On an Isomorphism between Specht Module and Left Cell of $\mathfrak{S}_n$

Abstract

an explicit isomorphism between the above two modules and show that the base change matrix is uni-triangular for some ordering of the base. Using this isomorphism, if a partition $\lambda$ satisfies certain condition, we can construct the W-graph of the left cell of $\mathfrak{S}_{n}$ (without some edges which connect vertices having the same I-set) corresponding to $\lambda$ not using the Kazhdan-Lusztig polynomials $P_{y,w}$ but using the relations in Specht module (the Garnir relations) inductively.