Log-Concavity of Multiplicities with Application to Characters ofU(∞)

✍ Scribed by Andrei Okounkov

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 380 KB
• Volume: 127
• Category: Article
• ISSN: 0001-8708
• DOI: 10.1006/aima.1997.1622

✦ Synopsis

The log-concavity of the reduction multiplicities for the classical groups of type A n , B n , C n is proved, moreover the skew Schur functions s *Â+ are shown to be logconcave coefficient by coefficient. The results are applied to the calculation of the characters of the infinite-dimensional classical groups. The log-concavity of density of the push-forward of the Liouville measure on coadjoint orbits under moment map is proved. 1997 Academic Press 1. LOG-CONCAVITY OF MULTIPLICITIES OF IRREDUCIBLE REPRESENTATIONS FOR UNITARY GROUPS 1.1. Given a compact Lie group G denote by W(G) its weight lattice and by W + (G) the semigroup of dominant weights. Suppose that * # W + (U(n)), and + # W + (U(k)). By Mult(* | +) denote the multiplicity of the irreducible representation of U(k) with highest weight + in the irreducible representation of U(n) with highest weight *. Put Mult(* | +)=0 if * or + is not a dominant weight. 1.2. A real function f on an Abelian semigroup L is called concave if for any a, b, c # L such that a+b=2c we have f(a)+ f (b) 2f (c).