The leading coefficient of certain Kazhdan-Lusztig polynomials of the permutation group σn

In this paper we show that the leading coefficient mu (y, w) of certain Kazhdan-Lusztig polynomials P-y,P-w of the permutation group G(n) of 1, 2, ..., n are not greater than 1. More precisely, we show that the leading coefficients mu(y w) are not greater than 1 whenever a(y) < a(w), where a : G(n) --> N is the function defined in [G. Lusztig, Cells in affine Weyl groups, in: Algebraic Groups and Related Topics, in: Adv. Stud. Pure Math., vol. 6, Kinokunia-North-Holland, 1985, pp. 255-287]. See 1.5 for a simple interpretation of the function a. (C) 2004 Elsevier Inc. All rights reserved.