Abstract
We consider the representation theory of a Hecke algebra H(n) of type A at a primitive lth root of unity, where l is a (not necessarily prime) positive integer. We assign a “Brauer character” to an H(n)-module. This is a class function on the set of l-regular elements of the symmetric group S n . We prove an analogue of Brauer’s orthogonality relations. As an application of the assignment of characters we show that the determinant of the Cartan matrix divides a power of l. A precise formula for this determinant has recently been obtained by Brundan and Kleshchev. They show in particular that this is a power of l, verifying a conjecture of A. Mathas.
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Donkin, S. (2003). Representations of Hecke Algebras and Characters of Symmetric Groups. In: Joseph, A., Melnikov, A., Rentschler, R. (eds) Studies in Memory of Issai Schur. Progress in Mathematics, vol 210. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0045-1_3
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DOI: https://doi.org/10.1007/978-1-4612-0045-1_3
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6587-0
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