Tensor product decompositions of representations of Lie algebras

introduction

• various method of tensor product decomposition
• Speiser method
• Bethe ansatz

Klimyk's formula

• http://mathoverflow.net/questions/85593/decompose-tensor-product-of-type-g-2-lie-algebras

memo

• http://mathoverflow.net/questions/163217/tensor-products-of-two-irreducible-representations-of-reductive-groups-and-their/163239#163239
• A Constructive Algorithm for Determining Branching Rules of Lie Group Representations

related items

• Littlewood–Richardson rule
• Parthasarathy-Ranga Rao-Varadarajan conjecture
• WZW Fusion rule, fusion ring and algebra

expositions

• Kumar, Shrawan. 2010. “Tensor Product Decomposition.” In Proceedings of the International Congress of Mathematicians. Volume III, 1226–1261. New Delhi: Hindustan Book Agency. http://www.ams.org/mathscinet-getitem?mr=2827839. http://www.unc.edu/math/Faculty/kumar/papers/kumar60.pdf
• Antoine, Jean-Pierre. 2006. “David Speiser’s Group Theory: From Stiefel’s Crystallographic Approach to Kac-Moody Algebras.” In Two Cultures, edited by Kim Williams, 13–23. Birkhäuser Basel. http://link.springer.com/chapter/10.1007/3-7643-7540-X_2.
• Walton, Mark A. 1990. “Fusion Rules in Wess-Zumino-Witten Models.” Nuclear Physics. B 340 (2-3): 777–790. doi:10.1016/0550-3213(90)90470-X.
  • Section 2
• D. Speiser, Theory of Compact Lie Groups and some Applications to Elementary Particle Physics, Group Theoretical Concepts and Methods in Elementary Particle Physics, NATO Summer School Istanbul 1962, pp. 201–276; F. Gürsey, ed. New York: Gordon and Breach, 1964.

articles

• Baldoni, Velleda, and Michèle Vergne. “Computation of Dilated Kronecker Coefficients.” arXiv:1601.04325 [math], January 17, 2016. http://arxiv.org/abs/1601.04325.
• Okada, Soichi. “Applications of Minor Summation Formulas to Rectangular-Shaped Representations of Classical Groups.” Journal of Algebra 205, no. 2 (July 15, 1998): 337–67. doi:10.1006/jabr.1997.7408.
• Koike, Kazuhiko. “On the Decomposition of Tensor Products of the Representations of the Classical Groups: By Means of the Universal Characters.” Advances in Mathematics 74, no. 1 (March 1989): 57–86. doi:10.1016/0001-8708(89)90004-2.
• Koike, Kazuhiko, and Itaru Terada. “Young-Diagrammatic Methods for the Representation Theory of the Classical Groups of Type Bn, Cn, Dn.” Journal of Algebra 107, no. 2 (May 1, 1987): 466–511. doi:10.1016/0021-8693(87)90099-8.
• Koike, Kazuhiko, and Itaru Terada. “Young Diagrammatic Methods for the Restriction of Representations of Complex Classical Lie Groups to Reductive Subgroups of Maximal Rank.” Advances in Mathematics 79, no. 1 (January 1990): 104–35. doi:10.1016/0001-8708(90)90059-V.
• Gungormez, Meltem, and Hasan R. Karadayi. “Explicit Calculations of Tensor Product Coefficients for E_7.” arXiv:1508.03956 [math-Ph], August 17, 2015. http://arxiv.org/abs/1508.03956.
• Kumar, Shrawan. “Components of V(\rho)\otimes V(\rho).” arXiv:1508.03071 [math], August 12, 2015. http://arxiv.org/abs/1508.03071.
• Coquereaux, Robert, and Jean-Bernard Zuber. 2014. “Conjugation Properties of Tensor Product Multiplicities.” arXiv:1405.4887 [hep-Th, Physics:math-Ph], May. http://arxiv.org/abs/1405.4887.
• De Loera, Jesús A., and Tyrrell B. McAllister. 2005. “On the Computation of Clebsch-Gordan Coefficients and the Dilation Effect.” arXiv:math/0501446, January. http://arxiv.org/abs/math/0501446.
• Berenstein, Arkady, and Andrei Zelevinsky. 2001. “Tensor Product Multiplicities, Canonical Bases and Totally Positive Varieties.” Inventiones Mathematicae 143 (1): 77–128. doi:10.1007/s002220000102.
• Grimm, S., and J. Patera. 1997. Decomposition of Tensor Products of the Fundamental Representations of \(E_8\). In Advances in Mathematical Sciences: CRM’s 25 Years (Montreal, PQ, 1994), 11:329–355. CRM Proc. Lecture Notes. Providence, RI: Amer. Math. Soc. http://www.ams.org/mathscinet-getitem?mr=1479683.
• Littelmann, Peter. 1995. “Crystal Graphs and Young Tableaux.” Journal of Algebra 175 (1): 65–87. doi:10.1006/jabr.1995.1175.
• Littelmann, Peter. 1994. “A Littlewood-Richardson Rule for Symmetrizable Kac-Moody Algebras.” Inventiones Mathematicae 116 (1-3): 329–346. doi:10.1007/BF01231564.
• Littelmann, Peter. 1990. “A Generalization of the Littlewood-Richardson Rule.” Journal of Algebra 130 (2): 328–368. doi:10.1016/0021-8693(90)90086-4.
• Kempf, George, and Linda Ness. 1988. “Tensor Products of Fundamental Representations.” Canadian Journal of Mathematics. Journal Canadien de Mathématiques 40 (3): 633–648. doi:10.4153/CJM-1988-027-1.
• Koike, Kazuhiko, and Itaru Terada. 1987. “Young-Diagrammatic Methods for the Representation Theory of the Classical Groups of Type \(B_n,\;C_n,\;D_n\).” Journal of Algebra 107 (2): 466–511. doi:10.1016/0021-8693(87)90099-8.