Young's Lattice - Significance

Significance

The traditional application of Young's lattice is to the description of the irreducible representations of symmetric groups S_n for all n, together with their branching properties, in characteristic zero. The equivalence classes of irreducible representations may be parametrized by partitions or Young diagrams, the restriction from S_{n + 1} to S_n is multiplicity-free, and the representation of S_n with partition p is contained in the representation of S_{n + 1} with partition q if and only if q covers p in Young's lattice. Iterating this procedure, one arrives at Young's semicanonical basis in the irreducible representation of S_n with partition p, which is indexed by the standard Young tableaux of shape p.