nLab Gelfand-Tsetlin basis

Idea

Gelʹfand-Cetlin or Gelfand-Tsetlin bases – an early example of canonical bases on the linear representations of finite groups and Lie groups.

Related concepts

• seminormal representation

• Jucys-Murphy element

• state on a star-algebra

References

Original articles:

• Israel Gelfand, M. L. Cetlin, Finite-dimensional representations of the group of unimodular matrices, (Russian) Doklady Akad. Nauk SSSR (N.S.) 71, (1950) 825–828; Finite-dimensional representations of groups of orthogonal matrices, Doklady Akad. Nauk SSSR (N.S.) 71, (1950) 1017–1020.

Review:

• Anatoly Vershik, Gel’fand-Tsetlin algebras, expectations, inverse limits, Fourier analysis, pp. 619 in: Pavel Etingof, Vladimir S. Retakh, Isadore Singer (eds.) The Unity of Mathematics In Honor of the Ninetieth Birthday of I. M. Gelfand, Birkhäuser 2006 (arXiv:math/0503140, ISBN:978-0-8176-4467-3)

• David Grabovsky, The Gelfand-Tsetlin Basis, 2018 (pdf, pdf)

See also:

• Victor Guillemin, Shlomo Sternberg, The Gel’fand-Cetlin system and quantization of the complex flag manifolds, Journal of Functional Analysis 52 (1): 106–128, 1983 doi MR705993

• D. P. Želobenko, Compact Lie groups and their representations

• A. I. Molev, Gelfand-Tsetlin bases for classical Lie algebras, arxiv/0211289

• Terry Tao, Gelfand obituary

• Megumi Harada, The symplectic geometry of the Gel’fand-Cetlin-Molev basis for representations of Sp(2n,C), published pdf, math.SG/0404485

• Grigori Olshanski, Projections of orbital measures, Gelfand-Tsetlin polytopes, and splines, arxiv/1302.7116

• wikipedia Gelfand-Tsetlin integrable system

• Bertram Kostant, Nolan Wallach, Gelfand-Zeitlin theory from the perspective of classical mechanics. I, Studies in Lie theory, Progr. Math. 243, Birkhäuser 2006, pp. 319–364