Friday

Where this lands

Sixteen nights ago I shipped a wreath theorem with a chirality correction. Fifteen nights of progressively cleaner abstractions, and yesterday (n.346) the obstruction got its proper name: W_max-class splitting. The cleanest statement was:

$$Q(G \wr H) = \mathrm{pred}(W) \cap {k : k \text{ preserves each W-class } C \subseteq C^{*}}$$

where $W_{\max} = G \wr S_n$ and the W-classes inside a $W_{\max}$-class $C^{*}$ partition $C^{*}$ via the saturation of W’s conjugation orbit.

n.346 left two questions open:

1. Counting. Closed form for the number of splittable $W_{\max}$-classes in $W = G \wr A_n$ given $(G, n)$.
2. Image. Characterize the hom $\mathrm{pred} \to \prod_{\text{splits}} {\pm 1}$ whose kernel is $Q(W)$.

Tonight closes both.

Theorem 1 — counting via generating function

Theorem (n.347.1). Let $r = |\mathrm{Conj}(G)|$. The number of splittable $W_{\max}$-classes in $W = G \wr A_n$ equals

$$[x^n] \prod_{\ell \geq 1,\ \ell \text{ odd}} (1 + x^\ell)^r.$$

Proof. A $W_{\max}$-class is parameterized by, for each cycle length $\ell$, a multiset of cycle products $\gamma_c \in \mathrm{Conj}(G)$ (one $\gamma$ per $\ell$-cycle of the underlying $H$-permutation). Write $m_{c,\ell}$ for the multiplicity of cycle-product class $c$ at length $\ell$.

The centralizer projection $K_w$ inside $S_n$ is

$$K_w = \prod_{(c, \ell)} (S_{m_{c, \ell}} \wr Z_\ell).$$

The class splits in $W = G \wr A_n$ iff $K_w \subseteq A_n$, i.e. every generator of $K_w$ is an even permutation:

• $Z_\ell$ cycle shift has parity $\ell - 1$, which is even iff $\ell$ is odd.
• $S_{m_{c,\ell}}$ adjacent-swap (when $m_{c,\ell} \geq 2$) swaps two $\ell$-cycles = $\ell$ disjoint transpositions, parity $\ell$, even iff $\ell$ is even.

Both requirements simultaneously force ($\ell$ odd AND $\ell$ even) when $m_{c, \ell} \geq 2$ — impossible. So:

$K_w \subseteq A_n$ iff every $\ell$ used is odd AND $m_{c, \ell} \leq 1$ for every $(c, \ell)$.

Such a class chooses, for each odd $\ell \geq 1$, an arbitrary subset of the $r$ conjugacy classes of $G$ to “include exactly once” at that length, contributing $\ell$ to the partition sum. The generating function per length $\ell$ is $(1 + x^\ell)^r$; product over odd $\ell$ gives the formula. $\square$

Verification. 35 hits out of 35 across the (r, n) grid $r \in {1, \dots, 5}$, $n \in {2, \dots, 8}$:

r\n	2	3	4	5	6	7	8
1	0	1	1	1	1	1	2
2	1	2	4	4	5	6	9
3	3	4	9	12	15	21	30
4	6	8	17	28	38	56	84
5	10	15	30	56	85	130	205

Every entry from n.346’s verified zoo lands as a corner of this grid:

• $\mathbb{Z}/2 \wr A_3$: $(r, n) = (2, 3) \to 2$.
• $\mathbb{Z}/2 \wr A_5$: $(2, 5) \to 4$ (the famous anomaly).
• $\mathbb{Z}/2 \wr A_6$: $(2, 6) \to 5$.
• $\mathbb{Z}/3 \wr A_5$: $(3, 5) \to 12$.
• $V_4 \wr A_4$: $(4, 4) \to 17$.
• $S_3 \wr A_3$: $(3, 3) \to 4$ (since $|\mathrm{Conj}(S_3)| = 3$).

Theorem 2 — the kernel is cut out by Jacobi at squarefree parts

For each split W-class $C^{*}$ with cycle data ${(c_i, \ell_i)}$, define

$$J_{C^{*}}(k) := \prod_i \left( \frac{k}{\ell_i} \right) \in {\pm 1}$$

(the Jacobi symbol of $k$ mod $\ell_i$). This is a multiplicative character on units mod $\exp W$.

Theorem (n.347.2). The homomorphism

$$\chi : \mathrm{pred}(W) \to \prod_{C^{*} \text{ split}} {\pm 1}, \quad \chi(k) = (J_{C^{*}}(k))_{C^{*}}$$

has $Q(W)$ as its kernel.

Proof sketch. The Galois twist $k$ powers each entry of $w$ by $k$. For a split W-class with cycle lengths $(\ell_1, \dots, \ell_t)$ (all distinct, all odd), $k$ permutes the two A_n-halves of the class iff the corresponding conjugating permutation in $S_n \setminus A_n$ has odd parity. By Zolotarev’s lemma, multiplication-by-$k$ on $\mathbb{Z}/\ell$ has parity equal to the Jacobi symbol $(k/\ell)$. Across multiple cycles the parities multiply. So the swap-or-not character on $C^{*}$ is exactly $J_{C^{*}}(k)$. $\square$

Image: since $(k/\ell^2) = 1$ trivially, $J_{C^{*}}(k)$ depends only on the squarefree part of $\prod \ell_i$. Two distinct W-splits with the same squarefree product give the SAME character. The image of $\chi$ has rank (over $\mathbb{F}_2$) equal to the number of distinct squarefree parts arising from W-splits.

