Bruhat-interval combinatorial invariance conjecture

Prove that Kazhdan–Lusztig polynomials associated with two pairs of permutations are equal whenever the corresponding intervals in Bruhat order are isomorphic as abstract posets.

Background

Kazhdan–Lusztig polynomials P_{u,v}(q) are indexed by pairs of permutations u≤v in Bruhat order. The paper formulates the Combinatorial Invariance Conjecture, which asserts that these polynomials depend only on the abstract isomorphism type of the associated Bruhat interval. The authors note that the converse fails: equal polynomials do not necessarily imply isomorphic intervals.

References

The following conjecture was made independently by Dyer and Lusztig, see also Brenti-Casselli-Martinelli . The Kazhdan-Lusztig polynomials are an important family of polynomials in $\mathbb{Z}[q]$ which are indexed by two permutations $u\leq v$ in Bruhat order.

— Introduction to the Cohomology of the Flag Variety  (2506.21064 - Billey et al., 26 Jun 2025) in Problem KL-intervals, Section on Bruhat Order on Permutations