Gelfand-Tsetlin bases

introduction

• In the fifties Gelfand posed the question of how to find a good basis for irreducible representations of reductive groups. In a paper with Zeitlin he defined special bases for irreducible representations of the general linear group and orthogonal groups.
• \(L(\lambda)\) : finite-dimensional irreducible representation of \(\mathfrak{gl}_n\) with the highest weight \(\lambda=(\lambda_1\geq \lambda_2\geq \cdots\geq\lambda_n\geq 0)\) of weakly decreasing non-negative integer sequence
• the set of all Gelfand-Zetlin patterns form a basis of \(L(\lambda)\)

identity

• LHS is the Weyl dimension formula for a representation of \(\mathfrak{gl}_n\), and RHS is the number of elements in Gelfand-Zetlin basis
• http://math.stackexchange.com/questions/1660/direct-proof-of-gelfand-zetlin-identity

related items

• Brauer-Weyl-Frobenius-Schur theory
• Crystal bases

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxZ1AzOXgwQ0FXX1E/edit

encyclopedia

• http://en.wikipedia.org/wiki/Gelfand%E2%80%93Zeitlin_integrable_system

expositions

• Molev, A. I. 2002. “Gelfand-Tsetlin Bases for Classical Lie Algebras”. ArXiv e-print math/0211289. http://arxiv.org/abs/math/0211289.

articles

• Igor Makhlin, Weyl's Formula as the Brion Theorem for Gelfand-Tsetlin Polytopes, arXiv:1409.7996 [math.RT], September 29 2014, http://arxiv.org/abs/1409.7996
• Gornitskii, A. A. “Essential Signatures and Canonical Bases for Irreducible Representations of \(D_4\).” arXiv:1507.07498 [math], July 27, 2015. http://arxiv.org/abs/1507.07498.
• Futorny, Vyacheslav, Dimitar Grantcharov, and Luis Enrique Ramirez. “Generic Irreducible Gelfand-Tsetlin Modules of Gl(n).” arXiv:1409.8413 [math], September 30, 2014. http://arxiv.org/abs/1409.8413.
• Makhlin, Igor. “Weyl’s Formula as the Brion Theorem for Gelfand-Tsetlin Polytopes.” arXiv:1409.7996 [math], September 29, 2014. http://arxiv.org/abs/1409.7996.
• Futorny, Vyacheslav, Dimitar Grantcharov, and Luis Enrique Ramirez. “Singular Gelfand-Tsetlin Modules of \(\mathfrak{gl}(n)\).” arXiv:1409.0550 [math], September 1, 2014. http://arxiv.org/abs/1409.0550.
• Hersh, Patricia, and Cristian Lenart. 2010. “Combinatorial Constructions of Weight Bases: The Gelfand-Tsetlin Basis.” Electronic Journal of Combinatorics 17 (1): Research Paper 33, 14. http://www.albany.edu/~lenart/articles/gt-basis1.pdf
• Refaghat, H., and M. Shahryari. 2010. “A Formula for the Number of Gelfand-Zetlin Patterns.” Journal of Generalized Lie Theory and Applications 4: Art. ID G100201, 8. doi:10.4303/jglta/G100201. http://www.ashdin.com/journals/jglta/2010/G100201.pdf
• Gel'fand, I. M., and M. L. Cetlin. 1950. “Finite-dimensional Representations of the Group of Unimodular Matrices.” Doklady Akad. Nauk SSSR (N.S.) 71: 825–828.