 Gelfand–Tsetlin basis

Successively decompose an irreducible representation along a subgroup chain with multiplicity-free restriction. A complete path selects a one-dimensional subspace; choosing one nonzero vector on each path gives a Gelfand–Tsetlin basis. For a symmetric group over the complex numbers, paths are standard Young tableaux, and the Young–Jucys–Murphy elements act diagonally with their cell contents.