Young's Lattice - Dihedral Symmetry

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

4

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.