2.2 (Cooper) — An invariant polynomial on a tensor product

Definitions and mathematical statement

Following the supplied research summary, let $U,V,W$ be complex vector spaces with dimensions $4,4,8$, and put $T=U\otimes V\otimes W$ and $G=\operatorname{SL}(U)\times\operatorname{SL}(V)\times\operatorname{SL}(W)$. The action is \[ (g_U,g_V,g_W)(u\otimes v\otimes w) =g_Uu\otimes g_Vv\otimes g_Ww. \] A polynomial $P\in{\mathbb{C}}[T]=\operatorname{Sym}(T^*)$ is invariant if $P(gt)=P(t)$ for every $g\in G$ and $t\in T$.

Cooper asks whether a nonzero invariant polynomial exists. Literally this is the condition ${\mathbb{C}}[T]^G\ne\{0\}$, which does not exclude nonzero constants. If the intended target is a nonconstant invariant, its precise homogeneous formulation is \[ \exists d\geq1:\qquad \left(\operatorname{Sym}^d(T^*)\right)^G\ne\{0\}. \] The positive-degree restriction is not printed in the source and therefore remains an explicit possible intended correction rather than a silently added hypothesis. The statement itself does not name the ground field; ${\mathbb{C}}$ is supplied by its research summary. The source relates the question to limits under conjugacy of the diagonal subgroup of $\operatorname{SL}(8,{\mathbb{R}})$, but provides no comparison construction in this record.

Short English statement

Ask for an invariant polynomial on a $4\times4\times8$ tensor space under the product of the three special linear groups. The source's word nonzero leaves unresolved whether a nonconstant, positive-degree invariant is intended.

Sources

• Cooper, attributed item 2.2 in Delp, Hoffoss and Manning, Problems In Groups, Geometry, and Three-Manifolds (2015), supplied PDF page 2. https://arxiv.org/pdf/1512.04620#page=2; status link https://arxiv.org/abs/1512.04620. Recorded access 2026-07-29; extraction source-tex checked against rendered PDF text, as reported in the record.
• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $10900007$, exact source code catalogue entry. Its OPEN-TRIAGE summary specifies the complex tensor space and reports no verified published answer, without an additional citation.