Symmetries of Solid Partitions

There is a a natural action of the permution group $S_4$ on solid partitions — it acts by permuting the four coordinates of the nodes in the Ferrers diagram for the solid partition. Given a solid partition, let $H$ denote the maximal sub-group of $S_4$ that acts trivially on the partition. Up to overall conjugation by an element of $S_4$, there are eleven symmetry classes which label by the order of $H$ using a second label to distinguish different classes with same order.

• Class 1A These are solid partitions that have the least symmetry — the action of $S_{4}$ leads to $24$ distinct solid partitions.
• Class 2A The symmetry group is generated by an order two element with cycle shape $1^2 2^1$. In other words, the symmetry corresponds to exchanging any two coordinates.
• Class 2B The order two element that generates this group has cycle shape $2^2$. It corresponds to simultaneously exchanging two pairs of coordinates.
• Class 3A The subgroup is isomorphic to $\mathbb{Z}_3$. It corresponds to a cyclic permutation of any three coordinates.
• Class 4A The symmetry is the order four subgroup $\mathbb{Z}_2\times\mathbb{Z}_2$ with both $\mathbb{Z}_2$ being generated by elements of cycle shape $1^22^1$.
• Class 4B The symmetry is the order four subgroup $\mathbb{Z}_2\times\mathbb{Z}_2$ with both $\mathbb{Z}_2$ being generated by elements of cycle shape $2^2$.
• Class 4C The symmetry is the order four subgroup $\mathbb{Z}_4$ generated by a cyclic permutation of the four coordinates.
• Class 6A This is the order six subgroup $S_3$ that leaves one coordinate untouched.
• Class 8A This is a subgroup that is isomorphic to the dihedral group with eight elements.
• Class 12A The symmetry group is isomorphic to the alternating group $A_4$. We have found no solid partition with this symmetry for all $n\leq 35$. We anticipate that a solid partition with this symmetry will first appear at $n=113$ — this partition can be obtained as follows. Take the node $(0,1,2,3)^T$ and add $11$ other nodes that are its $A_4$ transforms. Add all nodes that are needed to convert it into a valid solid partition.
• Class 24A These are partitions that are invariant under the full permutation group.

References

1. R. P. Stanley, Symmetries of Plane Partitions, J. of Comb. Theory Ser. A 43 (1986) 103-113.
2. S. Govindarajan, Symmetries of Solid Partitions, unpublished.