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 S n be the symmetric group on { 1 , 2 , … , n } , and let S = { s i | 1 ≤ i ≤ n − 1 } be the generating set of S n , where for 1 ≤ i ≤ n − 1 , s i is the adjacent transposition. For a subset J ⊆ S , let ( S n ) J be the parabolic subgroup generated by J , and let ( S n ) J be the set of minimal coset representatives for S n / ( S n ) J . For u ≤ v ∈ ( S n ) J in the Bruhat order and x ∈ { q , − 1 } , let R u , v J , x ( q ) denote the parabolic R -polynomial indexed by u and v . Brenti found a formula for R u , v J , x ( q ) when J = S ∖ { s i } , and obtained an expression for R u , v J , x ( q ) when J = S ∖ { s i − 1 , s i } . In this paper, we provide a formula for R u , v J , x ( q ) , where J = S ∖ { s i − 2 , s i − 1 , s i } and i appears after i − 1 in v . It should be noted that the condition that i appears after i − 1 in v is equivalent to that v is a permutation in ( S n ) S ∖ { s i − 2 , s i } . We also pose a conjecture for R u , v J , x ( q ) , where J = S ∖ { s k , s k + 1 , … , s i } with 1 ≤ k ≤ i ≤ n − 1 and v is a permutation in ( S n ) S ∖ { s k , s i } . [ABSTRACT FROM AUTHOR]