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In this thesis, we investigate the asymptotics of random partitions chosenaccording to probability measures coming from the representation theory of thesymmetric groups $S_n$ and of the finite Chevalley groups $GL(n,F_q)$ and$Sp(2n,F_q)$. More precisely, we prove laws of large numbers and central limittheorems for the $q$-Plancherel measures of type A and B, the Schur-Weylmeasures and the Gelfand measures. Using the RSK algorithm, it also givesresults on longest increasing subsequences in random words. We develop atechnique of moments (and cumulants) for random partitions, thereby using thepolynomial functions on Young diagrams in the sense of Kerov and Olshanski. Thealgebra of polynomial functions, or observables of Young diagrams is isomorphicto the algebra of partial permutations; in the last part of the thesis, we tryto generalize this beautiful construction.

Random partitions and asymptotic theory of symmetric groups, Hecke algebras and finite Chevalley groups