Rank Stratifications and KLR Actions on Flag Degenerations

Definitions and mathematical statement

Write $p:F_U\to U$ for the flat family of linear degenerations named in the supplied record, and put \[ Z=F_U\times_U F_U. \] Here $Z$ is the fiber product of the two copies of $p$, and $H_*^{\mathrm{BM}}(Z)$ denotes its Borel--Moore homology. The record does not define $U$, $F_U$, the underlying geometric category or ground field, or the homology coefficients. It also does not define the datum $A$, the locus $S(A)$, or the maps and rank conditions used to define that locus.

Subject to these unresolved definitions, the requested problem is to describe the stratification associated with the rank conditions defining $S(A)$, and to describe $H_*^{\mathrm{BM}}(Z)$ as a quotient of a quiver Hecke algebra, also called a Khovanov--Lauda--Rouquier (KLR) algebra, together with the standard representations indexed by Motzkin paths. An algebra quotient description would require specifying a KLR algebra $R$, an algebra structure on $H_*^{\mathrm{BM}}(Z)$, and a surjective algebra homomorphism \[ \Phi:R\twoheadrightarrow H_*^{\mathrm{BM}}(Z), \qquad R/\ker\Phi\cong H_*^{\mathrm{BM}}(Z). \] The record does not specify the quiver, KLR parameters, grading conventions, multiplication on the homology, or the intended meaning of standard representations.

Under the usual combinatorial convention, a Motzkin path is a finite lattice path from $(0,0)$ to $(n,0)$, for an integer $n\geq 0$, with steps $(1,1)$, $(1,0)$, and $(1,-1)$, remaining at nonnegative height. The record does not specify the length, additional restrictions, or the correspondence with representations. This conventional definition therefore does not determine the intended indexing set.

These omissions prevent a self-contained mathematical formulation beyond the conditional target above. In addition, the supplied literature assessment concerns $\operatorname{Aut}_Q(M)$-orbits, their finiteness, parametrization, and closure relation, whereas the supplied statement concerns rank stratifications and KLR algebras. No relationship between those targets is supplied.

Short English statement

The supplied problem asks for the rank stratification of a flat family of linear degenerations and a KLR algebra quotient description of the Borel--Moore homology of its fiber product, including standard representations indexed by Motzkin paths. Essential geometric and algebraic data are missing, and the accompanying literature assessment addresses a different orbit problem. This is a statement-only draft requiring mathematical review.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30004268$, exact source problem code catalogue entry, titled "Rank Stratifications and KLR Actions on Flag Degenerations," in the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2019.
• Supplied report citation: Mini-Workshop: Degeneration Techniques in Representation Theory (2020), Oberwolfach Reports, pp. 2869--2909, DOI 10.4171/owr/2019/46. Supplied source link: https://doi.org/10.4171/owr/2019/46. The citation's 2020 date and the registry's 2019 year are retained as supplied.
• Supplied literature source, labeled "OWR 2019 46": https://ems.press/content/serial-article-files/46825.
• Literature assessment dated exactly 2026-08-22: the record labels the problem open and reports that finiteness of $\operatorname{Aut}_Q(M)$-orbits, their parametrization, and their closure relation were posed as a problem, with no general answer located. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims. The assessment's relationship to the stated KLR problem is unresolved.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting self-containment and fidelity. These are supplied editorial claims; the missing definitions listed above remain unresolved in this draft.