Integrals over Grassmannians and random permutations
(2004) Advances in mathematics — Vol. 181, n° 1, p. 190-249 (2004)
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- 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 Painleve-like equations; in the complex case, they are solutions to genuine Painleve 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 less than or equal tok, 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. (C) 2003 Elsevier Inc. All rights reserved.
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Adler, M., & Van Moerbeke, P. (2004). Integrals over Grassmannians and random permutations. Advances in mathematics, 181(1), 190-249. https://doi.org/10.1016/j.aim.2003.07.008 (Original work published 2004)