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 $L(\lambda)$

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

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

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.