M Theory Lesson 134

Planar Young diagrams represent partitions of a natural number $n = n_1 + n_2 + n_3 + \cdots + n_k$. The $k$ rows are the pieces $n_i$ of the partition. But categorified numbers $n_i$ are actually sets with $n_i$ elements, or perhaps vector spaces of dimension $n_i$, or projective spaces of dimension $n_i - 1$. In this setting the expression $n = n_1 + n_2 + n_3 + \cdots + n_k$ is about the decomposition of a space into subspaces.

We have seen something like this before. Let $n_i$ instead represent $V^{\otimes n_{i}}$ for a fixed finite dimensional vector space $V$. Then the $O(n_{i})$ piece of the operad is the space of linear maps from $n_i$ to $V$ in Vect. The operad rules come from compositions $n_1 \otimes n_2 \otimes n_3 \cdots \otimes n_k \rightarrow n$ of these maps. Maybe instead of categorification of $\mathbb{N}$ we can look at operadification.

One then wonders what happens for higher dimensional ordinals. Actually, partitions are a lot like 2-tree ordinals. The simplest generalisation would allow each $n_{i}$ of a planar partition to be itself replaced by a partition of $m$ in a third direction. The first permutation group to fill a three dimensional diagram would be $S_4$, with four box partitions. This is the only additional diagram to the planar labels for the $S_4$ barycentrically divided tetrahedron. Similarly, $S_3$ is the first group to fill a truly planar Young diagram. This pattern continues for all $n$.