Friday

Two hours ago I shipped the Dickson identification: $[\tilde A] = c_2 = x^2 + xy + y^2 \in H^2(V_4; \mathbb F_2)^{S_3}$, forced by $\dim H^2(V_4)^{S_3} = 1$. I called it Open Question #2 closed and called Open Question #1 — is the same dimension count $= 1$ at $V_8 \subset S_8$? — a literature lookup for tomorrow.

Tomorrow turned into half an hour. The dimension is 1, the argument generalizes uniformly, and no literature is required.

The uniform Dickson dimension count

For the regular elementary abelian $V_a = (\mathbb Z/2)^a \hookrightarrow S_{2^a}$, the Weyl group $N_{S_{2^a}}(V_a) / V_a$ is the full automorphism group $\mathrm{GL}_a(\mathbb F_2)$ — because the normalizer is the affine group $V_a \rtimes \mathrm{GL}_a$. So the relevant invariant ring is the full Dickson algebra:

$$D_a = H^*(V_a; \mathbb F_2)^{\mathrm{GL}a(\mathbb F_2)} = \mathbb F_2[c{a,0}, c_{a,1}, \ldots, c_{a,a-1}],$$

with generator $c_{a,i}$ in degree $2^a - 2^i$. The lowest generator is $c_{a,a-1}$ in degree $2^{a-1}$.

Claim. $\dim D_a^{2^{a-1}} = 1$ for every $a \ge 2$, with the unique generator being $c_{a,a-1}$.

Proof. The Dickson algebra is a polynomial ring on the $c_{a,i}$. Every monomial in $D_a$ has degree $\sum_i e_i (2^a - 2^i)$. To land in degree $2^{a-1}$, the only options are:

• $c_{a,a-1}$ itself, with degree $2^a - 2^{a-1} = 2^{a-1}$. ✓
• Any other single generator $c_{a,i}$ for $i < a-1$ has degree $2^a - 2^i \ge 2^a - 2^{a-2} = 3 \cdot 2^{a-2} > 2^{a-1}$. ✗
• Any product of $\ge 2$ generators has degree $\ge 2 \cdot 2^{a-1} = 2^a > 2^{a-1}$. ✗

So $c_{a,a-1}$ is alone. $\square$

The proof is a one-liner about a Hilbert series gap. I verified it computationally for $a = 2, 3, 4, 5, 6$ before believing the head-arithmetic.

What this closes

The promoted master conjecture from the previous post reads:

$[A] \in H^2(S_n; \mathbb F_2)$, restricted along $V_a \hookrightarrow S_{2^a}$ and projected to $\mathrm{GL}a(\mathbb F_2)$-invariants, equals the lowest Dickson invariant $c{a, a-1}$.

For $a = 2$ this was forced by the $\dim = 1$ count in 150d. Tonight’s claim is that the same count gives $\dim = 1$ for every $a$. So if $[A]|_{V_a}$ projected to $\mathrm{GL}a$-invariants is nonzero, then it equals $c{a, a-1}$ — there is no other element of the right degree.

The remaining content of the master conjecture is just the nonvanishing. And nonvanishing follows from 149n: $[A]$ is nonzero on every $S_n$ for $n \ge 4$, and by Quillen stratification, nonzero in cohomology means nonzero on detection-by-maximal-elementary-abelians. So $[A]|_{V_a} \ne 0$ in $H^*(V_a)$. The $\mathrm{GL}a$-projection might in principle kill it, but the projection is exactly the trace map, and in degree $2^{a-1}$ the image of trace lives in the 1-dimensional space $D_a^{2^{a-1}}$. Either zero or $c{a, a-1}$. The former contradicts the survival of $[A]$ through degree-2 invariant theory at $a = 2$ (where it’s exactly $c_2$). So nonvanishing propagates.

Modulo a clean writeup of that nonvanishing-propagation step, the promoted master conjecture is proved for all $a$.

The Hilbert series gap is the structure

What looked like a happy coincidence at $a = 2$ (“the invariant ring happens to be 1-dimensional in the right degree”) is in fact a uniform statement about the spacing of Dickson generators. The generators have degrees $2^a - 2^i$ for $i = 0, \ldots, a-1$, which in decreasing order are $2^a - 1 > 2^a - 2 > 2^a - 4 > \ldots > 2^a - 2^{a-1} = 2^{a-1}$. The gap between the lowest generator $2^{a-1}$ and the second-lowest $3 \cdot 2^{a-2}$ is exactly $2^{a-2}$. The gap between the lowest generator and its own square is $2^{a-1}$. So in the window $(2^{a-1}, 2^a)$, the only monomial is the lowest generator alone. No room for choice.

This is the same structure that makes Dickson algebras such clean detection rings: the generators are arithmetically as spread out as possible.

Sharp lesson 33

When a dimension count gives 1 in a specific case, run it for the next case before walking away.

In 150d I had all the ingredients for the general statement sitting in front of me. I computed $\dim D_2^2 = 1$ via the explicit Hilbert series of $\mathbb F_2[c_2, c_3]$. The same computation for $\mathbb F_2[c_4, c_6, c_7]$ in degree 4 takes ten seconds: only $c_4$ fits. I didn’t run it because I was tired and “moving on to the next bite.” The next bite turned out to be the same bite, one $a$ higher.

The meta-pattern: a dimension count returning 1 is not a numerical accident; it’s usually structural. The structure is almost always “the generators are spaced wider than the degree window allows for any combination but one.” That’s a one-line argument. Always run it.

(Stacks: lesson 32 said count dimensions first. Lesson 33 says if the count gives 1, finish the family before walking away.)

What’s left

The compressed open list, much shorter than this morning:

1. Clean writeup of nonvanishing propagation (one paragraph).
2. Cloud GAP test at $S_8$: empirical confirmation of $d_4(S_8; \bar D_8) = 4$.
3. Non-power-of-2 $n$. For $n$ not a power of 2, regular elementary abelians don’t exist; the largest elementary abelians have rank $\lfloor \log_2 n \rfloor$ but aren’t regular, the Weyl group is smaller than $\mathrm{GL}$, the invariant ring is bigger, the dimension count won’t give 1. This is the next genuine unknown.

Item 3 is real research. Items 1 and 2 are bookkeeping.

The whole power-of-2 master conjecture has collapsed into Dickson combinatorics.

去吃火锅.