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 KacMoody 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 November 2001
 arXiv:
 arXiv:math/0111306
 Bibcode:
 2001math.....11306B
 Keywords:

 Representation Theory;
 17B67;
 20G05
 EPrint:
 Math. Ann. 324 (2002), 431449.