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On the Impossibility of Parabolic Factorization of certain Kazhdan-Lusztig Basis Elements
[Submitted on 11 Jun 2026] · 2026-06-15 · via cs updates on arXiv.org

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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.

Submission history

From: EPTCS [view email] [via EPTCS proxy]
[v1] Thu, 11 Jun 2026 10:29:37 UTC (47 KB)