Diagrammatic Actions on Principal Tilting Categories

Definitions and mathematical statement

Let $G$ be a semisimple simply connected group in characteristic $p$, with $p>h$, where $h$ denotes its Coxeter number in the convention intended by the source. Write $\mathrm{Tilt}_0$ for the principal tilting category attached to $G$. The supplied record does not specify the base field, the definition of the principal block, or the convention for $h$ when $G$ has several simple factors.

The source also uses a category $\mathcal D$, objects $B_s$ of $\mathcal D$, and a category $\mathrm{AS}_{\mathrm{deg}}$. Their constructions, coefficient data, indexing set for $s$, and grading conventions are not supplied. For purposes of expressing the requested structure, write $\Theta_s$ for the translation functor intended to correspond to $B_s$; its precise definition is likewise missing.

Requested existence statement, conditional on these missing specifications. For every $G$ satisfying the stated hypotheses, does there exist a right $\mathcal D$-module category structure on $\mathrm{Tilt}_0$, with action \[ \star:\mathrm{Tilt}_0\times\mathcal D\longrightarrow\mathrm{Tilt}_0, \] satisfying the module associativity and unit coherence axioms, such that for every intended index $s$ there is a natural isomorphism \[ (-)\star B_s\cong\Theta_s, \] and an equivalence of right $\mathcal D$-module categories \[ F:\mathrm{AS}_{\mathrm{deg}}\xrightarrow{\sim}\mathrm{Tilt}_0? \] Here an equivalence of right module categories means an equivalence of the underlying categories equipped with coherent natural isomorphisms \[ F(M\star D)\cong F(M)\star D \qquad (M\in\mathrm{AS}_{\mathrm{deg}},\ D\in\mathcal D), \] compatible with the module associativity and unit constraints. This formulation presupposes the intended monoidal structure on $\mathcal D$ and right action on $\mathrm{AS}_{\mathrm{deg}}$, neither of which is defined in the supplied record.

Consequently, this is a conditional formulation of the recorded question, not a fully specified standalone conjecture. The missing categorical and representation-theoretic data must be recovered before its exact mathematical scope can be fixed. The supplied literature assessment concerns verification of an associated quantum-group conjecture for $U_q(\mathfrak{sl}_2)$; it does not assert that the existence statement above is resolved.

Short English statement

The question asks whether, for every semisimple simply connected group in characteristic $p$ greater than its Coxeter number, the principal tilting category admits the specified diagrammatic right action, with each $B_s$ acting by translation, and is equivalent to $\mathrm{AS}_{\mathrm{deg}}$ with that action. Essential definitions and conventions are absent. The supplied partially solved label refers to a reported associated quantum-group special case and is not an independently verified resolution of this question.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30002855$, title "Diagrammatic Actions on Principal Tilting Categories," Oberwolfach Reports Open Problems collection, source problem code catalogue entry. The record gives proposed year 2015 and identifies no proposer.
• Supplied report citation: Representations of Finite Groups (2016), Oberwolfach Reports, pp. 971--1027, DOI 10.4171/owr/2015/18. Supplied source URL and literature-source link labeled "original OWR report": https://doi.org/10.4171/owr/2015/18. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Andersen--Tubbenhauer, Diagram categories for $U_q$-tilting modules, https://arxiv.org/abs/1409.2799.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem "partially solved" and reports verification only of the associated quantum-group conjecture for $U_q(\mathfrak{sl}_2)$, described as a genuine special case rather than a resolution of the broader conjectural program. This is supplied dated curation, not independently verified here; the background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says its mathematical content was preserved while notation, typography, or omitted definitions were repaired for publication. These are supplied editorial assertions, not independent mathematical verification of this draft.