Ext-Algebra Models for Crossingless Matchings

Definitions and mathematical statement

Fix the integer $k$ and the Slodowy-slice variety $S_{k,k}$ intended in the coherent-sheaf model of the report. Let $\mathcal M_k$ be its index set of crossingless matchings, conventionally noncrossing pairings of $2k$ ordered boundary points. For each matching $A$, write $i_A:A\hookrightarrow S_{k,k}$ for the corresponding geometric inclusion, and let $L_A=\Omega(A)^{1/2}$ be the specified square-root line bundle. The extraction does not define the geometric subvarieties, the meaning of $\Omega(A)$, or the square-root choices; in particular, these symbols must not be replaced without confirmation by an arbitrary cotangent bundle or arbitrary line bundle. Put $F_A=(i_A)_*L_A$.

The direct-sum Ext algebra is \[ \mathcal E_k=\bigoplus_{A,B\in\mathcal M_k}\ \bigoplus_{q\geq0} \operatorname{Ext}^q_{\operatorname{Coh}(S_{k,k})}(F_A,F_B), \] with $\operatorname{Ext}^q(F_A,F_B)=\operatorname{Hom}_{D^b\operatorname{Coh}(S_{k,k})}(F_A,F_B[q])$ in the intended derived setting. For \[ \alpha\in\operatorname{Ext}^p(F_A,F_B),\qquad \beta\in\operatorname{Ext}^q(F_B,F_C), \] the product is the Yoneda composition $\beta\circ\alpha\in\operatorname{Ext}^{p+q}(F_A,F_C)$; noncomposable blocks multiply to zero. Equivalently, for the finite index set, this is the Ext algebra of $\bigoplus_A F_A$.

Let $(H^\bullet,*)$ be the corresponding convolution algebra from the report. The conjecture asks for an algebra isomorphism \[ \Phi:\mathcal E_k\xrightarrow{\ \cong\ }H^\bullet, \qquad \Phi(\beta\circ\alpha)=\Phi(\beta)*\Phi(\alpha). \] The precise grading shifts, coefficient field, convolution correspondences, and block/idempotent identifications are not supplied. The notation suggests a graded comparison, but the extracted question explicitly requires an isomorphism as an algebra; whether a specified graded-vector-space identification, grading, or distinguished idempotents must be preserved requires confirmation.

The supplied assessment dated 2026-08-21 says that an identification of graded vector spaces is proved, but multiplicative compatibility with Yoneda product remained unresolved in the subsequent coherent-sheaf treatment located. A graded dimension match or vector-space isomorphism alone does not supply the algebra assertion. No multiplicative compatibility is inferred from that reported partial result.

Short English statement

The conjecture identifies the algebra of extensions between matching-indexed coherent sheaves with a convolution algebra. The essential issue is compatibility of multiplication, beyond the graded-vector-space identification reported as known. The geometric objects, bundle choices, and convolution conventions still require the full source.

Sources

• Local UnsolvedMath registry, permanent ID $30000930$, exact source code catalogue entry, proposed year 2008; no individual proposer is supplied.
• Representation Theory of Finite Dimensional Algebras (2008), Oberwolfach Reports, DOI 10.4171/owr/2008/09, pp. 401--472. https://doi.org/10.4171/owr/2008/09.
• Supplied reference: Stroppel--Webster, "coherent-sheaf algebra," https://doi.org/10.4171/CMH/261.
• The supplied corrected_verified note reports checking mathematical content against the official report. The partially solved label and assessment checked 2026-08-21 are dated supplied triage requiring independent verification of the precise known identification and unresolved multiplicative statement.