AN INVERSION FORMULA FOR (J, (J)over-cap)-RELATIVE KAZHDAN-LUSZTIG POLYNOMIALS

作者：Yin Yunchuan^*

来源：Communications in Algebra, 2009, 37(8): 2907-2919.

DOI：10.1080/00927870802623401

• Coxeter group
• Hecke algebra
• Kazhdan-Lusztig polynomial
• Solomon's ideals

The article proves a relative version of one of the results from the influential article [ 4] of Kazhdan and Lusztig which introduced the Kazhdan-Lusztig polynomials. Given a Coxeter group W and a set S of simple reflections, let H denote the corresponding Hecke algebra, it has a "standard" basis T(w) and another basis C(w) with many remarkable properties. The Kazhdan-Lusztig polynomials p(x, w) give the transition matrix between these bases. One of their results proved by Kazhdan and Lusztig is an inversion formula, which states that if W is finite with longest element w(0), then
Sigma({z vertical bar x <= z <= w}) epsilon(x)epsilon(z)P(x,z)P'(w0w,w0z) = delta(x,w)
for all x <= w in W.
The main result of this article generalizes this result to the following setting: for any subset J of S, we define elements eta(J), and xi((J) over cap), and consider the two " dual" ideals H eta(J)xi((J) over cap) and H eta((J) over cap)xi J, where their standard basis and a Kazhdan-Lusztig basis are, respectively, indexed by subsets E(J) and E((J) over cap) of W.

• 出版日期2009
• 单位上海大学