Representations of classical Lie groups and quantized free convolution

Abstract

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

43 pages, 4 figures. v3: relation to the Markov-Krein correspondence is updated and corrected

References in corpus (8)

• Limits of BC-type orthogonal polynomials as the number of variables goes to infinity
• Shuffling algorithm for boxed plane partitions
• Kerov's interlacing sequences and random matrices
• Anisotropic Young diagrams and infinite-dimensional diffusion processes with the Jack parameter
• Cut-off phenomenon in the uniform plane Kac walk
• Asymptotics of Plancherel measures for the infinite-dimensional unitary group
• An asymptotic log-Fourier interpretation of the R-transform
• Asymptotic fluctuations of representations of the unitary groups