Ben Spahiu | Searcharxiv

arXiv2026

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

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.

cs.DM↗

arXiv2026

On Kazhdan--Lusztig basis elements having no reversal factorization

For $w$ in the symmetric group $S_n$, let $\widetilde C_w$ 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 \begin{equation*} \widetilde C_w = \frac1{f(q)} \widetilde C_{v^{(1)}} \cdots \widetilde C_{v^{(r)}}, \end{equation*} with $v^{(1)},\dotsc,v^{(r)}$ maximal elements of parabolic subgroups of $S_n$ and $f(q) \in \mathbb N[q]$ depending on these, provides cancellation-free combinatorial interpretations of the polynomials $\{P_{v,w}(q) \,|\, v \in S_n \}$ appearing in the expansion $\sum_v P_{v,w}(q) T_v$ of $\widetilde C_w$ 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 $\widetilde C_w$ has not yet been characterized, we apply a result of Gaetz -- Gao [Adv. Math. 457 (2024) Paper No. 109941] to describe a set admitting no such factorization.

math.CO↗