Parthasarathy-Ranga Rao-Varadarajan conjecture

introduction

• PRV conjecture
• notations
  • $\nu$ integral weight
  • $\overline{\nu}$ the dominant integral weight of $W\cdot \nu$
  • $V(\overline{\nu})$ highest weight representation
• $\lambda,\mu$ dominant integral weights and $w\in W$, the module $V(\overline{\lambda+w\mu})$ occurs with multiplicity at least one in $V(\lambda)\otimes V(\mu)$

memo

• http://mathoverflow.net/questions/163217/tensor-products-of-two-irreducible-representations-of-reductive-groups-and-their/163239#163239

related items

• Tensor product decompositions of representations of Lie algebras

expositions

• Khare, Apoorva. 2012. “Representations of Complex Semi-simple Lie Groups and Lie Algebras.” arXiv:1208.0416 (August 2). http://arxiv.org/abs/1208.0416.
• 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

articles

• Montagard, P. L., B. Pasquier, and N. Ressayre. ‘Two Generalizations of the PRV Conjecture’. Compositio Mathematica 147, no. 04 (July 2011): 1321–36. doi:10.1112/S0010437X10005233.
• Kumar, Shrawan. 1988. “Proof of the Parthasarathy-Ranga Rao-Varadarajan Conjecture.” Inventiones Mathematicae 93 (1): 117–130. doi:10.1007/BF01393689. http://link.springer.com/article/10.1007%2FBF01393689
• Parthasarathy, K. R., R. Ranga Rao, and V. S. Varadarajan. ‘Representations of Complex Semi-Simple Lie Groups and Lie Algebras’. Annals of Mathematics, Second Series, 85, no. 3 (1 May 1967): 383–429. doi:10.2307/1970351.
• Parthasarathy, K. R., R. Ranga Rao, and V. S. Varadarajan. ‘Representations of Complex Semisimple Lie Groups and Lie Algebras’. Bulletin of the American Mathematical Society 72, no. 3 (1966): 522–25. doi:10.1090/S0002-9904-1966-11528-8.