Integrals over Grassmannians and Random permutations

Details Integrals over Grassmannians and Random permutations Integrals over Grassmannians and Random permutations

2001-10-25

arxiv.org/abs/math/0110281v1

Mathematics

Combinatorics

68 pages

Scientific paper

In testing the independence of two Gaussian populations, one computes the distribution of the sample canonical correlation coefficients, given that the actual correlation is zero. The "Laplace transform" of this distribution is not only an integral over the Grassmannian of p-dimensional planes in complex n-space, but is also related to a generalized hypergeometric function. Such integrals are solutions of Painlev\'e-like equations. They also have expansions, related to random words of length l formed with an alphabet of p letters. Given that each letter appears in the word, the maximal length of the disjoint union of p increasing subsequences of the word clearly equals l. But the maximal length of the disjoint union of p-1 increasing subsequences leads to a non-trivial distribution. It is precisely this probability which appears in the expansion above.

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