Low-Moment Detection of Reductive Subgroups

Definitions and mathematical statement

Conjecture 0.1, with subgroup notation separated. Let $H$ be a reductive subgroup of a classical group $G$ acting on $V$. Must either $H\supseteq[G,G]$, where $[G,G]$ is the derived subgroup, or\[M_{2k}(H,V)>M_{2k}(G,V)\quad\text{for some integer }1\le k\le4?\] The extraction uses the same letter for both groups; $H\le G$ resolves that collision without changing the alternatives. In the usual complex representation-theoretic convention, $M_{2k}(H,V)=\dim_{{\mathbb{C}}}\operatorname{End}_H(V^{\otimes k})=\dim_{{\mathbb{C}}}(V^{\otimes k}\otimes(V^*)^{\otimes k})^H$. For a compact form this is the normalized Haar integral of $|\operatorname{tr}_V(h)|^{2k}$. The excerpt does not itself define the moments or all conjectural dimension and field hypotheses, so this standard convention needs checking. Under it, $\operatorname{End}_G(V^{\otimes k})\subseteq\operatorname{End}_H(V^{\otimes k})$; the question asks whether equality through these low moments forces containment of $[G,G]$. It is not simply equality of one unspecified scalar character value.

Geometric applications retained from the source. For a smooth projective variety $X$ of dimension $n+1\ge1$ over a finite field $k$, the source considers the local system $\mathcal F_d$ over smooth degree-$d$ hypersurface sections and its Zariski-closed monodromy group $G_d$. Given such a hypersurface transverse to $X$, let $V$ be the vanishing-cycle subspace of $H^n((X\otimes_k\bar k)\cap H,{\mathbb{Q}}_\ell)$, with the usual $\ell\ne\operatorname{char}k$. Cup product gives a nondegenerate invariant bilinear form. The source attributes to Deligne [D, 4.4] the conclusion $G_d=\operatorname{Sp}(V)$ for odd $n$, and $G_d=\operatorname{O}(V)$ or finite $G_d$ for even $n\ge2$. It seeks additional hypotheses excluding the finite possibility. Katz [Ka1] is reported to give $G_d\le G=\operatorname{Sp}(V)$ or $\operatorname{O}(V)$ with $M_4(G_d,V)=M_4(G,V)$. The source also cites [Ka2] for geometric monodromy of families of character sums, with Larsen's conjecture and drop-ratio conjectures from [Ka2, Chapter 2] and its missing Theorem 5 playing a role. None of the omitted hypotheses or drop-ratio assertions is reconstructed.

For a finite $H\le\operatorname{O}(V)$ with $M_{2k}(H,V)=M_{2k}(G,V)$ for some $k\ge2$, the source further reports that symmetrized $H$-orbits are spherical $2k$-designs [LST], and that the orbit of every subspace $0\ne U<V$ is a Grassmannian $2k$-design in the convention of [BCN]. In the intended real Euclidean realization, a spherical design averages polynomial restrictions of degree at most the stated strength as does uniform sphere measure. If $H$ stabilizes an integral lattice $\Lambda$, the source says $\Lambda$ is strongly perfect in Venkov's sense, meaning its minimal vectors form a spherical $4$-design after normalization. The real-form, normalization, and Grassmannian conventions needed for these application claims are not supplied and are not promoted to unrestricted complex statements.

Truncated theorem and status. The excerpt begins Theorem 1 with $V={\mathbb{C}}^d$, $d\ge5$, $G=\operatorname{GL}(V),\operatorname{Sp}(V)$, or $\operatorname{O}(V)$, and $H\le G$ closed with reductive identity component $H^\circ$. It says one of several alternatives holds and gives only (i), $M_8(H,V)>M_8(G,V)$. The missing alternatives prevent treating this as a proof that (i) always holds or that Conjecture 0.1 is settled. The supplied triage checked 2026-08-21 labels the record open but says the incomplete extraction cannot reliably establish status; the later verification field reports confirming the conjecture's alternative and bound. Neither is independent current-status verification here.

Short English statement

Can moments through order eight distinguish a reductive subgroup from a classical group unless it contains the derived group? The subgroup notation is repaired, and the source's monodromy and design applications are retained with their missing hypotheses. A theorem cut off after its first alternative does not settle the conjecture.

Sources

• Local UnsolvedMath registry, permanent ID 30000165, code catalogue entry; Low-Moment Detection of Reductive Subgroups, Oberwolfach Reports Open Problems; source Conjecture 0.1, with Larsen's conjecture named in the applications, and truncated Theorem 1.
• Gitter und Anwendungen (2005), Oberwolfach Reports, DOI 10.4171/owr/2005/01, pp. 5--67. https://doi.org/10.4171/owr/2005/01. Proposed year 2005; no proposer field supplied.
• Original labels [Ka1], Katz's monodromy and fourth-moment application; [D, (4.2.4)] for vanishing cycles; [D, 4.4] for Deligne's monodromy conclusions; [Ka2], especially Chapter 2, and source Theorem 5 for the character-sum and drop-ratio discussion. Their bibliographic entries and Theorem 5 are absent.
• Original labels [LST] for spherical designs and [BCN] for Grassmannian designs; Venkov is named for strong perfection. Full citations are absent.
• Supplied literature label original OWR report: https://doi.org/10.4171/owr/2005/01. Literature triage checked 2026-08-21, status open with an incomplete-extraction warning; record updated 2026-09-16. The corrected_verified field reports direct source checking, not independently verified here.