Immanants and finite point processes (Q1584658)

Given a Hermitian, nonnegative definite kernel \(K\) and a character \(\chi\) of the symmetric group on \(n\) letters, the function \(K^\chi\) is symmetric and nonnegative, and, under suitable conditions, is also non-trivial and integrable with respect to the product measure \(\mu^{\otimes n}\) for a given measure \(\mu\). In this case, \(K^\chi\) can be normalized to be a symmetric probability density. The case where \(K\) gives rise to an orthogonal projection of \(L^2(\mu)\) onto a finite-dimensional subspace is studied here in detail. The representation theory of the symmetric group is used to compute the normalization constant and identify the \(k\)th-order marginal densities for \(1\leq k\leq n\) as linear combinations of analogously defined immanantal densities. Connections with inequalities for immanants, particularly the permanental dominance conjecture of \textit{E. H. Lieb} [J. Math. Mech. 16, 127-134 (1966; Zbl 0144.26802)] are considered, and asymptotics when the dimension of the subspace goes to infinity are presented.