Nonnegativity Of H-polynomials Coefficient Of Hecke Algebra Of Type H

Lusztig proved that a-function is constant on the two-sided cells of Weyl group W. His proof depends on three properties of W as follows:(1) a-function is up-bounded in Coxeter group.(2) Coefficients of KL polynomials are nonnegative.(3) Coefficients of h-polynomial are nonnegative.Dihedral group (I₂(m)) and Coxeter group of group H₃, H₄) are noncrystallographic in irreducible finite Coxeter group. Obviously I₂(m) satisfies those hypotheses. So we consider group H₃ and H₄. Because Coxeter group of type H is finite, so a-function is up-bounded. Coefficients of KL polynomials was proved by Alvis. So the aim of this paper is to prove coefficient of h-polynomial of group H₃ and H₄ is nonnegative.Results:(1) Coefficients of h-polynomial of group H₃ type are nonnegative.(2) Coefficients of h-polynomial of fully commutative elements of group H₄ type are nonnegative.And we check two statements :(1) a-function is constant on the two-sided cells of group H₃. We compute out all the constants.(2) There is just a distinguished involution on every left cell of group H₃. We find out all distinguished involutions.(3) a-function is constant on the two-sided cells of fully commutative elements of group H₃. We compute out all the constants.(4) There is just a distinguished involution on every left cell of fully commutative elements of group H₃. We find out all distinguished involutions.