Integrals over Grassmannians and random permutations

Abstract

Testing the independence of two Gaussian populations involves the distribution of the sample canonical correlation coefficients, given that the actual correlation is zero. The “Laplace transform” (as a function of x) of this distribution is not only an integral over the Grassmannian Gr(p, F n) of p-dimensional planes in real, complex or quaternion n-space F n , but is also related to a generalized hypergeometric function. Such integrals are solutions of Painlevé-like equations; in the complex case, they are solutions to genuine Painlevé equations. These integrals over Gr(p, C n) have remarkable expansions in x, related to random words of length ℓ formed with an alphabet of p letters 1,…, p. The coefficients of these expansions are given by the probability that a word ( i) contains a subsequence of letters p, p−1,…,1 in that order and ( ii) that the maximal length of the disjoint union of p−1 increasing subsequences of the word is ⩽ k, where k refers to the power of x. Note that, if each letter appears in the word, then the maximal length of the disjoint union of p increasing subsequences of the word is automatically =ℓ and is thus trivial.