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On multiplicities of irreducibles in large tensor product of representations of simple Lie algebras

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Abstract

In this paper, we study the asymptotics of multiplicities of irreducible representations in large tensor products of finite dimensional representations of simple Lie algebras and their statistics with respect to Plancherel and character probability measures. We derive the asymptotic distribution of irreducible components for the Plancherel measure, generalizing results of Biane and Tate and Zelditch. We also derive the asymptotics of the character measure for generic parameters and an intermediate scaling in the vicinity of the Plancherel measure. It is interesting that the asymptotic measure is universal and after suitable renormalization does not depend on which representations were multiplied but depends significantly on the degeneracy of the parameter in the character distribution.

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Notes

  1. If W is a finite dimensional representation of a simple Lie algebra and \(W\simeq \oplus _iV_i^{\oplus m_i}\) is its decomposition into irreducibles, then \(p_i=\frac{\dim V_im_i}{\dim W}\) is a probability measure on the set of irreducible components of W. We call it the Plancherel measure by the analogy with the Plancherel measure on the left regular representation of a finite group, or compact Lie group.

  2. This is a convenience assumption, and the irreducible characters restricted to the Cartan subgroup are invariant with respect to the action of the Weyl group, so we can always choose a representative of the orbit of W through t which is positive.

  3. Here, we use the basis of simple roots.

  4. This is proven in [19] for \(m=1\). The proof is based on the application of the steepest descent method and on the orthogonality of characters of irreducible representations of corresponding compact Lie group. For \(m>1\), the proof is completely similar.

  5. For a rigorous proof see [4, 19].

  6. There is an analogue of such asymptotics for every nonregular value of t (when \(t\subset S\in {\mathfrak {h}}^{\star }\ge 0\) where S is a stratum of the boundary of the cone \({\mathfrak {h}}^{\star }_{\ge 0}\)). In this case, transverse directions to S should be scaled as \(\sqrt{\epsilon }\).

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Acknowledgements

We thank A. Hammond, A. Nazarov and V. Serganova for stimulating discussions. We are grateful for E. Feigin, V. Gorin, M. Walter and S. Zelditch for important remarks and for pointing us to recent works on the subject. We are also grateful for referees for thorough reading of the paper and for important suggestions. This work was supported by RSF-18-11-00-297. The work of NR was partly supported by the Grant NSF DMS-1601947.

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Authors and Affiliations

  1. Laboratory of Mathematical Problems of Physics, St. Petersburg Department of Steklov Mathematical Institute, Fontanka 27, St. Petersburg, Russia, 191023

    Olga Postnova

  2. Department of Mathematics, University of California, Berkeley, CA, 94720, USA

    Nicolai Reshetikhin

  3. Department of Physics, Saint Petersburg State University, Staryi Petergoff, Ulyanovskaya st., 1, St. Petersburg, Russia

    Nicolai Reshetikhin

  4. KdV Institute for Mathematics, University of Amsterdam, Science Park 904, 1098 XH, Amsterdam, The Netherlands

    Nicolai Reshetikhin

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  1. Olga Postnova
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Postnova, O., Reshetikhin, N. On multiplicities of irreducibles in large tensor product of representations of simple Lie algebras. Lett Math Phys 110, 147–178 (2020). https://doi.org/10.1007/s11005-019-01217-4

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