Atypicality Detection by Ambidextrous Modified Dimensions

Definitions and mathematical statement

Let $\mathfrak g$ be a basic classical Lie superalgebra, and let $\mathcal C$ denote the intended tensor category of $\mathfrak g$-supermodules. The supplied record does not specify the ground field, dimensional restrictions on objects, allowed morphisms, or pivotal structure; these must be fixed before the question has a fully precise interpretation.

The notation in the question has the following conventional interpretation, subject to those missing choices. The tensor ideal $\mathcal I_J$ generated by an object $J$ consists of direct summands of finite direct sums of objects $J\otimes X$, with $X\in\mathcal C$, in the usual symmetric tensor setting for supermodules. A modified trace on this ideal is a family of linear maps \[ t_V:\operatorname{End}_{\mathcal C}(V)\longrightarrow k, \qquad V\in\mathcal I_J, \] where $k$ is the ground field, satisfying cyclicity and compatibility with the categorical partial trace. Explicitly, for $f:U\to V$, $g:V\to U$, and $h\in\operatorname{End}_{\mathcal C}(U\otimes X)$, these identities are \[ t_V(fg)=t_U(gf),\qquad t_{U\otimes X}(h)=t_U\left(\operatorname{ptr}_X(h)\right). \] Here $U,V\in\mathcal I_J$, $X\in\mathcal C$, and $\operatorname{ptr}_X$ uses the chosen categorical duality and pivotal structure. An ambidextrous trace on $J$ is a trace functional satisfying the compatibility between left and right partial traces required for the intended extension to $\mathcal I_J$. For the resulting family, the modified dimension is conventionally \[ d_J(V)=t_V(\operatorname{id}_V). \] The record does not supply the precise ambidextrous-trace convention, extension hypotheses, or nonzero normalization. A nonzero scalar rescaling of a chosen trace family does not change whether $d_J(V)$ vanishes, whereas the zero trace would change the question.

For a simple highest-weight supermodule $M$ of highest weight $\lambda$, its usual root-theoretic atypicality $\operatorname{atyp}(M)$ is the maximum cardinality of a linearly independent, pairwise orthogonal set of isotropic odd positive roots $\alpha$ satisfying $(\lambda+\rho,\alpha)=0$. Here the positive roots, invariant bilinear form, and $\rho$, the half-sum of positive even roots minus the half-sum of positive odd roots, belong to the applicable highest-weight setup. The record does not specify that setup or explicitly restrict its modules to a class for which this description applies.

Subject to the intended categorical and highest-weight conventions, the supplied question is: for every basic classical Lie superalgebra $\mathfrak g$, every simple $\mathfrak g$-supermodule $J$ admitting the intended ambidextrous trace, and every simple $L\in\mathcal I_J$, does \[ \operatorname{atyp}(L)=\operatorname{atyp}(J) \quad\Longleftrightarrow\quad d_J(L)\ne 0 \] hold? This is a statement-only formulation of the supplied question; its unresolved ambient hypotheses and trace conventions require source clarification.

Short English statement

The question asks whether, among simple supermodules in the tensor ideal generated by a simple ambidextrous object $J$, the modified dimension is nonzero exactly for those with the same atypicality as $J$. The ground field, module category, and trace conventions remain unspecified. The supplied partially solved label is dated curation, not an independently verified mathematical conclusion.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $30001451$, source problem code catalogue entry, title "Atypicality Detection by Ambidextrous Modified Dimensions." The record belongs to the Oberwolfach Reports Open Problems collection and gives the proposed year as 2010; no individual proposer is supplied.
• Supplied original report citation: Combinatorial Representation Theory (2010), Oberwolfach Reports, pp. 799--882, DOI 10.4171/owr/2010/15. Original OWR report: https://doi.org/10.4171/owr/2010/15. The collection describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied statement-verification label: corrected_verified. The accompanying registry note says that the mathematical content was checked against the official Oberwolfach report and that notation, context, and question form were normalized without changing the source-level claim. This is supplied verification metadata, not a verification performed for this draft.
• Supplied literature source, labelled "generalized Kac--Wakimoto for osp": https://arxiv.org/abs/1112.3384.
• Supplied literature source, labelled "Kac--Wakimoto for the periplectic Lie superalgebra": https://arxiv.org/abs/1905.04712.
• Literature assessment checked at 2026-08-21: the supplied status is partially solved. The supplied assessment states that the ordinary Kac--Wakimoto conjecture and generalized versions are proved in important cases, including basic classical, general-linear, or orthosymplectic settings, while the broad generalized ambidextrous-trace formulation in the extracted record is not fully settled. This is supplied dated curation derived from the OWR collection, not independently verified here; its time-sensitive status and cited claims require independent verification.