Brunn-Minkowski inequality for multiplicities
type: Item
instance of: scholarly article
title: Brunn-Minkowski inequality for multiplicities
zbMATH Open document ID: 0893.52004
published in: Inventiones Mathematicae
publication date: 1997-01-05T00:00:00Z
review text: 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 \(P\bigotimes_\mathbb Z\mathbb R\).
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 13H15
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A07
zbMATH DE Number: 931976
zbMATH Keywords: Brunn-Minkowski inequality
zbMATH Keywords: multiplicity theory
zbMATH Keywords: logarithmic convexity
zbMATH Keywords: log-concavity of multiplicities
zbMATH Keywords: moment mappings
zbMATH Keywords: convexity properties of the moment mapping
Wikidata QID: Q56602748
author: Andrei Okounkov
reviewed by: Serguey M. Pokas
MaRDI profile type: Publication
full work available at URL: s002220050081
OpenAlex ID: W2094594105
DOI: 10.1007/S002220050081
description: scientific article; zbMATH DE number 931976