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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Received: 20 November 2001 / Revised version: 7 February 2002 / Published online: 27 June 2002
Second author partially supported by the NSF (grant no. DMS-9900134).
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Brundan, J., Kleshchev, A. Cartan determinants and Shapovalov forms. Math Ann 324, 431–449 (2002). https://doi.org/10.1007/s00208-002-0346-0
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DOI: https://doi.org/10.1007/s00208-002-0346-0