Geodesic
Spectral properties of the periodic Coxeter Laplacian in the two-row ferromagnetic case
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XVIII, Tome 378 (2010), pp. 111-132 Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper is a part of the project suggested by A. M. Vershik and the author and aimed to combine the known results on the representation theory of finite and infinite symmetric groups and a circle of results related to the quantum inverse scattering method and Bethe ansatz. In this first part, we consider the simplest spectral properties of a distinguished operator in the group algebra of the symmetric group, which we call the periodic Coxeter Laplacian. Namely, we study this operator in the two-row representations of symmetric groups and in the “ferromagnetic” asymptotic mode. Bibl. 11 titles. 
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N. V. Tsilevich. Spectral properties of the periodic Coxeter Laplacian in the two-row ferromagnetic case. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XVIII, Tome 378 (2010), pp. 111-132. http://geodesic.mathdoc.fr/item/ZNSL_2010_378_a8/

[1] L. Babai, “Spectra of Cayley graphs”, J. Combin. Theory B, 27 (1979), 180–189 | DOI | MR | Zbl

[2] C. M. Bender, D. C. Brody, B. K. Meister, “Powers of Bessel functions”, J. Math. Phys., 44:1 (2003), 309–314 | DOI | MR | Zbl

[3] I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York–London, 1965 | MR

[4] R. M. Gray, Toeplitz and Circulant Matrices: A Review, Now Publishers, Norwell, Massachusetts, 2006

[5] Yu. A. Izyumov, Yu. N. Skryabin, Statistical Mechanics of Magnetically Ordered Systems, Consultants Bureau, New York, 1988 | MR

[6] J. C. Mason, D. C. Handscomb, Chebyshev Polynomials, Chapman Hall/CRC, Boca Raton, Florida, 2003 | MR | Zbl

[7] E. Mukhin, V. Tarasov, A. Varchenko, “Bethe algebra of homogeneous XXX Heisenberg model has simple spectrum”, Comm. Math. Phys., 288:1 (2009), 1–42 | DOI | MR | Zbl

[8] L. A. Takhtadzhyan, L. D. Faddeev, “The spectrum and scattering of excitations in the one-dimensional isotropic Heisenberg model”, Zap. Nauchn. Semin. LOMI, 109, 1981, 134–178 | MR | Zbl

[9] N. V. Tsilevich, A. M. Vershik, “Induced representations of the infinite symmetric group”, Pure Appl. Math. Quart., 3:4 (2007), 1005–1026 | DOI | MR | Zbl

[10] A. M. Vershik, S. V. Kerov, “Asymptotic theory of the characters of a symmetric group”, Funkts. Anal. i Prilozhen., 15:4 (1981), 15–27 | MR | Zbl

[11] J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965 | MR | Zbl