Irreducibility of Restricted Classical-Group Tensor Representations

Definitions and mathematical statement

Let $W$ be the natural module for the simple classical algebraic group $\mathcal G=\operatorname{Cl}(W)$ intended by the source, and let $G\le\mathcal G$ be Zariski closed. Consider, in the three separate cases, the $\mathcal G$-modules \[ W\otimes W^*,\qquad\operatorname{Sym}^k(W),\qquad\bigwedge^k(W), \] for the source's "small" values of $k$. Here $W^*$ is the dual module, and a composition factor is a simple subquotient in a composition series, not necessarily a direct summand.

Let $V$ be the source's largest composition factor of the chosen module. Determine when \[ V\mathbin{\downarrow}_G \] is irreducible, meaning that the same vector space with the restricted $G$-action has no nonzero proper invariant linear subspace. The classification must retain the selected simple factor; it is not replaced by irreducibility of the entire tensor, symmetric power, or exterior power.

The field, characteristic, classical-group type and isogeny convention, dimension of $W$, and numerical range of small $k$ are not supplied. Nor is "largest" defined as largest dimension, distinguished highest weight, or another choice, including what to do if several factors tie. These missing data matter especially when the ambient modules are not semisimple. No characteristic-zero or algebraically closed field is imposed as an explicit source assumption.

The subgroup may be finite or positive dimensional and connected or disconnected as written; no properness or connectedness restriction is added. The intended classification equivalence for subgroups also remains unspecified. The assessment reports a solution of the Kollár--Larsen symmetric-power problem for $k\ge4$ by Guralnick and Tiep, but says the broader Problem 2 and extracted small-$k$ cases remain partially classified. That is supplied literature context, not a theorem here about every selected composition factor.

Short English statement

The problem asks which Zariski-closed subgroups keep a specified large simple factor of a classical tensor, symmetric, or exterior power irreducible. The field, small-power range, and rule selecting the largest factor are missing; the whole module must not be substituted for that factor.

Sources

• Local UnsolvedMath registry, permanent ID 30000809, code catalogue entry, "Irreducibility of Restricted Classical-Group Tensor Representations." Proposed year 2007; proposer unspecified.
• Permutation Groups (2008), Oberwolfach Reports, pp. 2173--2236, DOI 10.4171/owr/2007/37; https://doi.org/10.4171/owr/2007/37. The displayed report year 2008 differs from the DOI and proposed year 2007.
• Supplied "Guralnick--Tiep, symmetric powers": https://arxiv.org/abs/0810.0853.
• Statement status corrected_verified; the supplied note calls the irreducibility question complete and says classical-group, tensor, exterior-power, and restriction notation was restored. The missing field and selection conventions still require source review.
• Literature triage checked 2026-08-21: partially solved. It reports a Guralnick--Tiep solution of the Kollár--Larsen symmetric-power problem for $k\ge4$ and partial classification of broader Problem 2 and small-$k$ cases. Exact theorem hypotheses and relation to the selected factors require independent verification. Created 2026-08-25; updated 2026-09-16.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; underlying report and literature provenance remain as supplied. Unreviewed statement-only draft.