Part 1: Vertex Operators

Background

A partition of an integer $n$ is a sequence $(a_1, a_2, ... )$ of weakly decreasing nonnegative integers $a_i$ with $\sum_i a_i = n$.

Ex: $(5, 3, 2, 2, 1) = (5, 3, 2, 2, 1, 0, 0, ...)$ is a partition of $13$.

We usually treat these as top-left-justified sets of squares in a grid, so that $a_i$ is the length of row $i$, and we freely identify partitions with their Young diagrams.

Asymptotic Young Diagrams and Hooks

A Young diagram asymptotic to a partition $\lambda$ is the collection of boxes $\mathbb{N}^2 \setminus \lambda$.

The hook corresponding to a box in a Young diagram is the right-angled collection of all boxes directly below and to the right of it. The hook length $h(i, j)$ of a box is the total number of boxes in its hook, and the corner box $(i, j)$ is called the pivot of the hook.

In an asymptotic Young diagram, hooks extend up and left (i.e. always toward the boundary).

Plane Partitions

A reverse plane partition (or RPP) is a placement of nonnegative integers into a Young diagram, such that the rows and columns weakly increase. We can also think of them as a stacks of boxes in the corner of a room.

A skew plane partition (or SPP) is identical, but we place nonnegative integers into an asymptotic Young diagram $\mathbb{N} \setminus \lambda$ that decrease weakly along rows and columns. When $\lambda = \emptyset$, we just call the object a plane partition.

The weight of an RPP or SPP $\pi$, written $|\pi|$, is just the sum of its entries, or equivalently the total number of boxes stacked.

Interlacing

While the rows and columns of RPPs and SPPs contain valuable information, the diagonals of both turn out to be much more useful.

Given partitions $\lambda$ and $\mu$, we say $\lambda$ interlaces $\mu$, written $\lambda \succ \mu$, if $\lambda_1 \geq \mu_1 \geq \lambda_2 \geq \mu_2 \geq \cdots$ — in other words, if $\lambda$ and $\mu$ satisfy the plane partition inequalities when placed next to one another as diagonal slices with $\lambda$ coming first.

Vertex Operators

Given a partition $\mu$, the set of partitions $\lambda$ with $\lambda \succ \mu$ is a poset product of intervals in $\mathbb{N}$: each $\lambda_i$ is bounded below by $\mu_i$ and above by $\mu_{i - 1}$ (or unbounded for $i = 1$). We use this to build the generating functions for RPPs and SPPs diagonal by diagonal.

We define a formal $\mathbb{Q}$-linear vector space $\Lambda$ whose basis consists of vectors $\ket{\lambda}$ for partitions $\lambda$, and denote the corresponding dual basis elements by $\bra{\lambda}$.

We also define a weighing operator $Q$ by $Q(q)\,\ket{\lambda} = q^{|\lambda|}\,\ket{\lambda}$.

Interlacing Operators

While the formula for $\Gamma_+$ nominally contains an infinite sum, we can avoid convergence issues in practice by adding $Q$ operators to produce a generating function.

A Generating Function for Plane Partitions

To convert this into a more useful formula, we need to commute the $Q$ and $\Gamma$ operators.

Commutation Relations

Toggles

More Toggles

More broadly, we define the toggle of $\nu$ when it has any interlacing relationship with $\lambda$ and $\mu$.

When $\lambda \prec \nu \succ \mu$, we need to handle $\nu_1$ differently, since its poset of possible entries is unbounded above. We instead “pop off” the value $\nu_1 - \max\left\{ \lambda_1, \mu_1 \right\}$ in addition to producing the toggled partition $T(\nu)$, which then satisfies $\lambda \succ T(\nu) \prec \mu$.

Similarly, if $\lambda \succ \nu \prec \mu$, then the toggle map takes in both $\nu$ and a nonnegative integer $n$ to produce a partition $T(\nu, n)$, where $T(\nu, n)_1 = n + \max\left\{ \lambda_1, \mu_1 \right\}$. This new partition then satisfies $\lambda \prec T(\nu, n) \succ \mu$.

Bijectivizing $\Gamma$ Commutation

This proposition bijectivizes the following established commutation relations for $\Gamma$ operators:

$\displaystyle \Gamma_-(a)\Gamma_+(b) = \frac{1}{1 - ab}\Gamma_+(b)\Gamma_-(a)$

$\displaystyle \Gamma_+(a)\Gamma_+(b) = \Gamma_+(b)\Gamma_+(a)$

$\displaystyle \Gamma_-(a)\Gamma_-(b) = \Gamma_-(b)\Gamma_-(a).$

Specifically, in a generating function $z_1(q)$ equal to a product of $\Gamma$ operators, commuting any two adjacent operators in $z_1$ results in a generating function $z_2$ that counts objects given by toggling the corresponding diagonal of the objects counted by $z_1$.

Producing MacMahon's Function

This corresponds to toggling the main diagonal of the plane partitions counted by $M(q)$ and recording the popped numbers separately. We now have two choices of opposite-sign commutations, corresponding to the two diagonals of the plane partition that interlace their neighbors.

