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Cartan Determinants and Shapovalov forms

Jonathan Brundan·Alexander Kleshchev
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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.

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BibTeXRIS

Jonathan Brundan, Alexander Kleshchev. 2001-11-29. Cartan Determinants and Shapovalov forms. https://arxiv.org/abs/math/0111306

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