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We study the decompositions into irreducible components of tensor productsand restrictions of irreducible representations of classical Lie groups as therank of the group goes to infinity. We prove the Law of Large Numbers for therandom counting measures describing the decomposition. This leads to twooperations on measures which are deformations of the notions of the freeconvolution and the free projection. We further prove that if one replacescounting measures with others coming from the work of Perelomov and Popov onthe higher order Casimir operators for classical groups, then the operations onthe measures turn into the free convolution and projection themselves.  We also explain the relation between our results and limit shape theorems foruniformly random lozenge tilings with and without axial symmetry.

Representations of classical Lie groups and quantized free convolution