On parabolic Kazhdan–Lusztig R-polynomials for the symmetric group

Parabolic R-polynomials were introduced by Deodhar as parabolic analogues of ordinary R-polynomials defined by Kazhdan and Lusztig. In this paper, we are concerned with the computation of parabolic R  -polynomials for the symmetric group. Let SnSn be the symmetric group on {1,2,…,n}{1,2,…,n}, and let S={si|1≤i≤n−1} be the generating set of SnSn, where for 1≤i≤n−11≤i≤n−1, sisi is the adjacent transposition. For a subset J⊆SJ⊆S, let (Sn)J(Sn)J be the parabolic subgroup generated by J  , and let (Sn)J(Sn)J be the set of minimal coset representatives for Sn/(Sn)JSn/(Sn)J. For u≤v∈(Sn)Ju≤v∈(Sn)J in the Bruhat order and x∈{q,−1}x∈{q,−1}, let Ru,vJ,x(q) denote the parabolic R-polynomial indexed by u and v  . Brenti found a formula for Ru,vJ,x(q) when J=S∖{si}J=S∖{si}, and obtained an expression for Ru,vJ,x(q) when J=S∖{si−1,si}J=S∖{si−1,si}. In this paper, we provide a formula for Ru,vJ,x(q), where J=S∖{si−2,si−1,si}J=S∖{si−2,si−1,si} and i   appears after i−1i−1 in v. It should be noted that the condition that i   appears after i−1i−1 in v is equivalent to that v   is a permutation in (Sn)S∖{si−2,si}(Sn)S∖{si−2,si}. We also pose a conjecture for Ru,vJ,x(q), where J=S∖{sk,sk+1,…,si}J=S∖{sk,sk+1,…,si} with 1≤k≤i≤n−11≤k≤i≤n−1 and v   is a permutation in (Sn)S∖{sk,si}(Sn)S∖{sk,si}.