Closed formula for Kazhdan–Lusztig polynomial degree

Derive a closed formula for the degree of the Kazhdan–Lusztig polynomial P_{x,w}(q) indexed by a pair of permutations x and w.

Background

Kazhdan–Lusztig polynomials are integer polynomials indexed by pairs of permutations and have important connections to Schubert calculus, Hecke algebras, and symmetric-group representation theory. Although they can be computed recursively, the paper emphasizes that their structure remains poorly understood. In particular, the degree of P_{x,w}(q) is identified as a quantity for which no closed formula is currently available.

References

They can be computed via a recursive formula , nevertheless, in many ways they remain mysterious. For instance, there is no known closed formula for the degree of $P_{x,w}(q)$.

— Machine Learning meets Algebraic Combinatorics: A Suite of Datasets Capturing Research-level Conjecturing Ability in Pure Mathematics  (2503.06366 - Chau et al., 9 Mar 2025) in Section 3, subsection “The Coefficients of Kazhdan-Lusztig Polynomials (Open Problem)”