Editing Restricted representation (section)

==Gelfand–Tsetlin basis==<!-- Linked from [[Gelfand–Tsetlin integrable system]] -->
{{see also|Gelfand–Tsetlin integrable system}}
Since the branching rules from <math>U(N)</math> to <math>U(N-1)</matH> or <math>SO(N)</math> to <matH>SO(N-1)</math> have multiplicity one, the irreducible summands corresponding to smaller and smaller ''N'' will eventually terminate in one-dimensional subspaces. In this way [[I. M. Gelfand|Gelfand]] and [[Michael Tsetlin|Tsetlin]] were able to obtain a basis of any irreducible representation of <math>U(N)</math> or <math>SO(N)</math> labelled by a chain of interleaved signatures, called a '''Gelfand–Tsetlin pattern'''.
Explicit formulas for the action of the Lie algebra on the '''Gelfand–Tsetlin basis''' are given in {{harvtxt|Želobenko|1973}}. Specifically, for <math>N=3</math>, the Gelfand-Testlin basis of the irreducible representation of <math>SO(3)</math> with dimension <math>2l+1</math> is given by the complex [[spherical harmonics]] <math>\{Y_m^l | -l\leq m\leq l\}</math>.

For the remaining classical group <math>Sp(N)</math>, the branching is no longer multiplicity free, so that if ''V'' and ''W'' are irreducible representation of <math>Sp(N-1)</math> and <math>Sp(N)</math> the space of intertwiners <math>Hom_{Sp(N-1)}(V,W)</math> can have dimension greater than one. It turns out that the [[Yangian]] <math>Y(\mathfrak{gl}_2)</math>, a [[Hopf algebra]] introduced by [[Ludwig Faddeev]] and [[LOMI|collaborators]], acts irreducibly on this  multiplicity space, a fact which enabled {{harvtxt|Molev|2006}} to extend the construction of Gelfand–Tsetlin bases to <math>Sp(N)</math>.<ref>G. I. Olshanski had shown that the twisted Yangian <math>Y^-(\mathfrak{gl}_2)</math>, a sub-Hopf algebra of <math>Y(\mathfrak{gl}_2)</math>, acts naturally on the space of intertwiners. Its natural irreducible representations correspond to tensor products of the composition of point evaluations with irreducible representations of <math>\mathfrak{gl}</math><sub>2</sub>. These extend to the Yangian <math>Y(\mathfrak{gl})</math> and give a representation theoretic explanation of the product form of the branching coefficients.</ref>