We consider the Plancherel measure on irreducible components of tensor powers of the spinor representation of so2n+1. With respect to this measure, the probability of an irreducible representation is the product of its multiplicity and dimension, divided by the total dimension of the tensor product. We study the limit shape of the highest weight as the tensor power N and the rank n of the algebra tend to infinity with N/n fixed.
Similar content being viewed by others
References
P. Biane, “Approximate factorization and concentration for characters of symmetric groups,” Int. Math. Res. Not., 2001, No. 4, 179–192 (2001).
P. Biane, “Representations of symmetric groups and free probability,” Adv. Math., 138, No. 1, 126–181 (1998).
A. Borodin, V. Gorin, and A. Guionnet, “Gaussian asymptotics of discrete β-ensembles,” Publ. Math. Inst. Hautes Études Sci., 125, No. 1, 1–78 (2017).
J. Breuer and M. Duits, “Central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients,” J. Amer. Math. Soc., 30, No. 1, 27–66 (2017).
P. Deift, Orthogonal Polynomials and Random Matrices: a Riemann–Hilbert Approach, Amer. Math. Soc. (1999).
D. V. Giang, “Finite Hilbert transforms logarithmic potentials and singular integral equations,” arXiv:1003.3070.
A. Guionnet, Asymptotics of Random Matrices and Related Models: The Uses of Dyson–Schwinger Equations, Amer. Math. Soc. (2019).
J. Hong and S.-J. Kang, Introduction to Quantum Groups and Crystal Bases, Amer. Math. Soc. (2002).
M. Kashiwara and T. Nakashima, “Crystal graphs for representations of the q-analogue of classical Lie algebras,” J. Algebra, 165, No. 2, 295–345 (1994).
S. V. Kerov, “On asymptotic distribution of symmetry types of high rank tensors,” Zap. Nauchn. Semin. POMI, 155, 181–186 (1986).
P. P. Kulish, V. D. Lyakhovsky, and O. V. Postnova, “Tensor powers for non-simply laced Lie algebras B2-case,” J. Phys. Conf. Ser., 346, No. 1, 012012 (2012).
P. P. Kulish, V. D. Lyakhovsky, and O. V. Postnova, “Multiplicity function for tensor powers of modules of the An algebra,” Theor. Math. Phys., 171, No. 2, 666–674 (2012).
P. P. Kulish, V. D. Lyakhovsky, and O. V. Postnova, “Multiplicity functions for tensor powers. An-case,” J. Phys. Conf. Ser., 343, No. 1, 012070 (2012).
P. P. Kulish, V. D. Lyakhovsky, and O. V. Postnova, “Tensor power decomposition. Bn case,” J. Phys. Conf. Ser. 343, No. 1, 012095 (2012).
T. Nakashima, “Crystal base and a generalization of the Littlewood–Richardson rule for the classical Lie algebras,” Comm. Math. Phys., 154, No. 2, 215–243 (1993).
A. Nazarov, P. Nikitin, and O. Postnova, “Limit shape for infinite rank limit of non simply-laced Lie algebras of series so2n+1,” arxiv:2010.16383.
A. A. Nazarov and O. V. Postnova, “The limit shape of a probability measure on a tensor product of modules of the Bn algebra,” J. Math. Sci., 240, No. 5, 556–566 (2019).
O. Postnova and N. Reshetikhin, “On multiplicities of irreducibles in large tensor product of representations of simple Lie algebras,” Lett. Math. Phys., 110, 147–178 (2020).
D. Romik, The Surprising Mathematics of Longest Increasing Subsequences, Cambridge Univ. Press (2015).
T. Tate and S. Zelditch, “Lattice path combinatorics and asymptotics of multiplicities of weights in tensor powers,” J. Funct. Anal., 217, No 2, 402–447 (2004).
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to the memory of Vladimir Dmitrievich Lyakhovsky (1942–2020)
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 507, 2021, pp. 99–113.
Rights and permissions
About this article
Cite this article
Nazarov, A.A., Nikitin, P.P. & Postnova, O.V. Statistics of Irreducible Components in Large Tensor Powers of the Spinor Representation for so2n+1 as n→∞. J Math Sci 261, 658–668 (2022). https://doi.org/10.1007/s10958-022-05778-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-022-05778-z