Log-concavity of multiplicities with application to characters of \(U(\infty)\)
Let \(L\) be an abelian semigroup. A real function \(f\) on \(L\) is called log-concave if, for all \(a,b,c\in L\) with \(2c=a+b\), the inequality \(f(a)f(b)\leq f(c)^2\) holds. Now let \(L\) be the lattice of weights of the compact Lie group \(U(n)\times U(k)\), where \(n\), \(k\) are natural numbers with \(n\geq k\). For dominant \((\lambda,\mu)\), denote by \(\text{Mult}(\lambda, \mu)\) the multiplicity of the irreducible \(U(k)\)-representation with highest weight \(\mu\) in the restriction of the irreducible \(U(n)\)-representation with highest weight \(\lambda\) to its natural \(U(k)\)-subgroup, and set \(\text{Mult}(\lambda, \mu)=0\) for non-dominant \((\lambda, \mu)\). The author shows that \(\text{Mult}(\lambda, \mu)\) is log-concave on \(L\). The proof uses combinatorial properties of the Gelfand-Tsetlin patterns and actually shows that the skew Schur functions \(s_{\lambda/\mu}\) on \(L\) are log-concave coefficient by coefficient. Similar results are proved for \((Sp(2n),Sp(2k))\) and \((SO(2n+1),SO(2k+1))\) instead of \((U(n),U(k))\). The author gives two applications of the result for unitary groups: a proof of the (known) fact that the Duistermaat-Heckman measure on a coadjoint orbit of \(\lambda\) has log-concave density, and limit computations related to a direct proof [cf. \textit{S. V. Kerov} and \textit{M. Vershik}, Sov. Math., Dokl. 26, 570-574 (1982); translation from Dokl. Akad. Nauk SSSR 267, 272-276 (1982; Zbl 0524.22017)] of an explicit (known) description of the characters of the infinite unitary group \(U(\infty)\).
scientific article; zbMATH DE number 1076914
scientific article