Young's lattice

A Hasse diagram of 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 Sn 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 Sn + 1 to Sn is multiplicity-free, and the representation of Sn with partition p is contained in the representation of Sn + 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 Sn with partition p, which is indexed by the standard Young tableaux of shape p.


Dihedral symmetry

The portion of Young's lattice lying below 1 + 1 + 1 + 1, 2 + 2 + 2, 3 + 3, and 4
Conventional diagram with partitions of the same rank at the same height
Diagram showing dihedral symmetry

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.

The partition

of the nth triangular number has a Ferrers diagram that looks like a staircase. The largest elements whose Ferrers diagrams are rectangular that lie under the staircase are these:

Partitions of this form are the only ones that have only one element immediately below them in Young's lattice. Suter showed that the set of all elements less than or equal to these particular partitions has not only the bilateral symmetry that one expects of Young's lattice, but also rotational symmetry: the rotation group of order n + 1 acts on this poset. Since this set has both bilateral symmetry and rotational symmetry, it must have dihedral symmetry: the (n + 1)th dihedral group acts faithfully on this set. The size of this set is 2n.

For example, when n = 4, then the maximal element under the "staircase" that have rectangular Ferrers diagrams are

1 + 1 + 1 + 1
2 + 2 + 2
3 + 3

The subset of Young's lattice lying below these partitions has both bilateral symmetry and 5-fold rotational symmetry. Hence the dihedral group D5 acts faithfully on this subset of Young's lattice.

See also


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