Positivity in Affine Kazhdan–Lusztig Theory

Definitions and mathematical statement

The supplied question concerns affine type $A$ and uses the symbols $N_{\lambda_{i+1}}$ and $N_{\mu_i}$, without defining these objects, their ambient algebra or module, their coefficient ring, or their indexing sets. The subscripts below reproduce the supplied cleaned statement; their intended mathematical meaning is unresolved.

In general, given a coefficient ring $R$, a designated set $R_{\geq 0}\subseteq R$ of nonnegative coefficients, and a specified expansion \[ N_{\lambda_{i+1}}=\sum_{\mu}c_{\lambda,\mu,i}N_{\mu_i}, \] positivity of that expansion means \[ c_{\lambda,\mu,i}\in R_{\geq 0} \quad\text{for every index $\mu$ occurring in the expansion.} \] This is a conditional explanation of the word "positive," not a specification of the missing coefficient ring, index ranges, basis properties, or permitted sums.

Source question. In affine type $A$, is the decomposition of $N_{\lambda_{i+1}}$ in terms of the $N_{\mu_i}$ positive?

A fully quantified conjecture cannot be recovered from this record: the affine rank, admissible values of $i,\lambda,\mu$, meaning of the indexed $N$-objects, and intended notion of coefficient positivity are not supplied. In particular, the record does not determine whether $i$ denotes a level of a family of pre-canonical bases or forms part of a weight subscript. The supplied literature assessment associates the question with pre-canonical bases, but does not provide their definitions. The corrupted original excerpt also mentions a second problem concerning positivity properties of Kazhdan--Lusztig polynomials and then reports an Elias--Williamson positivity result. Those passages do not specify an additional unresolved mathematical assertion and cannot be used to identify the displayed conjecture with ordinary Kazhdan--Lusztig polynomial positivity.

Short English statement

The record asks whether certain expansion coefficients in affine type $A$ are nonnegative. It does not define the elements, indices, coefficient ring, or meaning of nonnegativity, so the question remains a statement template requiring source clarification and human mathematical review. The supplied dated assessment distinguishes this pre-canonical-basis question from the reported positivity theorem for Kazhdan--Lusztig polynomials.

Sources

• Supplied local registry record: permanent UnsolvedMath ID $30005218$, title "Positivity in Affine Kazhdan--Lusztig Theory," exact source problem code catalogue entry. The registry supplies the labels partially_solved and corrected_verified; these are source labels, not independent verification of the mathematics.
• Supplied report citation: Character Theory and Categorification (2023), Oberwolfach Reports, pp. 2267--2333, DOI https://doi.org/10.4171/owr/2022/39. The registry separately gives proposed year $2022$. The supplied description says that Oberwolfach Reports document mathematical workshops and their open-problem discussions.
• Attribution appearing in the original excerpt: "Some aspects of Kazhdan--Lusztig polynomials," Nicolas Libedinsky, joint work with Leonardo Patimo, David Plaza and Geordie Williamson. The record does not separately identify a proposer of this conjecture. It attributes the reported Kazhdan--Lusztig positivity breakthrough to Ben Elias and Geordie Williamson.
• Supplied literature reference: Patimo--Plaza--Williamson, pre-canonical bases, https://doi.org/10.1016/j.aim.2022.108255.
• Supplied literature reference: Elias--Williamson, Soergel Hodge theory, https://doi.org/10.4007/annals.2014.180.3.6.
• Supplied literature reference: original Oberwolfach report, https://doi.org/10.4171/owr/2022/39.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved and states that affine type $A$ pre-canonical-basis positivity remains open beyond verified ranges, while Elias and Williamson proved positivity of Kazhdan--Lusztig polynomials in full generality. The verified ranges are not identified. This is supplied dated curation derived from the supplied OWR material, not an independently verified literature assessment; its time-sensitive status and cited claims require independent checking.