On the Impossibility of Parabolic Factorization of certain Kazhdan-Lusztig Basis Elements
Abstract
For w in the symmetric group Sn, let Cw be the corresponding modified, signless Kazhdan-Lusztig basis element of the type-A Hecke algebra Hn(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)Cw = Cv1 ... Cvr, with v1, . . . , vr maximal elements of parabolic subgroups of Sn and f(q) in N[q] depending on these, provides cancellation-free combinatorial interpretations of the polynomials (Pv,w(q) | v in Sn) appearing in the expansion of Cw in terms of the natural basis (Tv | v in Sn) of Hn(q). While the set of permutations w in Sn admitting such a factorization of Cw 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.
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