2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/169253We provide an algorithm to calculate the invariant factors of the Shapovalov form on the standard $\Z$-lattice inside the basic representation of a Kac-Moody algebra of $ADE$ type, and give explicit formulae in some cases. The techniques developed reduce the problem to finding the invariant factors of a family of bilinear forms on the ring of symmetric functions, having Jack's symmetric functions as an orthonormal basis. These results have applications to the representation theory of Iwahori-Hecke algebras at roots of unity and the modular representation theory of symmetric groups.Representation TheoryGroup Theory17B67; 20G05text