Let (W, S) be a Weyl group. Let A = Z[u, u−1] be the Laurent polynomial ring in an indeterminate u. Kazhdan and Lusztig [Invent. Math. 33 (1979) 165–184] introduced two A-bases {Tw}w∈W and {Cw}w∈W for the Hecke algebra H Then associated to W. Let Yw =∑y≤w ul(w)−l(y)Ty . {Yw}w∈W is also an A-basis for the Hecke algebra. In this paper we assume W of type Dn and we express certain Kazhdan–Lusztig basis elements Cw as A-linear combination of Yx’s. This in turn gives an explicit expression for certain Kazhdan–Lusztig basis elements Cw as A-linear combination of Tx ’s. Thus we describe explicitly the Kazhdan–Lusztig polynomials for certain pairs of elements of W. We also study the joint relation among some elements in W. In particular, we find certain distinguished involutions in the two-sided cell Ωt of W with a-value 1/2(n2 − n + 4t2 − 2t) for 1 ≤ 2t ≤ n and n even (1/2 (n2 − n + 4t2 + 2t) for n odd), where the two-sided cell Ωt does not contain the longest element (w0)J in subgroup WJ of W for any J ⊂ S.

Let (W, S) be the finite Weyl group with S as its Coxeter generating set. For w ∈W, let R(w) = {si ∈ S | l(wsi) < l(w)} and L(w) = {si ∈ S | l(siw) < l(w)}, where we denote by l(w) the minimal length of an expression of w as a product of simple reflections. To any Weyl group one can associate a corresponding finite-dimensional algebra called 0-Hecke algebra H where K is any field. Norton [J. Austral. Math. Soc. Ser. A 27 (1979)337–357] pointed out that the principal indecomposable modules and the irreduciblemodules over the 0-Hecke algebra H parametrized by a subset J of S. We denote by U(Jˆ) and M(Jˆ) respectively the principal indecomposable module and the irreducible module parametrized by J . For two subset J , L of S, let CJL = the number of times M(L) is a composition factor of U(Jˆ). Norton [J. Austral. Math. Soc. Ser. A 27 (1979)337–357] shown that CJL = |YL ∩ (YJ )−1| where YL = {w ∈W|R(w) = L} and(YJ)−1 = {w ∈W | L(w) = J}. In this article, we describe explicitly CJL for the 0-Hecke algebra of type F4 by applying the canonical expression of every element in theWeyl group of type F4. Thus we determine the Cartan matrix over the 0-Hecke algebra of type F4.

Let (W, S) be aWeyl 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) 165–184] introduced two A-bases {Tw}w∈W and {Cw}w∈W for the Hecke algebra H associated to W. Let Yw =∑y≤w ul(w)−l(y)Ty. Then {Yw}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 Cw as A-linear combination of Yx’s in the Hecke algebra of type Dn. In fact, this gives also an explicit expression for certain Kazhdan–Lusztig basis elements Cw as A-linear combination of Tx ’s in the Hecke algebra of type Dn. 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 n2 − 3n + 3 in the Weyl group of type Dn.

In this paper we determine expression of Kazhdan–Lusztig basis elements Cw over the Hecke Algebra of Type An where w = a(j, i) is an element in Weyl group of type An. We also find joined relation of some elements in W. 2002 Elsevier Science (USA). All rights reserved.