Why would multiplicities be log-concave? (English) Zbl 1063.22024

Okounkov, Andrei

Duval, Christian (ed.) et al., The orbit method in geometry and physics. In honor of A. A. Kirillov. Papers from the international conference, Marseille, France, December 4–8, 2000. Boston, MA: Birkhäuser (ISBN 0-8176-4232-3/hbk). Prog. Math. 213, 329-347 (2003).

Summary: A basic property of entropy in statistical physics is that it is concave as a function of energy. The analog of this in representation theory would be the concavity of the logarithm of the multiplicity of an irreducible representation as a function of its highest weight. We discuss various situations where such concavity can be established or reasonably conjectured and consider some implications of this concavity.
For the entire collection see [Zbl 1027.00015].

MSC:

22E70	Applications of Lie groups to the sciences; explicit representations
81R10	Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations
82B10	Quantum equilibrium statistical mechanics (general)

Keywords:

entropy; representation theory; logarithm; multiplicity; irreducible representation; highest weight