Symmetries Of Plane Partitions

The Ferrers diagram of plane partitions has a natural action of $S_3$. The $S_3$ arises as the group of permutations of the three coordinates. Unrestricted plane partitions can be classified by conjugacy classes of subgroups of $S_3$. There are four classes of unrestricted partitions[1].

1. Non-symmetric: Those that have no symmetry i.e., the action of $S_3$ creates five other Ferrers diagrams. (One can, in principle, replace this class with one that includes all symmetries. Stanley calls the class, any.
2. Symmetric: The Ferrers diagram is invariant under the exchange of (any fixed) two coordinates i.e., an $S_2\subset S_3$.
3. Cyclically Symmetric: The Ferrers diagram is invariant under a cyclic permutation of the coordinates i.e., $C_3\subset S_3$.
4. Totally Symmetric: The Ferrers diagram is invariant under the action of $S_3$.

Stanley extended these considerations to the case of plane partitions restricted to a box. The box is chosen to respect the symmetry. The work of Mills, Robbins and Rumsey on Alternating Sign Matrices lead Stanley to introduce a new symmetry operation called complementation to classify restricted partitions. Let $\lambda$ be a partition that fits in a box $B(r,s,t)$ i.e, all nodes $\mathbf{y}=(y_1,y_2,y_3) \in \lambda$ are such that $0\leq y_1 < r,\ 0\leq y_2 < s,\ 0\leq y_3 < t$. The complement of a plane partition $\lambda,\ \lambda^c$ is defined as the plane partition constructed as follows. It is constructed from all the points in $B(r,s,t)$ that are not in $\lambda$.

(1)
\begin{align} \lambda^c =\left\{(r-y_1-1,s-y_2-1,t-y_3-1)~\Big|~ (y_1,y_2,y_3)\in B(r,s,t) \ \mathrm{and}\ (y_1,y_2,y_3)\notin\lambda\right\}\ . \end{align}

It is easy to see that $\lambda^c$ is another plane partition. Further, one has $|\lambda|+|\lambda^c|=rst$ and $(\lambda^c)^c=\lambda$. This operation enlarges $S_3$ to a larger group with order twelve1, $T=S_3 \times \mathbb{Z}_2$, which has ten conjugacy class of sub-groups. This leads to ten classes (six new ones) of symmetries for restricted plane partitions. We reproduce the table of symmetry classes from Ref. [1].

Case Box Type Class
1 $B(r,s,t)$ Any
2 $B(r,r,t)$ Symmetric
3 $B(r,r,r)$ Cyclically Symmetric
4 $B(r,r,r)$ Totally Symmetric
5 $B(r,s,t)$ Self-complementary
6 $B(r,r,t)$ Complement=Transpose
7 $B(r,r,t)$ Symmetric and Self-complementary
8 $B(r,r,r)$ Cyclically Symmetric and Complement=Transpose
9 $B(r,r,r)$ Cylically Symmetric and Self-complemenatary
10 $B(r,r,r)$ Totally Symmetric and Self-complementary

### Generating functions for unrestricted partitions

This section is incomplete and may have incorrect statements.

The generating function for unrestricted partitions is given by the MacMahon formula

(2)
\begin{align} \sum_{n=0}^\infty \textrm{PL}(n)\ q^n = \prod_{m=1}^\infty \frac1{(1-q^m)^m} \end{align}

### Generating functions for restricted partitions

Stanley collected, conjectured and proved in some cases, for the numbers of plane partitions in all ten symmetry classes of restricted partitions[1]. As a preparation, we define the height of a node $\mathbf{y}=(y_1,y_2,y_3)$ to be

(3)
\begin{align} ht(\mathbf{y})= (y_1+y_2+y_3+1)\ . \end{align}

If $G$ acts on the box $B$ and belongs to cases 1-4, then for any node $\mathbf{y}\in B$, let $\eta$ denote the orbit of nodes obtained by the action of $G$ on $\mathbf{y}$.

Case Box Type No. of Plane Partitions
1 $B(r,s,t)$ $\displaystyle{\prod_{\mathbf{y}\in B} \frac{1+ht(\mathbf{y})}{ht(\mathbf{y}}}$
2 $B(r,r,t)$
3 $B(r,r,r)$
4 $B(r,r,r)$
5 $B(r,s,t)$
6 $B(r,r,t)$
7 $B(r,r,t)$
8 $B(r,r,r)$
9 $B(r,r,r)$
10 $B(r,r,r)$

Let $G$ denote any one of the ten conjugacy classes of sub-groups of $T=S_3 \times \mathbb{Z}_2$ and $B$ be a $G$-invariant box. Consider the following generating function:

(4)
\begin{align} N_G(B;q) = \sum_{\substack{\lambda\subseteq B\\ \lambda^G =\lambda}} q^{|\lambda|} \end{align}

where $|\lambda|$ is the number of nodes in the Ferrers diagram $\lambda$ and $\lambda^G=\lambda$ implies that we sum over plane partitions that are $G$-invariant. Consider another generating function

(5)
\begin{align} N'_G(B;q) = \sum_{\substack{\lambda\subseteq B\\ \lambda^G =\lambda}} q^{|\lambda/G|} \end{align}

where $|\lambda/G|$ counts the number of $G$-orbits formed by the nodes of $\lambda$ by the action of $G$.

### References

1. R. P. Stanley, Symmetries of Plane Partitions, J. of Comb. Theory, Ser. A, 43 (1986) 103-113.
2. G. Kuperberg, Symmetries of plane partitions and the permanent-determinant method, J. Combin. Theory Ser. A 68 (1994), no. 1, 115-151 arXiv:math/9410224

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