Kazhdan-Lusztig polynomials, tight quotients and Dyck superpartitions.
The aim of the present paper is to study the parabolic Kazhdan-Lusztig polynomials (or KL-polynomials in short) for the tight quotients of the symmetric group \(S_n\). The tight quotients were introduced by \textit{J. R. Stembridge} [in Electron. J. Comb. 11, No. 2, Research paper R14 (2004; Zbl 1067.20053)] who classified them for finite Coxeter groups. Denote \([k]:=\{1,2,\dots,k\}\) for any \(k\in\mathbb{N}\). Let \(S=\{s_i\mid i\in[n-1]\}\) with \(s_i=(i,i+1)\) be the transposition of \(i\) and \(i+1\). Then the non-trivial tight quotients of \(S_n\) have the form \(W^J\), the set of minimal left coset representatives of \(S_n\) with respect to the parabolic subgroup generated by \(J\) with either \(|J|=n-2\) or \(J=S\setminus\{s_i,s_{i+1}\}\), \(i\in[n-2]\). The parabolic KL-polynomials for the tight quotients \(W^J\) of \(S_n\), \(|J|=n-2\), were studied by \textit{F. Brenti} [in Pac. J. Math. 207, No. 2, 257-286 (2002; Zbl 1059.20007)]. The present paper completes the study for all the tight quotients \(W^J\) of \(S_n\). An explicit closed combinatorial formula is given for the parabolic KL-polynomials of all tight quotients of \(S_n\). For \(J=S\setminus\{s_i,s_{i+1}\}\), \(i\in[n-2]\), each of the parabolic KL-polynomials for the tight quotients \(W^J\) of \(S_n\) is encoded in a pair of superpartitions, which is either zero or a monic monomial. From the formula given in the present paper, one can derive some consequences including the formula of Brenti [loc. cit.] and some new identities for the ordinary KL-polynomials and for their leading terms.
scientific article; zbMATH DE number 5955939
scientific article