The distinguished involutions with a-value n2−3n+3 in the Weyl group of type Dn

✍ Scribed by Chen Cheng Dong; Liu Jia Chun

• Book ID: 104140650
• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 114 KB
• Volume: 265
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1016/s0021-8693(03)00147-9

✦ Synopsis

Let (W, S) be a Weyl group and H its associated Hecke algebra. Let A = Z[u, u -1 ] be the Laurent polynomial ring. Kazhdan and Lusztig [Representation of Coxeter groups and Hecke algebras, Invent. Math. 53 (1979) [165][166][167][168][169][170][171][172][173][174][175][176][177][178][179][180][181][182][183][184] introduced two A-bases {T w } w∈W and {C w } w∈W for the Hecke algebra H associated to W . Let Y w = y w u l(w)-l(y) T y . Then {Y w } w∈W is also an A-base for the Hecke algebra. In this paper we give an explicit expression for certain Kazhdan-Lusztig basis elements C w as A-linear combination of Y x 's in the Hecke algebra of type D n . In fact, this gives also an explicit expression for certain Kazhdan-Lusztig basis elements C w as A-linear combination of T x 's in the Hecke algebra of type D n . Thus we describe also explicitly the Kazhdan-Lusztig polynomials for certain elements of the Weyl group. We study also the joint relation among some elements in W and some distinguished involutions with a-value n 2 -3n + 3 in the Weyl group of type D n .