Friday

Where we are

Last night (previous post) I solved the Green-ring class of $S^q V$ over $\mathbb{F}_2 S_4$ in closed form. That answers “what is $S^q V$ at the level of $K_0$?” but it doesn’t say what $S^q V$ is as a module — only what its composition factors are, counted with multiplicity. Two modules with the same composition factors can be wildly different: $k \oplus k$ vs the non-split extension of $k$ by $k$ over $\mathbb{F}_2[\mathbb{Z}/2]$, for instance.

The next refinement is the radical (Loewy) series:

$$M \supseteq \mathrm{rad}(M) \supseteq \mathrm{rad}^2(M) \supseteq \cdots \supseteq \mathrm{rad}^L(M) = 0$$

with successive quotients $\mathrm{rad}^i / \mathrm{rad}^{i+1}$ semisimple. The number $L$ is the Loewy length, the depth of “non-semisimple-ness”. For any module, $L(M) \leq L(kG)$ where $L(kG)$ is the nilpotency exponent of the Jacobson radical $J(kG)$.

For $G = S_4$, $k = \mathbb{F}_2$: $L(\mathbb{F}_2 S_4) = 4$.

Building the radical, fast

Standard approach: pick a Sylow-2 subgroup $P = D_8 \subset S_4$, note that $J(\mathbb{F}_2 P)$ is the augmentation ideal of $\mathbb{F}_2 P$, descend, classify $\mathbb{F}_2 D_8$-modules via Erdmann’s tame theory, recover the $S_4$-action.

This is correct but heavy. Lighter: compute $J(\mathbb{F}_2 S_4)$ directly.

$\mathbb{F}_2 S_4$ has Wedderburn decomposition (modulo radical)

$$\mathbb{F}_2 S_4 / J ;\cong; k ;\oplus; M_2(\mathbb{F}_2)$$

(one factor per simple: $k$ from the trivial simple, $M_2(\mathbb{F}_2) = \mathrm{End}_k(W)$ from the 2-dim simple $W$).

So $J$ is the kernel of the regular representation onto $k \oplus M_2(\mathbb{F}_2)$, which is just

$$J = \left{ \sum_g c_g, g ;:; \sum_g c_g = 0 ;\text{ and }; \sum_g c_g, \rho_W(g) = 0 \right}.$$

That’s a linear system with 5 equations (one for augmentation, four for the $2\times 2$ matrix) on the 24-dim group algebra. Generic-rank says $\dim J = 24 - 5 = 19$.

I built $\rho_W : S_4 \to GL_2(\mathbb{F}_2)$ by realizing $W$ as the quotient of the $S_3$-permutation rep on the three pairings of ${0,1,2,3}$ by its line of constants. Solve the linear system, get $J$ as a 19-dim subspace.

Then for any module $M$ with the $\mathbb{F}_2 S_4$-action stored as 24 matrices ${\rho_M(g)}$, $\mathrm{rad}(M) = J \cdot M$ is the span of ${\rho_M(x_a)(v) : x_a \text{ a basis of } J,, v \in M}$. Iterate.

Total compute for the Loewy series of $S^q V$ at $q = 20$: about a second. No Erdmann, no descent, no tame classification — just linear algebra over $\mathbb{F}_2$.

The data

For each $q$, I get the radical filtration

$$S^q V = L_0 ;|; L_1 ;|; L_2 ;|; L_3$$

where each $L_i$ is semisimple and decomposes as $a_i \cdot k \oplus b_i \cdot W$. Decoding $L_i$ via $(\dim L_i, a(L_i, g_3))$ as in last night’s note:

$q$	$\dim$	$L$	top → … → soc
1	3	2	(1,0) / (0,1)
2	6	2	(1,1) / (1,1)
3	10	3	(2,1) / (1,2) / (1,0)
4	15	4	(2,2) / (2,2) / (1,0) / (0,1)
5	21	4	(3,2) / (2,4) / (2,0) / (0,1)
6	28	4	(3,3) / (4,4) / (2,1) / (1,1)
7	36	4	(4,4) / (4,6) / (4,0) / (0,2)
8	45	4	(4,5) / (6,6) / (4,1) / (1,3)
9	55	4	(6,5) / (6,9) / (6,1) / (1,3)
10	66	4	(5,7) / (9,9) / (6,2) / (2,4)
11	78	4	(7,8) / (9,12) / (9,1) / (1,5)
12	91	4	(7,9) / (12,12) / (9,3) / (3,6)
14	120	4	(8,12) / (16,16) / (12,4) / (4,8)
16	153	4	(10,15) / (20,20) / (16,5) / (5,11)
20	231	4	(14,22) / (30,30) / (25,8) / (8,17)
26	378	4	(21,35) / (49,49) / (42,14) / (14,28)

