Cartan Determinants and Shapovalov forms

2001/11/29 by Jonathan Brundan, Alexander Kleshchev
Mathematics · #math.RT #msc:17B67 #msc:20G05

paper · pdf

published as Math. Ann. 324 (2002), 431-449.

arxiv created 2001/11/29 · arxiv updated 2009/11/30

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.

Related

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• Lattice of dominant weights of affine Kac-Moody algebras
• Graded Cartan determinants of the symmetric groups
• Shapovalov elements of classical and quantum groups
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