Immanants and Finite Point Processes

Givena Hermitian,non-negativede nitekernelK and a character of the symmetric group on n letters, de ne the corresponding immanant function K x1;::: ;xn := P Qn i=1 Kxi;x i, where the sum is over all permutations of f1;::: ;ng. When is thesigncharacterresp. the trivial character, then K is a determinant resp. permanent. The function K is symmetricand non-negative,and, under suitableconditions,is also non-trivial and integrable with respect to the product measure n for a given measure . In this case, K can be normalised to be a symmetric probability density. The determinantal and permanental cases or this construction correspond to the fermion and boson point processes which have been studied extensively in the literature. The case where K gives rise to an orthogonal projection of L2 onto a nite dimensional subspace is studied here in detail. The determinantal instance of this special case has a substantial literature because of its role in several problems in mathematical physics, par...

• Type: Journal
• Year: 2000
• Where: JCT
• Authors: Persi Diaconis, Steven N. Evans
• Keywords: Corresponding Immanant Function, JCT 2000, Symmetric Group, Symmetric Probability Density