Cartan determinants and Shapovalov forms (Q1865568)

The authors 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, they 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 root of unity.