The Trace Formula of the Weyl Group Representations of the Symmetric Group

Abstract

Let g be a complex simple Lie algebra, h its Cartan subalgebra, and Σ and $\prod =\left\{ {{\beta }_{1}},\ldots ,{{\beta }_{i}} \right\}$the root system of g and a basis of the root system, respectively. It is known that the Weyl group W(g) is a finite group generated by reflections [3]

$${w_{{\rm B}i}}:\xi \in h,\xi \to \xi - \frac{{2\left( {\xi ,} \right.}}{{\left( {{\beta _{i,}}} \right.}}\frac{{\left. {{\beta _i}} \right)}}{{\left. {{\beta _i}} \right)}}{\beta _{i \bullet }}$$