Kazhdan-Lusztig basis for generic Specht modules

In this paper, we let $\text{Hecke}$ be the Hecke algebra associated with a finite Coxeter group $W$ and with one-parameter, over the ring of scalars $\text{Alg}=\mathbb{Z}(q,q^{-1})$. With an elementary method, we introduce a cellular basis of $\text{Hecke}$ indexed by the sets $E_J(J\subseteq S)$ and obtain a general theory of “Specht modules”. We provide an algorithm for $W\text{\hspace{0.17em}⁣}$-graphs for the “generic Specht module”, which associates with the Kazhdan and Lusztig cell ( or more generally, a union of cells of $W$ ) containing the longest element of a parabolic subgroup $W_J$ for appropriate $J\subseteq S$. As an example of applications, we show a construction of $W\text{\hspace{0.17em}⁣}$-graphs for the Hecke algebra of type $A$.