Kazhdan-Lusztig basis for generic Specht modules

TL;DR

This paper develops a cellular basis for Hecke algebras of finite Coxeter groups, introduces a general theory of Specht modules, and provides algorithms for W-graphs, with applications to type A.

Contribution

It introduces a new cellular basis for Hecke algebras and a general framework for Specht modules, along with algorithms for W-graphs in the generic case.

Findings

Constructed a cellular basis indexed by subsets of simple reflections.

Developed an algorithm for W-graphs associated with Kazhdan-Lusztig cells.

Applied the theory to construct W-graphs for type A Hecke algebras.

Abstract

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 $A$.