Stabilizer-dimension problem for concise tensors

For each integer $n\ge 1$ , let $V_1,V_2,V_3$ be $n$ -dimensional vector spaces and let $T\in V_1\otimes V_2\otimes V_3$ be concise, meaning that each contraction map $V_i^*\to V_j\otimes V_k$ is injective. Determine the maximum of $\dim\operatorname{Stab}(T)$ over all concise tensors $T$ , where $\operatorname{Stab}(T)=\{(g_1,g_2,g_3)\in\operatorname{GL}(V_1)\times\operatorname{GL}(V_2)\times\operatorname{GL}(V_3):(g_1\otimes g_2\otimes g_3)T=T\}$ . Classify the concise tensors attaining this maximum.

References

Primary source

arXiv

Additional references

Concise tensors with maximal symmetries — arXiv — Annika Holtrup, Jeroen Zuiddam

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A new unrefereed preprint claims to settle the extremal question for concise tensors by identifying the largest possible symmetry dimension, but the result has not been independently verified.

The problem asks for the largest stabilizer dimension of a concise tensor. Annika Holtrup and Jeroen Zuiddam report a complete extremal classification, including the tensors attaining the maximum.

September 2026 preprint

The preprint claims that the maximum stabilizer dimension is $n^2+1$ , with maximizers including null algebra tensors and the skew-symmetric tensor $e_1\wedge e_2\wedge e_3$ . It also gives bounds for related matrix-tuple actions. The result is unrefereed and therefore remains unverified.

Current status (as of September 2026): Holtrup and Zuiddam claim the extremal problem is solved with maximum $n^2+1$ , but independent verification is not recorded.

Sources

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