Variance coefficients as hook multiplicities

Definitions and mathematical statement

Jordan Ellenberg's AIM question asks for a geometric interpretation of a variance expression printed as $\operatorname{variance}\Lambda(n)\sim q^{?}\,n$ as $q\to\infty$. The supplied report says the exponent question mark is present in the official source, not merely an OCR loss. The averaging ensemble, normalization, and meaning of $\Lambda(n)$ are unspecified, so the exponent cannot be silently filled in.

The report proposes a common representation-theoretic synthesis for two interpretations. For monic $f\in{\mathbb{F}}_q[T]$, define $\Lambda(f)=\deg P$ if $f=P^a$ for a monic irreducible $P$, and zero otherwise. On the squarefree degree-$n$ configuration space this is $n$ for irreducible polynomials and zero otherwise. Let $V_{\mathrm{std}}$ be the $(n-1)$-dimensional standard representation of $S_n$ and put \[ L_n=\lambda_{-1}(V_{\mathrm{std}}) =\sum_{j=0}^{n-1}(-1)^j\bigwedge^jV_{\mathrm{std}}. \] The report asserts that its virtual character is $n$ on $n$-cycles and zero on other conjugacy classes, hence realizes $\Lambda$ on squarefree Frobenius configurations. The summands correspond to the hooks $(n-j,1^j)$.

The claimed raw and centered global leading coefficients are respectively \[ \frac1{n!}\sum_{\sigma\in S_n}\chi_{L_n}(\sigma)^2=n,\qquad \frac1{n!}\sum_{\sigma\in S_n}(\chi_{L_n}(\sigma)-1)^2=n-1. \] The report interprets these as top-cohomology invariant multiplicities. In an unnormalized global polynomial sum the associated leading terms are $nq^n$ and $(n-1)q^n$; averaging over $q^n$ monic polynomials removes that factor. Thus a raw second moment must not be identified with a centered variance or with its normalized version.

For the short-interval interpretation, intervals are naturally of the form $I(A;h)=\{f\text{ monic of degree }n:\deg(f-A)\leq h\}$, of size $q^{h+1}$ for $0\leq h<n$. Pairing $L_n\otimes L_n$ with the Hast--Matei top-weight cohomology for pairs of nearby polynomials is said to retain exactly the hooks indexed by \[ h+2\leq j\leq n-1, \] giving coefficient $n-h-2$ in the Keating--Rudnick short-interval variance. The summary does not state the precise admissible $(n,h)$ range, characteristic conditions, or variance normalization, so no complete asymptotic theorem with a fixed power of $q$ is supplied for this interpretation.

The candidate synthesis therefore distinguishes three truncations of the same virtual hook object: all hooks give $n$, all nontrivial hooks give $n-1$, and the stated short-interval hooks give $n-h-2$. It does not uniquely determine which expression Ellenberg intended by the incomplete original variance notation.

Short English statement

The original variance expression leaves its exponent and normalization unspecified. The report proposes a geometric hook-representation explanation for three distinct coefficients: a raw moment, a centered global variance, and a short-interval variance.

Sources

• AIM workshop "Arithmetic statistics over finite fields and function fields," Question 23, attributed to Jordan Ellenberg: https://aimath.org/pastworkshops/arithstatffieldproblems.pdf. No PDF was accessed here.
• Local canonical registry aim-algebraic-number-theory-notes.json, notes[52], tag question; permanent UnsolvedMath ID $20000401$, source code catalogue entry. Original asks to describe geometrically the right side of a variance relation with $q^{?}$. Original remarks and literature field are empty.
• Supplied research attempt 1 invokes Hast--Matei top-weight cohomology and Keating--Rudnick short-interval variance, and proposes the virtual-hook synthesis. No full authoritative bibliographic references or complete asymptotic hypotheses are supplied. The report is machine generated, not peer reviewed, and requires independent verification before citation.