2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62761We compute the determinant of the Gram matrix of the Shapovalov form on weight spaces of the basic representation of an affine Kac-Moody algebra of ADE type (possibly twisted). As a consequence, we obtain explicit formulae for the determinants of the Cartan matrices of $p$-blocks of the symmetric group and its double cover, and of the associated Hecke algebras at roots of unity.Representation Theory17B67; 20G05text