Integrals over Grassmannians and random permutations |
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Authors: | M Adler P van Moerbeke |
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Institution: | a Department of Mathematics, Brandeis University, Waltham, MA 02454, USA b Department of Mathematics, Université de Louvain, 1348 Louvain-la-Neuve, Belgium |
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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 of p-dimensional planes in real, complex or quaternion n-space , 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 have remarkable expansions in x, related to random words of length ? 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 =? and is thus trivial. |
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