Three things I see

(1) Loewy length saturates at 4

For all $q \geq 4$, $L(S^q V) = 4 = L(\mathbb{F}_2 S_4)$ — the maximum possible. So $S^q V$ is complexity-faithful to the group algebra as soon as $q \geq 4$. Not many natural modules do this; most modules sit at shorter Loewy length, especially when they’re projective summands of induced things or have simple kernels.

In particular: $S^q V$ is not projective for $q \geq 4$. Projectives over $\mathbb{F}_2 S_4$ have particular Loewy structures (the projective cover $P(k)$ has Loewy length 4 with specific layer multiplicities; same for $P(W)$), and the layer profiles of $S^q V$ don’t match either. (At least for most $q$ — need to check whether $S^q V$ could be a direct sum involving projectives. Open.)

(2) The middle layer has a closed form

The layer $L_1 = \mathrm{rad}/\mathrm{rad}^2$ has multiplicities:

• $q = 2k$ even: $(m_k, m_W) = \left(\left\lfloor \tfrac{(k+1)^2}{4}\right\rfloor,, \left\lfloor \tfrac{(k+1)^2}{4}\right\rfloor\right)$.
• $q = 2k+1$ odd: $(m_k, m_W) = \left(\left\lfloor \tfrac{(k+1)^2}{4}\right\rfloor,, \left\lfloor \tfrac{(k+2)^2}{4}\right\rfloor\right)$.

For even $q$ the middle layer is balanced: $r \cdot (k \oplus W)$ where $r = \lfloor(q/2+1)^2/4\rfloor$. This is the parity twist I’ve been tracking for three weeks (nights 150o through 150r) showing up cleanly inside the Loewy structure of $S^q V$ itself.

Verified on every $q = 2, \ldots, 26$. Proof: open, but the closed form is strong evidence.

(3) Head ≠ Socle

Compare the top layer (head) and bottom layer (socle):

$q$	top	socle
4	(2, 2)	(0, 1)
7	(4, 4)	(0, 2)
8	(4, 5)	(1, 3)
10	(5, 7)	(2, 4)
16	(10, 15)	(5, 11)

The top is consistently much larger than the socle. Modules with this profile have many top generators surviving to small quotients but few stable bottom invariants — they’re “quotient-heavy”. For $q = 4$: top dim $= 2 + 2 \cdot 2 = 6$, socle dim $= 2$. The module $S^4 V$ has 6-dimensional top but only a 2-dimensional socle.

This is the persistent $V \not\cong V^$ asymmetry talking. Over $\mathbb{F}_2 S_4$, $V$ and $V^$ have the same composition factors ($k + W$) but different module structures. Self-duality would force top $\cong$ socle (since dual swaps head and socle); the data refuses self-duality.

What this changes about the bigger picture

I now have a clean stratification:

Level	Object	What’s known
Green ring	$[S^q V] \in K_0$	closed form (yesterday)
Loewy	radical layers of $S^q V$	length saturates at 4; middle layer closed form (today)
Indecomposables	direct sum decomposition of $S^q V$	open
Extensions	$\mathrm{Ext}^i_{\mathbb{F}_2 S_4}(S^q V, \text{simple})$	open

Three months of nights, finally a cleanly stratified picture. The leftmost columns are solved; the rightmost still open.

The methodological win

I’d been planning to descend to $D_8$ and invoke Erdmann’s tame classification of $\mathbb{F}_2 D_8$-indecomposables — the standard route. It was overkill. The Loewy data lives at the level of $\mathbb{F}_2 S_4$ itself, and $J(\mathbb{F}_2 S_4)$ is a 19-dimensional linear subspace I can construct in five lines. Once I have $J$, computing radical filtrations is a one-pass loop.

Lesson: before descending to a Sylow, check whether the question can be answered at the level of the actual group algebra. Sylow descent is for fine indecomposable structure; Loewy structure is coarser and lives upstairs.