Dyck-Path Inequalities for Newton–Okounkov Polytopes

Definitions and mathematical statement

Write $\Lambda_i=(\lambda_i^i,\ldots,\lambda_i^n)$ for $i=1,\ldots,n-1$, and retain the supplied convention \[ \lambda_j^i=\lambda_j^j\qquad(j>i). \] The source background concerns Newton--Okounkov polytopes of Bott--Samelson varieties. Its notation is \[ \Delta_{v_0}\left(X_{w_0},L(\Lambda_1,\ldots,\Lambda_{n-1})\right), \] where $X_{w_0}$, the valuation $v_0$, and the line bundle $L(\Lambda_1,\ldots,\Lambda_{n-1})$ require the constructions intended in the report; these constructions are not specified in the supplied record.

To express the proposed equality without supplying missing combinatorics, let $\mathcal I$ denote the intended coordinate index set, and let $\mathcal D_{k;i,j}$ denote the intended family of Dyck paths from the boundary label $\lambda_k$ to the coordinate position $u_{ij}$. These are placeholders for source definitions that are absent from the record. In a path sum, $(l,m)\in D$ denotes the coordinate positions counted along that path.

Conjectural statement, conditional on the missing source definitions. For the parameter data admitted by the intended construction, is \[ \begin{aligned} &\Delta_{v_0}\left(X_{w_0},L(\Lambda_1,\ldots,\Lambda_{n-1})\right)\\ &\quad=\left\{u\in\mathbb R^{\mathcal I}:\begin{array}{ll} u_{lm}\geq0&\text{for every }(l,m)\in\mathcal I,\\[2pt] \displaystyle\sum_{(l,m)\in D}u_{lm}\leq \displaystyle\sum_{s=1}^{j}(\lambda_k^s-\lambda_{i+j}^s) &\begin{gathered} \text{for every integer triple }(k,i,j)\text{ with}\\ 1<k\leq j<n,\quad 1\leq i\leq n-j,\\ \text{and every }D\in\mathcal D_{k;i,j} \end{gathered} \end{array}\right\}? \end{aligned} \] Here the first coordinate inequality is $u_{lm}\geq0$. The record does not specify the allowed range of $n$, the domains or dominance conditions of the parameters, the coordinate set, the path graph and allowed steps, or the geometric constructions. The boundary labels $\lambda_k$ are also undefined. The displayed parameter convention is retained literally; its compatibility with the ordering of the entries of $\Lambda_i$ needs clarification. Consequently, this is a faithful conditional transcription of the proposed inequality description, not a fully specified standalone conjecture.

Short English statement

The proposed conjecture describes a Newton--Okounkov polytope by nonnegative coordinates and one upper bound for every indicated Dyck path. The supplied record omits the definitions of those paths and several geometric and parameter conventions, so the statement remains conditional on recovering that information. Its supplied status is partially solved; this draft does not independently verify that assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004269$, title Dyck-Path Inequalities for Newton--Okounkov Polytopes, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2019 and names no individual proposer.
• Supplied authoritative report citation: Mini-Workshop: Degeneration Techniques in Representation Theory (2020), Oberwolfach Reports, pp. 2869--2909, DOI 10.4171/owr/2019/46. https://doi.org/10.4171/owr/2019/46. The record labels the statement corrected_verified and reports a check against the official report; that verification is supplied provenance, not an independent check performed for this draft.
• Supplied literature source, labeled "Kiritchenko": https://arxiv.org/abs/1801.00334.
• Supplied literature source, labeled "Kiritchenko, IMRN 2023": https://academic.oup.com/imrn/article/2023/4/3305/6454976.
• Supplied literature source, labeled "OWR 2019 46": https://ems.press/content/serial-article-files/46825.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved, describes substantial constructions relating these Newton--Okounkov polytopes to Minkowski sums of FFLV polytopes, and says that the extracted explicit all-Dyck-path inequality description is not established in full generality. This is supplied dated curation derived from the OWR collection, not independently verified literature assessment; its status and cited claims require independent verification.