Birational Equivalence of Twisted Algebraic-Group Representations

Definitions and mathematical statement

Let $k$ be the source's base field, let $G$ be a connected semisimple algebraic group over $k$, and let $\rho:G\to\operatorname{GL}(V)$ be a finite-dimensional algebraic representation with trivial stabilizers in general position. In the standard geometric interpretation this means that stabilizers are trivial on a dense open subset after the intended base change. The field, characteristic, and scheme-theoretic versus pointwise stabilizer convention are not supplied.

For a $k$-defined algebraic automorphism $\sigma:G\to G$, use the twist convention \[ g\star v=\rho(\sigma(g))v\qquad\text{on }V^{(\sigma)}. \] The general question is whether there exists a birational map with rational inverse \[ \Phi:V\dashrightarrow V^{(\sigma)},\qquad \Phi(\rho(g)v)=\rho(\sigma(g))\Phi(v) \] where the expressions are defined, for every indicated $G,V,\sigma$. This is birational $G$-isomorphism of the actual representations, not merely stable equivalence after adding trivial factors or an isomorphism of their invariant fields. Some sources precompose by $\sigma^{-1}$; the chosen convention must be matched to the report.

The separate explicit question, identified as Problem 4 by supplied verification, takes \[ G=\operatorname{SL}_d/\mu_d,\qquad V=\operatorname{Sym}^d(k^d),\qquad \sigma([g])=[(g^{\mathsf T})^{-1}]. \] Here $\mu_d$ is the central subgroup of scalar $d$-th roots of unity. Its action on $\operatorname{Sym}^d(k^d)$ is trivial, giving the stated quotient-group action. Is this $V$ birationally $G$-isomorphic to its specified inverse-transpose twist? The record reports trivial stabilizers in general position for $d>3$ in the intended field setting, making that range an instance of the general question. It does not thereby assert the same stabilizer property for $d\leq3$, nor exclude those dimensions from the separately posed specialization without a source range.

The supplied literature triage checked 2026-08-21 reports no authoritative complete resolution of the extracted pair of problems. No additional field hypotheses or partial theorem are given, so the generic-stabilizer assertion is retained as supplied rather than certified uniformly in every characteristic.

Short English statement

Ask whether a generically free representation of a connected semisimple group is birationally equivariantly isomorphic to every automorphism twist, especially the stated symmetric-power example with inverse-transpose twist. The base-field and stabilizer conventions remain essential.

Sources

• Local UnsolvedMath registry, permanent ID 30000666, catalogue entry, Birational Equivalence of Twisted Algebraic-Group Representations.
• Affine Algebraic Geometry (2007), Oberwolfach Reports, pp. 5--82, DOI https://doi.org/10.4171/owr/2007/01; supplied verification identifies the general question as Problem 3 and specialization as Problem 4.
• Supplied literature triage checked 2026-08-21 cites only that report; no separate theorem or precise field assumptions for generic freeness are supplied.