Support Ideals in Constructible-Function Convolution Algebras

Definitions and mathematical statement

Let $\mathcal A(\Pi)=\bigoplus_d\mathcal A(\Pi)_d$ denote the graded convolution algebra of constructible functions referred to in the supplied record, and write $*$ for its multiplication. The index $d$ ranges over the homogeneous degrees intended in that source; its admissible values are not supplied. For each such degree, retain the source's notation and set \[ I_d=\left\{f\in\mathcal A(\Pi)_d:\dim\operatorname{supp}(f)<\dim G(d)-q_D^C(d/D)\right\}, \qquad I=\bigoplus_d I_d. \] Support Conjecture. The source describes the sets $I_d$ as homogeneous subspaces and conjectures that $I$ is a two-sided ideal of $\mathcal A(\Pi)$. Explicitly, the assertion is \[ \forall a\in\mathcal A(\Pi)\;\forall f\in I,\qquad a*f\in I\quad\text{and}\quad f*a\in I. \] This is a conditional transcription of the algebraic target, not a complete specification of its geometric setting. The supplied record does not define the quiver data $\Pi$, the spaces supporting the constructible functions, their coefficient ring, the convolution construction, the grading, the group $G(d)$, the parameters $C,D$, the expression $d/D$, or the function $q_D^C$. It also does not specify the support-dimension convention, including its value for the zero function. These definitions and the hypotheses under which they apply must be recovered from the cited source before the conjecture becomes a fully determined mathematical statement. The strict inequality is retained exactly.

Short English statement

The support conjecture asks whether the direct sum of the homogeneous spaces of constructible functions whose support dimension is strictly below $\dim G(d)-q_D^C(d/D)$ is closed under convolution on both sides by every element of the algebra. The supplied record labels this conjecture open, but omits essential definitions needed to identify the algebra and the dimension bound.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30003421$, exact source problem code catalogue entry, title Support Ideals in Constructible-Function Convolution Algebras. The record attributes the problem to the Oberwolfach Reports collection and supplies proposed year $2017$; no individual proposer is identified.
• Supplied report citation: Representation Theory of Quivers and Finite Dimensional Algebras (2018), Oberwolfach Reports, DOI 10.4171/owr/2017/12, pp. 591--681. https://doi.org/10.4171/owr/2017/12. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: OWR Representation Theory report. https://ems.press/content/serial-article-files/46676.
• Supplied literature source: Quivers with loops and Lagrangian subvarieties. https://arxiv.org/abs/1311.5396.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and states that the support conjecture remains an unproved premise for the stated Hall-algebra realization; later convolution and canonical-basis results are described as providing related support theorems but not this exact ideal assertion. This is supplied dated literature triage derived from the OWR curation, not an independently verified assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.