Admissible Gelfand–Zetlin Faces Representing Smooth Schubert Cycles

Definitions and mathematical statement

Let $G=GL_n({\mathbb{C}})$, let $B\subseteq G$ be a Borel subgroup, and let $X=G/B$ be the complete flag variety. For $w\in S_n$, choose a permutation-matrix representative $\dot w$, and write \[ C_w=B\dot wB/B,\qquad Y_w=\overline{C_w}\subseteq X. \] Thus $C_w$ is a Schubert cell, $Y_w$ is its Schubert variety, and $[Y_w]$ denotes its fundamental cycle. The smoothness hypothesis concerns the entire closure $Y_w$.

For clarity of notation, the usual Gelfand--Zetlin array convention for a real tuple $\lambda_1\geq\cdots\geq\lambda_n$ describes $Q_\lambda$ by coordinates $x_{i,j}$, with $1\leq j\leq i\leq n-1$, fixed top row $x_{n,j}=\lambda_j$, and inequalities \[ x_{i+1,j}\geq x_{i,j}\geq x_{i+1,j+1} \qquad(1\leq j\leq i\leq n-1). \] This convention explains the polytope notation; it does not resolve the source's missing hypotheses on $\lambda$, including whether strict inequalities or integrality are required.

Source question, with unresolved terminology retained. For the intended parameters $n$ and $\lambda$, does every Schubert cell whose closure is smooth have its Schubert cycle represented by an admissible face of $Q_\lambda$? Schematically, the requested assertion is \[ \forall w\in S_n,\qquad Y_w\text{ smooth} \ \Longrightarrow\ \exists\,\Gamma\text{ a face of }Q_\lambda \text{ such that }\Gamma\text{ is admissible and represents }[Y_w]. \] Here "admissible" and "represents" retain their intended source-specific meanings: the supplied record does not define admissibility or supply a map from faces to Schubert cycles. Consequently, this is a formulation schema requiring those definitions before it becomes a fully specified mathematical conjecture. The phrase that face combinatorics should "capture the cycle's geometry" supplies no additional precise condition. The record also does not specify the allowed parameters or whether the assertion must hold for every permitted $\lambda$.

Short English statement

The question asks whether every Schubert variety with smooth closure in the complete flag variety for $GL_n({\mathbb{C}})$ has its fundamental cycle represented by an admissible face of a Gelfand--Zetlin polytope. The source leaves admissibility, the face-to-cycle correspondence, and the hypotheses and quantifiers on $\lambda$ unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001132$, source problem code catalogue entry, titled "Admissible Gelfand--Zetlin Faces Representing Smooth Schubert Cycles." The collection is Oberwolfach Reports Open Problems; the supplied proposed year is 2009. No individual proposer is identified.
• Supplied report citation: Toric Geometry (2009), Oberwolfach Reports, pp. 5--74, DOI 10.4171/owr/2009/01. Listed also as "original OWR report": https://doi.org/10.4171/owr/2009/01. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Gelfand--Zetlin polytopes and flag varieties, https://pure.mpg.de/rest/items/item_3122679/component/file_3122680/content. No author, publication details, or pinpoint location is supplied for this reference.
• Literature assessment dated exactly 2026-08-21: the supplied curation labels the problem open and states that the smooth-closure-to-admissible-face assertion remains a conjecture in the primary Gelfand--Zetlin treatment, while known examples and smoothness criteria do not prove it in general. This is supplied dated curation, not independently verified source consultation or a new literature assessment. The background expressly requests independent verification of time-sensitive status and cited claims.