Jonathan Brundan and Alexander Kleshchev

"Cartan determinants and Shapovalov forms"

Abstract: We 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.