Proof of Two Conjectures of Brenti and Simion on Kazhdan-Lusztig Polynomials
نویسنده
چکیده
We find an explicit formula for the Kazhdan-Lusztig polynomials Pui,a ,vi of the symmetric group S(n) where, for a, i, n ∈ N such that 1 ≤ a ≤ i ≤ n, we denote by ui,a = sasa+1 · · · si−1 and by vi the element ofS(n) obtained by inserting n in position i in any permutation ofS(n −1) allowed to rise only in the first and in the last place. Our result implies, in particular, the validity of two conjectures of Brenti and Simion [4, Conjectures 4.2 and 4.3], and includes as a special case a result of Shapiro, Shapiro and Vainshtein [13, Theorem 1]. All the proofs are purely combinatorial and make no use of the geometry of the corresponding Schubert varieties.
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