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Representations of classical Lie groups and quantized free convolution

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Abstract

We study the decompositions into irreducible components of tensor products and restrictions of irreducible representations for all series 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.

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Authors and Affiliations

  1. Department of Mathematics Higher School of Economics, Moscow, Russia

    Alexey Bufetov

  2. Institute for Information Transmission Problems of Russian Academy of Sciences, Moscow, Russia

    Alexey Bufetov & Vadim Gorin

  3. Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

    Vadim Gorin

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  1. Alexey Bufetov
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  2. Vadim Gorin
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Correspondence to Vadim Gorin.

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Bufetov, A., Gorin, V. Representations of classical Lie groups and quantized free convolution. Geom. Funct. Anal. 25, 763–814 (2015). https://doi.org/10.1007/s00039-015-0323-x

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