  We present equalities in law between the spectra of the minors of a GUE
matrix and some maximal functionals of independent Brownian motions. In turn,
these results allow to recover the limiting shape (properly centered and
scaled) of the RSK Young diagrams associated with a random word as a function
of the spectra of these minors. Since the length of the top row of the diagrams
is the length of the longest increasing subsequence of the random word, the
corresponding limiting law also follows.
