Characterization of special matchings for arbitrary Coxeter groups

Establish a characterization of special matchings of arbitrary Bruhat intervals in arbitrary Coxeter groups, extending the characterization known for Bruhat intervals in symmetric groups.

Background

The paper studies special matchings of Bruhat intervals in the symmetric group and presents a complete characterization in that setting. Such matchings yield recursive formulas for Kazhdan–Lusztig R-polynomials and, consequently, Kazhdan–Lusztig polynomials. The authors explicitly identify the corresponding characterization for arbitrary Coxeter groups as unresolved, although special matchings are known to compute Kazhdan–Lusztig polynomials for intervals beginning at the identity element.

References

This is a consequence of the main achievement of , which provides a characterization of special matchings of arbitrary Bruhat intervals in the symmetric group, a result which is still unknown for arbitrary Coxeter groups.

— A simple classification of the special matchings in the symmetric group  (2610.21647 - Caselli et al., 15 Oct 2026) in Section 1, Introduction