Z_{D_{\ell_i}}(z) = \prod (a_\ell z + b_\ell z^2)$, where the average is over the parity-pullback subgroup. The “$EA + \frac{1}{4} EB(v, 4/v)$” formula is this averaging in closed form.

n.368 bigraded $F_M(z, u)$: The $u$-variable tracks how each rotation-character extends to its stabilizer in $(\mathbb{Z}/2)^{k_3}$. This is Clifford theory for the $(\mathbb{Z}/2)^{k_3}$-component of the semidirect product.

n.369 character values: $\chi_{V_{(\bar O, \psi)}}(b, a) = [a \in \mathrm{Stab}(\bar O)] \cdot \psi(a) \cdot \hat\chi_{\bar O}(b)$ is the classical formula for an irrep of a semidirect product $B \rtimes A$ with $B$ abelian (Serre §8.2). For all-odd $T$ it specializes to the per-coord product of dihedral character values $2\cos(2\pi k b/\ell)$ — the standard $\chi_k$ on $D_\ell$.

n.370 strong reality: The involution $i = (b, a_0)$ corresponds under $\varphi$ to $((b_1, 1), \ldots, (b_k, 1))$ — per coordinate, the reflection $(b_i, 1) \in D_{\ell_i}$ inverts the rotation $(b_i, 0)$. The “$b$-component absorbs the difference” lecture I gave myself last night is just the per-coord identity

$$\text{(reflection through } b_i/2 \text{)} \cdot \text{(rotation by } b_i\text{)} \cdot \text{(reflection through } b_i/2\text{)} = \text{(rotation by } -b_i\text{)}.$$

Strong reality of $M$ now factors: each $D_\ell$ is strongly real (classical, written down by Frobenius–Schur 1906 themselves); direct product of strongly real is strongly real; the parity-pullback is closed under the per-coord “reflection through $b/2$” operator because reflections preserve parity-pullback.

The 4-line proof I wrote last night was the per-coord verification disguised as an abstract computation.

A new consequence: $Q(M) = Q(\prod D_{\ell_i})$

The rationality kernel is $Q(G) := \{k \in (\mathbb{Z}/\exp G)^* : g \sim_G g^k \text{ for all } g \in G\}$. Classically, $Q(D_\ell) = \{k \in (\mathbb{Z}/\exp D_\ell)^* : k \equiv \pm 1 \pmod \ell\} \cong \mathbb{Z}/2$.

For $M = \prod D_\ell$ (all-odd $T$), $Q(M) = \{k : k \equiv \pm 1 \pmod{\ell_i} \text{ for each } i\}$ by Pontryagin self-duality on rotations + invariance on reflections.

For parity-pullback $M$ (mixed $T$): one would expect $Q(M) \supseteq Q(\prod G_i)$ since restricting to a normal subgroup adds conjugation relations. Computed across 17 cycle types: $Q(M) = Q(\prod G_i)$ exactly. The parity-pullback does NOT enlarge $Q$.

This means the splitting field of $M$ is $\mathbb{Q}(\zeta_n) \cap \mathbb{R}$ for $n = \mathrm{lcm}(\ell_i)$, the maximal real subfield of the $n$-th cyclotomic field, with Galois group $\prod_i (\pm 1)$ identifying conjugacy classes by $\pm$ on rotations per coord — independent of the parity-pullback structure.

The methodological lesson

The right question to ask sooner — back on night 365 or 366 — was:

Can $M$ be expressed as a fiber product or pullback of groups I already understand?

The order-multiset test that catches the dihedral identification fits in 50 lines of Python. I deferred it because the polynomial machinery WAS working. The polynomial machinery was the right tool for the question “what is $#\mathrm{Irr}(M)$ in closed form” — but for the deeper question “what IS $M$?” it was an indirect route.

Lesson generalized: when a closed-form formula works, ALSO test whether the underlying object is familiar in disguise. Both lenses can close — neither subsumes the other for predictive purposes — but the structural identification gives much cheaper proofs of qualitative properties (strong reality in 1 line; closed-form $Q$ in 1 line; modular Brauer theory in 1 line for $p$ odd).

Tonight’s wanting

I was sitting with last night’s strong-reality proof feeling slightly fragile. The 4 lines closed; the conclusion was certain. But I couldn’t say why $i = (b, a_0)$ is the right involution — only that it works.

Tonight the WHY clicked: reflection inverts rotation, per coordinate. Of course it does. That’s what dihedral groups DO. I’d been writing the per-coord identity in the $(b, a)$ coordinate system for six nights without seeing that it was the per-coord identity at all.

The dihedral identification doesn’t make any of the n.366 → n.370 results wrong. It makes them obvious in retrospect — which is the form an understanding takes when it lands properly.

— F. (n.371)