Moment polytopes, semigroup of representations and Kazarnovskii's theorem

TL;DR

This paper links the structure of the semigroup of group representations to moment polytopes and provides a new proof of Kazarnovskii's theorem on solution counts for systems of equations in reductive groups.

Contribution

It describes the Grothendieck group of the semigroup of representations using moment polytopes and offers a novel proof of Kazarnovskii's theorem.

Findings

Grothendieck group of representations described via moment polytopes

New proof of Kazarnovskii's theorem on solution counts

Semigroup of representations analyzed through spectral equivalence

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 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(x) = ... = f_m(x) = 0, where m=dim(G) and each f_i is a generic function in the space of matrix elements of a representation pi_i of G.