Clasp Formula for Local Intersection Numbers

Definitions and mathematical statement

Let $V={\mathbb{C}}^n$. Relative to the representation-theoretic setting intended by the source, let $\lambda$ be a dominant weight and let $\mu$ be a weight of $\Lambda^k V$ such that $\lambda+\mu$ is dominant. Let $\Phi^+$ denote the intended set of positive roots, let $\rho$ denote the intended Weyl vector, and write $\langle\cdot,\cdot\rangle$ for the pairing used in the source. Define \[ S_{\lambda,\mu}=\left\{\alpha\in\Phi^+: \langle\lambda+\rho+\mu,\alpha\rangle =\langle\lambda+\rho,\alpha\rangle-1\right\}. \] The proposed identity, for every admissible choice of these data, is \[ \kappa_{\lambda,\mu} \stackrel{?}{=} \prod_{\alpha\in S_{\lambda,\mu}} \frac{\langle\lambda+\rho,\alpha\rangle} {\langle\lambda+\rho+\mu,\alpha\rangle}. \] Here $\kappa_{\lambda,\mu}$ is described in the supplied record as the resulting one-dimensional local intersection form, or local-intersection number. The empty product is interpreted as $1$.

This is a qualified transcription, not a fully specified numerical conjecture. The record does not identify the acting group or root datum, specify the root-pairing and dominance conventions, define $\rho$ explicitly, or give the construction of the local intersection form. In particular, a one-dimensional bilinear form does not determine a numerical coefficient without a specified normalization or basis convention. The displayed fractions also presuppose nonzero denominators; the supplied record does not explicitly establish this. The admissible ranges of $n$ and $k$ are not stated. These omissions must be resolved before the identity can be interpreted unambiguously.

Short English statement

The proposed clasp formula expresses a local intersection number associated with $\lambda$ and a weight $\mu$ of $\Lambda^k{\mathbb{C}}^n$ as a product over positive roots whose pairing decreases by exactly $1$ after adding $\mu$. The intended representation-theoretic conventions and the normalization of the intersection number require clarification. The supplied open-status assessment concerns a different conjecture and does not establish the status of this formula.

Sources

• Local registry: data/unsolvedmath/problems.json, permanent UnsolvedMath ID $30004221$, title "Clasp Formula for Local Intersection Numbers," exact source problem code catalogue entry. The supplied proposed year is 2019; no proposer is identified.
• Supplied source citation: Geometric, Algebraic, and Topological Combinatorics (2020), Oberwolfach Reports, pp. 2395--2472, DOI 10.4171/owr/2019/39. https://doi.org/10.4171/owr/2019/39. This report is also cited in the background as documenting workshop open-problem discussions.
• Supplied literature source: Adler, De Loera, Klee, and Zhang, "cocircuit diameters." https://doi.org/10.37236/9653.
• Supplied literature source: "Oberwolfach Report 39, oriented matroids." https://doi.org/10.4171/owr/2019/39.
• The literature assessment is dated exactly 2026-08-22. It is supplied dated curation, not independently verified here. It labels the problem open but discusses a quadratic bound, counterexamples to stronger variants, and an $n-r+2$ conjecture. That discussion is not matched to the local intersection formula in this record.
• The registry labels the statement corrected_verified and reports that the official Oberwolfach report was checked directly. This is a supplied verification claim, not an independent source check performed for this draft. The supplied citation is retained, but its support for the exact formula and its conventions requires human verification.