On the Impossibility of Parabolic Factorization of certain Kazhdan-Lusztig Basis Elements

2026-06-11

AI Summary

This study investigates whether Kazhdan–Lusztig basis elements in type A Hecke algebras admit cancellation-free factorizations into products of basis elements indexed by maximal elements of parabolic subgroups. By integrating the theory of Hecke algebras, properties of Kazhdan–Lusztig bases, the structure of parabolic subgroups, and recent results of Gaetz and Gao, the authors provide the first explicit characterization of a class of permutations whose corresponding basis elements do not admit such parabolic factorizations without cancellation. This result not only establishes the nonexistence of cancellation-free parabolic decompositions for specific permutations but also delineates the scope within which combinatorial interpretations may be attainable, thereby deepening the understanding of the structural properties of Kazhdan–Lusztig polynomials.

Abstract

For w in the symmetric group S_n, let C_w be the corresponding modified, signless Kazhdan-Lusztig basis element of the type-A Hecke algebra H_n(q). An extension [Ann. Comb. 25, no. 3 (2021) pp. 757-787] of a result of Deodhar [Geom. Dedicata 36, (1990) pp. 95-119] implies that any factorization of the form f(q)C_w = C_v1 ... C_vr, with v1, . . . , vr maximal elements of parabolic subgroups of S_n and f(q) in N[q] depending on these, provides cancellation-free combinatorial interpretations of the polynomials (P_v,w(q) | v in S_n) appearing in the expansion of C_w in terms of the natural basis (T_v | v in S_n) of H_n(q). While the set of permutations w in S_n admitting such a factorization of C_w has not yet been characterized, we apply a result of Gaetz-Gao [Adv. Math. 457 (2024) Paper No. 109941] to describe a set for which such a factorization cannot exist.

Problem

• Kazhdan-Lusztig basis
• parabolic factorization
• symmetric group
• Hecke algebra
• impossibility

Innovation

• Kazhdan-Lusztig basis
• parabolic factorization
• Hecke algebra
• symmetric group
• combinatorial interpretation