Gelfand-Tsetlin bases

introduction

• $L(\lambda)$ : 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

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

• 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.

• 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.