Kazhdan-Lusztig basis for generic Specht modules

In this paper, we let \(\Hecke\) be the Hecke algebra associated with a finite Coxeter group \(W\) and with one-parameter, over the ring of scalars \(\Alg=\mathbb{Z}(q, q^{-1})\). With an elementary method, we introduce a cellular basis of \(\Hecke\) indexed by the sets \(E_J (J\subseteq S)\) and obtain a general theory of "Specht modules". We provide an algorithm for \(W\!\)-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 \(W\!\)-graphs for the Hecke algebra of type \(A\).