Young's lattice

In mathematics, Young's lattice is a partially ordered set and a lattice that is formed by all integer partitions. It is named after Alfred Young, who, in a series of papers On quantitative substitutional analysis, developed representation theory of the symmetric group. In Young's theory, the objects now called Young diagrams and the partial order on them played a key, even decisive, role. Young's lattice prominently figures in algebraic combinatorics, forming the simplest example of a differential poset in the sense of Stanley (1988). It is also closely connected with the crystal bases for affine Lie algebras.

Young's lattice is a partially ordered set Y formed by all integer partitions ordered by inclusion of their Young diagrams (or Ferrers diagrams).

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.

Conventionally, Young's lattice is depicted in a Hasse diagram with all elements of the same rank shown at the same height above the bottom. Suter (2002) has shown that a different way of depicting some subsets of Young's lattice shows some unexpected symmetries.

...
Wikipedia