On H <sub> 3 </ Sub> Kl Polynomial Combination Of Invariance

There is a conjecture regarding the Kazhdan-Lusztig polynomials, the so-called combinatorial invariance conjecture. This conjecture, proposed by Lusztig and independently by Dyer, states that if W₁ and W₂ are two Coxeter groups, u,v∈W₁,u 2,xu,v(q) = P_x,y(q).In other words, the Kazhdan-Lusztig polynomial of u,v supposedly depends only on the unlabeled abstract poset [u,v]. This conjecture is known to be true if [u,v] is a lattice and holds for intervals of rank≤4 .In fact, the combinatorial invariance conjecture is equivalent to the analogous statement for the R - polynomials(includes u = x = e ), and the R-polynomials are equivalent to the (?)-polynomials. Since it's easy to compute the (?) - polynomials, so we only study the (?) -polynomials.Our aim in this paper is to study and classify the intervals of length 5 in H₃ .We mainly study some easy intervals, list their (?) - polynomials and graphs (in Bruhat order). Therefore, we obtain while the graphs (so do intervals) are isomorphic, the corresponding (?) - polynomials are same, which is to say the combinatorial invariance is checked in these cases. On the contrary, it is not always true.