Immanants and finite point processes

March, 2000
Report Number: 
Persi Diaconis & Steven N. Evans
J. Comb. Theory A. 98,175-191 (2002)

Given a Hermitian, non-negative definite kernel $K$ and a character $\chi$ of the symmetric group on $n$ letters, define the corresponding immanant function $K^\chi[x_1, \ldots, x_n] := \sum_{\sigma} \chi(\sigma) \prod_{i=1}^n K(x_i, x_{\sigma(i)})$, where the sum is over all permutations $\sigma$ of $\{1, \ldots, n\}$. When $\chi$ is the sign character (resp. the trivial character), then $K^\chi$ is a determinant (resp. permanent). The function $K^\chi$ is symmetric and non-negative, 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 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 $L^2(\mu)$ onto a finite--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, particularly as the distribution of eigenvalues for various models of random matrices. The representation theory of the symmetric group is used to compute the normalisation constant and identify the $k^{\mathrm{th}}$--order marginal densities for $1 \le k \le n$ as linear combinations of analogously defined immanantal densities. Connections with inequalities for immanants, particularly the permanental dominance conjecture of Lieb, are considered, and asymptotics when the dimension of the subspace goes to infinity are presented.

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