Restricted Representation - Gelfand-Tsetlin Basis

Gelfand-Tsetlin Basis

Since the branching rules from U(N) to U(N–1) or SO(N) to SO(N–1) have multiplicity one, the irreducible summands corresponding to smaller and smaller N will eventually terminate in one dimensional subspaces. In this way Gelfand and Tsetlin were able to obtain a basis of any irreducible representation of U(N) or SO(N) 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 Želobenko (1973).

For the remaining classical group Sp(N), the branching is no longer multiplicity free, so that if V and W are irreducible representation of Sp(N–1) and Sp(N) the space of intertwiners Hom_Sp(N–1)(V,W) can have dimension greater than one. It turns out that the Yangian Y(₂), a Hopf algebra introduced by Ludwig Faddeev and collaborators, acts irreducibly on this multiplicity space, a fact which enabled Molev (2006) to extend the construction of Gelfand-Tsetlin bases to Sp(N).