Cayley Degrees of Simple Algebraic Groups

Definitions and mathematical statement

Let $G$ be a connected reductive algebraic group over the field $k$ intended by the report, and write $\mathfrak g=\operatorname{Lie}(G)$. For a dominant $G$-equivariant rational map \[ f:G\dashrightarrow\mathfrak g, \] use conjugation on $G$ and the adjoint action on $\mathfrak g$, so \[ f(aga^{-1})=\operatorname{Ad}(a)f(g) \] where defined. Since the source and target have the same dimension, a dominant such map is generically finite. Its degree is the function-field extension degree $[k(G):f^*k(\mathfrak g)]$ under the usual convention. Define \[ \operatorname{Cay}(G)=\min_f\deg(f) \] over the intended maps.

The main task is to determine $\operatorname{Cay}(G)$ for every connected simple algebraic group $G$. The specified test case is to determine $\operatorname{Cay}(\operatorname{SL}_5)$. The source verification identifies these as Problems 7 and 8, respectively. This is an exact minimum-degree problem, not merely existence of a rational map or a numerical upper bound. The group $\operatorname{SL}_5$ is not replaced by its adjoint quotient.

The field, characteristic, splitness, form and isogeny conventions for simple groups are not supplied. The minimum also presupposes the relevant nonempty class of maps; this source convention must be confirmed rather than extended to unprovided field classes. In positive characteristic the degree/separability convention would need explicit attention.

The original extraction additionally contains a different question: for $d>3$, with $G=\operatorname{SL}_d/\mu_d$, $V=\operatorname{Sym}^d(k^d)$, and $\sigma$ induced by $g\mapsto(g^{\mathsf T})^{-1}$, are $V$ and the representation twisted by $\sigma$ birationally $G$-isomorphic? The supplied verification explicitly assigns that question to OWR-1452-030. It is retained here only as extraction provenance, not as an additional target of this Cayley-degree record.

Short English statement

Determine the smallest degree of a dominant conjugation-to-adjoint equivariant rational map for every connected simple algebraic group, especially $\operatorname{SL}_5$. A symmetric-power twist question appended in the original belongs to a separate source entry and is retained only for provenance.

Sources

• Local UnsolvedMath registry, permanent ID 30000667, code catalogue entry, "Cayley Degrees of Simple Algebraic Groups." Proposed year 2007; proposer unspecified.
• Affine Algebraic Geometry (2007), Oberwolfach Reports, pp. 5--82, DOI 10.4171/owr/2007/01; https://doi.org/10.4171/owr/2007/01. This is the sole supplied literature reference.
• Statement status corrected_verified; the supplied verification identifies the classification task as Problem 7 and the $\operatorname{SL}_5$ case as Problem 8, and assigns the unrelated symmetric-power twist clause to OWR-1452-030. That entry's permanent numeric ID is not supplied here.
• Literature triage checked 2026-08-21: open. It generically calls the extracted claim unresolved and reports no authoritative complete resolution from the supplied check, without separate group cases or a computed $\operatorname{SL}_5$ value. Created 2026-08-25; updated 2026-09-16. Current status and field conventions require independent verification.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; the underlying report retains its supplied provenance. Unreviewed statement-only draft.