Geometry behind a Plethystic Representation Isomorphism

Definitions and mathematical statement

For each integer $\ell\ge 2$, define the partitions \[ \lambda_\ell=(3\ell-3,\underbrace{2,\ldots,2}_{\ell-1\text{ entries}}),\qquad \mu_\ell=(\ell+1,\underbrace{1,\ldots,1}_{\ell-2\text{ entries}}). \] For $\ell=2$, the second repeated list is empty, so $\mu_2=(3)$. For a vector space $E$ over a specified field, $\operatorname{Sym}^{d}E$ denotes its $d$-th symmetric power. The notation $\nabla_\lambda$ is retained from the source for its partition-indexed representation construction; its precise convention is not supplied.

The source asks for a geometric or invariant-theoretic interpretation, for every integer $\ell\ge2$, of the displayed isomorphism \[ \nabla_{\lambda_\ell}\left(\operatorname{Sym}^{\ell}E\right) \cong \nabla_{\mu_\ell}\left(\operatorname{Sym}^{3\ell-4}E\right). \] This is a request to explain the supplied isomorphism, rather than a separately specified theorem to prove. The record does not identify the ground field or its characteristic, the dimension of $E$, the acting group, the category in which the isomorphism holds, or the convention for $\nabla_\lambda$. It also gives no criterion for what constitutes the requested interpretation. Consequently, the display is preserved as a source assertion with unresolved hypotheses; it is not asserted here for arbitrary vector spaces or representation categories.

Short English statement

For every integer $\ell\ge2$, seek a geometric or invariant-theoretic explanation of the stated isomorphism between two partition-indexed constructions on symmetric powers of $E$. The setting of the representations and the required form of the explanation are unspecified and require clarification.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30004424$, source problem code catalogue entry, title: Geometry behind a Plethystic Representation Isomorphism. The supplied record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is identified; the supplied proposed year is 2020.
• Supplied source citation: Mini-Workshop: Kronecker, Plethysm, and Sylow Branching Coefficients and their Applications to Complexity Theory (2021), Oberwolfach Reports, DOI 10.4171/owr/2020/9, pp. 517--568. Source URL: https://doi.org/10.4171/owr/2020/9. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source, labelled "original OWR report": https://doi.org/10.4171/owr/2020/9.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem open because the requested geometric or invariant-theoretic interpretation has no precise target theorem and remains unresolved or unclear. This is supplied dated curation, not independently verified literature assessment. The source explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review claiming that the clean statement is self-contained and preserves its target without unsupported additions. That supplied review label does not resolve the missing representation-theoretic hypotheses identified here.