Hall Multiplication on Principal Quiver Grassmannians

Definitions and mathematical statement

For a quiver $Q$, a representation $M$ over a field $k$, and a dimension vector $e=(e_v)_{v\in Q_0}$, the quiver Grassmannian $\operatorname{Gr}_e(M)$ parametrizes subrepresentations $N\subseteq M$ satisfying $\dim_k N_v=e_v$ at every vertex $v$. The group $\operatorname{Aut}_Q(M)$ of invertible morphisms of representations acts by $N\mapsto gN$. Under an invariant-function interpretation, invariance means $f(gN)=f(N)$ for every $g\in\operatorname{Aut}_Q(M)$ and every point $N$ in the chosen domain of the function.

The supplied record specifies the space \[ A=\bigoplus_{[P],[I],e} \mathbb{Q}^{\operatorname{Aut}_Q(P\oplus I)} [\operatorname{Gr}_e(P\oplus I)]. \] Here $[P]$ and $[I]$ denote isomorphism classes of representations in the intended indexing families. Although the notation suggests projective representations $P$ and injective representations $I$, those families and the meaning of "principal" are not defined in the supplied record. The notation for each invariant-function summand is also retained without choosing between all functions, constructible functions, or another function space.

Requested construction. Define a Hall-algebra-type multiplication on $A$ that makes it a graded associative algebra and induces an isomorphism \[ \Phi:U_q(\mathfrak b^+)\otimes U_q(\mathfrak b^+) \longrightarrow A. \] Under a $\mathbb Q$-vector-space interpretation of the summands, the requested multiplication would be a bilinear map $\mu:A\otimes_{\mathbb Q}A\to A$ satisfying \[ \mu(\mu(a,b),c)=\mu(a,\mu(b,c)) \qquad(a,b,c\in A), \] and, once a grading $A=\bigoplus_\gamma A_\gamma$ and its index operation are specified, \[ \mu(A_\gamma,A_\delta)\subseteq A_{\gamma+\delta}. \] The symbol $U_q(\mathfrak b^+)$ is the intended quantum enveloping algebra of a positive Borel subalgebra; neither that subalgebra nor its relation to $Q$ is specified. The class of quivers, field, indexing families, function theory, grading, parameter $q$, coefficient ring and tensor-product base, Hall-type compatibility conditions, construction of $\Phi$, and required category of isomorphism remain unresolved. Thus these formulas record the construction problem but do not supply a fully determined existence statement or a universal quantifier over quivers.

Short English statement

The problem asks for a Hall-algebra-type product on invariant functions on principal quiver Grassmannians, giving an associative graded algebra identified with two tensor factors of $U_q(\mathfrak b^+)$. The supplied record leaves essential hypotheses and the meaning of this identification unspecified; these require mathematical review.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30004264$, title "Hall Multiplication on Principal Quiver Grassmannians," source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record lists proposed year 2019 and names no individual proposer.
• Supplied source citation: Mini-Workshop: Degeneration Techniques in Representation Theory (2020), Oberwolfach Reports, DOI 10.4171/owr/2019/46, pp. 2869--2909. https://doi.org/10.4171/owr/2019/46. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: "OWR 2019 46," https://ems.press/content/serial-article-files/46825.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports that the proposed Hall-algebra-type multiplication is posed as a problem and that no general construction with the stated properties was found. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment; time-sensitive status and cited claims require independent verification.