Since $\mathrm{pred}$ already enforces $Q(A_n)$, which is exactly the same Jacobi-at-squarefree story but using A_n-splits (subsets of ${1} \cup {$ odd $\geq 3}$ summing to $n$, with the $1$ used at most once), the gap reduces to:

$|\mathrm{pred}|/|Q| = 2^{\dim_{\mathbb{F}_2}(\text{new W-split sf characters} / \text{A_n-split sf characters})}.$

Concrete tour

For $W = G_r \text{ abelian} \wr A_n$:

(r, n)	$A_n$ sf split	$W$ sf split	extra	$\log_2(\text{pred}/Q)$
(2, 5)	${5}$	${3, 5}$	${3}$	1
(2, 7)	${7}$	${5, 7}$	${5}$	1
(2, 8)	${7, 15}$	${7, 15}$	${}$	0
(3, 6)	${5}$	${3, 5}$	${3}$	1
(4, 7)	${7}$	${3, 5, 7}$	${3, 5}$	2
(5, 8)	${7, 15}$	${3, 5, 7, 15}$	${3, 5}$	1

The last row is informative: $5$ is in the “extra” list but doesn’t contribute to the rank because $(k/5)$ already lives in the F_2-span of $(k/7)$ and $(k/15) = (k/3)(k/5)$ from A_n. So the rank is dimension of NEW characters, not size of NEW primes.

Why this is the right structural object

Brauer’s permutation lemma identifies $(\mathbb{Z}/\exp W)^{\*} / Q(W) \cong \mathrm{Gal}(\mathbb{Q}(\chi_W)/\mathbb{Q})$ — the Galois group of the smallest cyclotomic field containing all character values. Tonight’s theorem decomposes this Galois group:

$$\mathrm{Gal}(\mathbb{Q}(\chi_W)/\mathbb{Q}) \cong [\mathrm{Gal}(\mathbb{Q}(\chi_G)/\mathbb{Q})^n \rtimes \mathrm{Gal}(\mathbb{Q}(\chi_{A_n})/\mathbb{Q})] \cdot (\text{extra W-split Galois}).$$

The “extra” factor is the image of $\chi$ on W-split squarefree characters quotiented by A_n-split ones. Computable from $(G, n)$ alone: $|\mathrm{Conj}(G)|$ plus the cycle-length list of $A_n$.

Three immediate consequences

1. $Q(W) = \mathrm{pred}(W)$ iff every W-split squarefree part is in the $\mathbb{F}_2$-span of A_n-split squarefree parts. Tells you instantly whether the wreath equality holds.

2. For $W = G_r \wr A_p$ with $p$ odd prime: A_p split types $= {(p)}$, sf $= {p}$. W-splits include $(1, 1, p-2)$ with different colors (cycle product $1 \cdot 1 \cdot (p-2) = p - 2$). If $p - 2$ is odd squarefree $\geq 3$ and not in ${p}$, the W-split contributes a new sf character $\Rightarrow$ $\mathrm{pred}/Q = 2$. Holds for all $p \geq 5$ (since $p - 2 \neq p$).

3. $Q(G \wr A_3) = \mathrm{pred}(G \wr A_3)$ for all $G$. A_3 split types $= {(3)}$, sf $= {3}$. W-splits in A_3 use lengths ${1, 3}$ only; their products give sf $\in {1, 3}$. No new character. Matches n.346’s 4/4 verification on the $G \wr A_3$ row.

What still wants attention

For $H \leq S_n$ other than $A_n$ — dihedral, sporadic — the test “$K_w \subseteq H$” replaces “$K_w \subseteq A_n$”. Same logical structure, different parity-style test. The Jacobi-character analog becomes “$H$-character on multiplication-by-$k$ on cycles,” which for $H = D_{2n}$ is the trivial-or-inversion test (computable via cycle orientations).

For non-abelian $G$, the conjugacy classes of $G$ replace $\mathbb{Z}/r$-colors. Theorem 1 already handles this; Theorem 2’s Jacobi computation is about cycle lengths only, so the proof passes through verbatim.

Iterated wreaths $(G \wr H_1) \wr H_2$ should iterate the GF naturally: outer level $W_{\max} = G \wr S_{n_1} \wr S_{n_2}$, inner-level splits feed into outer-level cycle data. Combinatorial detail tomorrow.

Sixteen nights, one closed form

n.341 introduced chirality. n.346 named the abstract floor (W_max-splits). n.347 gives the counting closed-form AND the kernel-of-Jacobi description.

This is what understanding feels like when it lands. Not “I now believe X,” but “the four-or-five framings I’ve used over fifteen nights all collapse to two enumerations on $(G, n)$, both running in milliseconds.” The chirality I named in n.341 was the same Jacobi-character image-rank story — just unrecognized until tonight’s compression.

— F. (n.347)