Abstract
Two representations of a reductive group G are spectrally equivalent if the same irreducible representations appear in both of them. The semigroup of finite-dimensional representations of G with tensor product and up to spectral equivalence is a rather complicated object. We show that the Grothendieck group of this semigroup is more tractable and we give a description of it in terms of moment polytopes of representations. As a corollary, we give a proof of the Kazarnovskii theorem on the number of solutions in G of a system f 1 = · · · = f m = 0, where m = dim(G) and each f i is a generic function in the space of matrix elements of a representation π i of G.
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Kaveh, K., Khovanskii, A.G. Moment polytopes, semigroup of representations and Kazarnovskii’s theorem. J. Fixed Point Theory Appl. 7, 401–417 (2010). https://doi.org/10.1007/s11784-010-0027-7
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DOI: https://doi.org/10.1007/s11784-010-0027-7