Cartan Determinants and Shapovalov forms

Details Cartan Determinants and Shapovalov forms Cartan Determinants and Shapovalov forms

2001-11-29

arxiv.org/abs/math/0111306v1

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

Mathematics

Representation Theory

Scientific paper

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.

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