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Brunn-Minkowski inequality for multiplicities. (English) Zbl 0893.52004
Let a connected reductive group \(G\) act in a vector space \(V\). Suppose \(X\) is a closed \(G\)-stable irreducible subvariety of \(\mathbb P(V)\). Let \(F[X] =\bigoplus_mF[X]_m\) be the homogeneous coordinate ring of \(X\). Consider the decomposition of \(F[X]_m\) as \(G\)-module \(F[X]_m =\bigoplus_{\lambda}\mu_m(\lambda)V^\lambda\), where \(V^\lambda\) is the irreducible \(G\)-module with highest weight \(\lambda\) and \(\mu_m(\lambda)\) are the multiplicities. Let us consider \(\mu_m\) as a measure supported on the weight lattice \(P\) of \(G\). Put \(\varGamma=\text{Convex hull}\left(\bigcup_m(\text{supp}\mu_m)/m\right)\). This is a convex subset of the real vector space \(P\bigotimes_{\mathbb Z}\mathbb R\). It is known that the total mass of \(\mu_m\) is a polynomial in \(m\) for sufficiently large \(m\) (denote by \(k\) its degree) and \(m^{\dim\varGamma -k}\mu_m(m\cdot\lambda)\overset\text{weak}\longrightarrow\mu(\lambda)d\gamma\), where \(d\gamma\) is the Lebesgue measure supported on \(\varGamma\) and the density \(\mu(\lambda)\) is a piecewise-polynomial function.
The aim of this paper is to prove the following: the function \(\mu^{1/(k-\dim\varGamma)}\) is concave on \(\varGamma\); the function \(\log \mu\) is concave on \(P\bigotimes_\mathbb Z\mathbb R\).
Reviewer: S.M.Pokas (Odessa)

MSC:
52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
13H15 Multiplicity theory and related topics
52A40 Inequalities and extremum problems involving convexity in convex geometry
52A07 Convex sets in topological vector spaces (aspects of convex geometry)
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