Multideterminantal measures

TL;DR

This paper introduces multideterminantal probability measures on $[k]^n$, generalizing determinantal measures, and characterizes their kernels, with applications to the Grassmannian, dimer, spanning tree models, and permutation groups.

Contribution

It defines and characterizes multideterminantal measures, extending determinantal measures to a broader setting with explicit kernel descriptions and applications.

Findings

Characterization of kernels of pure $k$-determinantal measures.

Construction of all kernels of pure determinantal measures via Grassmannian elements.

Complete characterization of determinantal measures on the permutation group.

Abstract

We define multideterminantal probability measures, a family of probability measures on $[k]^n$ where $[k]=\{1,2,\dots,k\}$, generalizing determinantal measures (which correspond to the case $k=2$). We give examples coming from the positive Grassmannian, from the dimer model and from the spanning tree model. We characterize kernels of \emph{pure} $k$-determinantal measures as those arising from $k$-tuples of Grassmannian elements whose maximal minors have certain sign restrictions. As a special case we construct all kernels of pure determinantal measures via a pair of elements of $Gr_{n_1,n}$ having corresponding Pl\"ucker coordinates of the same signs. We also define and completely characterize determinantal probability measures on the permutation group $S_n$.