Comptes Rendus
Lie algebras
On the SO(n + 3) to SO(n) branching multiplicity space
Comptes Rendus. Mathématique, Volume 356 (2018) no. 11-12, pp. 1112-1124.

We study the decomposition as an SO(3)-module of the multiplicity space corresponding to the branching from SO(n+3) to SO(n). Here, SO(n) (resp. SO(3)) is considered embedded in SO(n+3) in the upper left-hand block (resp. lower right-hand block). We show that when the highest weight of the irreducible representation of SO(n) interlaces the highest weight of the irreducible representation of SO(n+3), then the multiplicity space decomposes as a tensor product of (n+2)/2 reducible representations of SO(3).

Nous étudions la décomposition de l'espace de multiplicité, comme SO(3)-module, correspondant au branchement de SO(n+3) vers SO(n). Ici, SO(n) (resp. SO(3)) est considéré comme plongé dans SO(n+3) dans le bloc en haut à gauche (resp. le bloc en bas à droite). Nous montrons que, lorsque le plus grand poids de la représentation irréductible de SO(n) s'entrelace avec le plus grand poids de la représentation irréductible de SO(n+3), alors l'espace de multiplicité se décompose en un produit tensoriel de (n+2)/2 représentations réductibles de SO(3).

Published online:
DOI: 10.1016/j.crma.2018.09.004

Emilio A. Lauret 1; Fiorela Rossi Bertone 1

1 CIEM–FaMAF (CONICET), Universidad Nacional de Córdoba, Medina Allende s/n, Ciudad Universitaria, 5000 Córdoba, Argentina
     author = {Emilio A. Lauret and Fiorela Rossi Bertone},
     title = {On the {SO(\protect\emph{n} + 3)} to {SO(\protect\emph{n})} branching multiplicity space},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1112--1124},
     publisher = {Elsevier},
     volume = {356},
     number = {11-12},
     year = {2018},
     doi = {10.1016/j.crma.2018.09.004},
     language = {en},
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Emilio A. Lauret; Fiorela Rossi Bertone. On the SO(n + 3) to SO(n) branching multiplicity space. Comptes Rendus. Mathématique, Volume 356 (2018) no. 11-12, pp. 1112-1124. doi : 10.1016/j.crma.2018.09.004.

[1] F. El Chami Spectra of the Laplace operator on Grassmann manifolds, Int. J. Pure Appl. Math., Volume 12 (2004) no. 4, pp. 395-418

[2] F. El Chami A branching law from Sp(n) to Sp(q) × Sp(n-q) and an application to Laplace operator spectra, Indian J. Pure Appl. Math., Volume 43 (2012) no. 1, pp. 71-86 | DOI

[3] H.W. Galbraith; J.D. Louck Canonical solution of the SU(3)SO(3) reduction problem from the SU(3) pattern calculus, Acta Appl. Math., Volume 24 (1991) no. 1, pp. 59-108

[4] I.M. Gelfand; M.L. Cetlin Finite-dimensional representations of the group of unimodular matrices, Dokl. Akad. Nauk SSSR (N.S.), Volume 71 (1950), pp. 825-828

[5] R. Goodman; N. Wallach Symmetry, Representations, and Invariants, Grad. Texts in Math., vol. 255, Springer-Verlag, New York, 2009 | DOI

[6] K. Gross; R. Kunze Finite-dimensional induction and new results on invariants for classical groups, I, Amer. J. Math., Volume 106 (1984) no. 4, pp. 893-974 | DOI

[7] R. Howe; E.-C. Tan; J.F. Willenbring Stable branching rules for classical symmetric pairs, Trans. Amer. Math. Soc., Volume 357 (2005) no. 4, pp. 1601-1626 | DOI

[8] S. Kim Tiling branching multiplicity spaces with GL2 pattern blocks, J. Aust. Math. Soc., Volume 94 (2013) no. 3, pp. 362-374 | DOI

[9] S. Kim; O. Yacobi A basis for the symplectic group branching algebra, J. Algebraic Comb., Volume 35 (2012) no. 2, pp. 269-290 | DOI

[10] A.W. Knapp Branching theorems for compact symmetric spaces, Represent. Theory, Volume 5 (2001), pp. 404-463 | DOI

[11] A.W. Knapp Lie Groups Beyond an Introduction, Progr. Math., vol. 140, Birkhäuser Boston Inc., 2002

[12] J. Lepowsky Multiplicity formulas for certain semisimple Lie groups, Bull. Amer. Math. Soc., Volume 77 (1971), pp. 601-605 | DOI

[13] A.I. Molev Gelfand–Tsetlin bases for classical Lie algebras, Handbook of Algebra, Handb. Algebr., vol. 4, Elsevier/North-Holland, Amsterdam, 2006, pp. 109-170 | DOI

[14] C. Tsukamoto Spectra of Laplace–Beltrami operators on SO(n+2)/SO(2)×SO(n) and Sp(n+1)/Sp(1)×Sp(n), Osaka J. Math., Volume 18 (1981) no. 2, pp. 407-426

[15] C. Tsukamoto Branching rules for SO(n+3)/SO(3)×SO(n), Bull. Fac. Tex. Sci., Kyoto Inst. Technol., Volume 30 (2005), pp. 11-20

[16] N. Wallach; O. Yacobi A multiplicity formula for tensor products of SL2 modules and an explicit Sp2n to Sp2n2×Sp2 branching formula, Symmetry in Mathematics and Physics, Contemp. Math., vol. 490, Amer. Math. Soc., Providence, RI, USA, 2009, pp. 151-155 | DOI

[17] O. Yacobi An analysis of the multiplicity spaces in branching of symplectic groups, Sel. Math. (N.S.), Volume 16 (2010) no. 4, pp. 819-855 | DOI

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This research was partially supported by grants from CONICET and Agencia Nacional de Promoción Científica y Tecnológica (PICT-2015-0274 and PICT-2014-2706). The first named author was supported by the Alexander von Humboldt Foundation.

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