The resulting map is a weight-preserving bijection $\tau$ between plane partitions and hook-length-weighted tableaux. It is functionally identical to a map introduced by $\pak$ that operates on RPPs; what is new with this presentation is a more clear explanation of why it is independent of the order of

A Generating Function for RPPs

A famous result (due to Stanley in his Ph.D. thesis) expresses the generating function for RPPs of shape $\lambda$ as a product involving hook lengths:

Thm: $\displaystyle \sum_\pi q^{|\pi|} = \prod_{(i, j) \in \lambda} \frac{1}{1 - q^{h(i, j)}},$

where the sum is taken over all RPPs $\pi$ of shape $\lambda$. Since boxes and hooks are in bijection, the combinatorial interpretation is that RPPs of shape $\lambda$ are in bijection with multi-sets of hooks in a diagram of shape $\lambda$.

Hillman–Grassl

The first bijective proof of the hook length formula is due to Hillman and Grassl in 1975. Given a multi-set of hooks, it is not possible to place them unmodified to construct an RPP: for example, with $\lambda = (2, 2)$, including just the hook with pivot $(1, 1)$ makes the tableau (and invalid RPP)

$\begin{array}{cc} 1 & 1 \\ 1 & 0 \end{array}$

Instead, each hook is reshaped in an invertible manner before it is placed. We can view this process decomposing an RPP into a multi-set of hooks and back again.

Pak

In 2002, Igor Pak found a different bijection between RPPs and multisets of hooks. In contrast to Hillman-Grassl, Pak’s bijection operates on the level of boxes rather than hooks.

For ease of notation, let $\lambda$ be a partition and $x \in \lambda$ (i.e. a box in the Young diagram), and define $\mathbf{n}x$, $\mathbf{s}x$, $\mathbf{e}x$, and $\mathbf{w}x$ to be the boxes above, below, right, and left of $x$, respectively.

Given an RPP $\pi$, select an outer corner of the diagram and call it $y$. For every $x$ in the same diagonal as $y$ other than $y$ itself, define the toggle of $\pi(x)$ to be

$\displaystyle \pi'(x) = \max(\pi(\mathbf{n}x), \pi(\mathbf{w}x)) + \min(\pi(\mathbf{s}x), \pi(\mathbf{e}x)) - \pi(x).$

In other words, the toggle of $\pi(x)$ is the max of the adjacent entries smaller than it, plus the min of the adjacent entries larger than it, minus itself.

We now replace $\pi(x)$ with $\pi'(x)$ for all $x$ in the diagonal, excluding $y$.

We handle the outer corner $y$ differently: it is completely removed from $\pi$, and the value

$\displaystyle \pi(y) - \max(\pi(\mathbf{n}y), \pi(\mathbf{w}y)),$

which is how much larger $\pi(y)$ is than strictly necessary, is stored in a tableau at the same location. The shape of the RPP is now one box smaller, and we repeat the previous procedure, choosing a new outer corner, until the RPP is empty.

The RPP in the next animation has been rotated 180$^\circ$, so outer corners are in the top-left. For clarity, toggled corners remain in the same diagram, but they are given color.

It can be shown that the order in which corners are chosen does not impact the final result of Pak, and (more easily) that entries in the output tableau contribute weight equal to their hook length.

Toggling a non-corner entry is an involution, and knowing the entry recorded to the tableau from a toggled corner lets us recover the corner’s entry in the RPP. Therefore, the inverse bijection is well-defined.

While the algorithm is simple enough, the fact that it outputs multisets of hooks seems to hint at a description closer in style to Hillman-Grassl.

Sulzgruber

In 2017, Robin Sulzgruber found exactly that: an equivalent formulation of Pak’s bijection that removes one hook at a time.

Given a partition $\lambda$, we divide its Young diagram into four regions:

$\mathcal{O}$ consists of diagonals ending in an outer corner.

$\mathcal{I}$ consists of diagonals ending in an inner corner.

$\mathcal{A}$ consists of diagonals ending in a horizontal edge.

$\mathcal{B}$ consists of diagonals ending in a vertical edge.

Given an RPP $\pi$, we define the set of candidates $C(\pi)$ to be the cells $x \in \mathcal{O}$ with $\pi(\mathbf{w}u) < \pi(u)$, along with the cells $x \in \mathcal{A}$ with $\pi(\mathbf{w}u) < \pi(u) > \pi(\mathbf{n}u)$.

We select the minimum candidate $x$ with respect to content order — that is, whose diagonal is as north-east as possible, with preference within a diagonal given to the south-east end. With $x$ as a starting position, we build a path as follows:

If the current cell $x$ belongs to $\mathcal{O} \cup \mathcal{B}$ and $\pi(\mathbf{nx}) = \pi(x)$, we move one cell north. Otherwise, we move cell east unless that move leaves the diagram, in which case we stop. As with Hillman-Grassl, we decrement this path, store its pivot in a tableau, and repeat until the RPP is empty.

Thm: This process is equivalent to $\operatorname{PS}$.

We can describe $\operatorname{HG}$ in an identical way by changing the labeling order to be right to left, top to bottom, and then changing the hook ordering to match $\operatorname{HG}$’s insertion order. The result is a much more clear connection between the two bijections.

What’s the Use?

My research uses these bijections in order to develop new ones between generalizations of plane partitions. A central result deals with plane partitions asymptotic to a Young diagram $\lambda$, which are analogous to skew semi-standard Young tableaux.

Thm: With $\operatorname{M}(q)$, $\operatorname{RPP}_\lambda(q)$, and $\operatorname{APP}_\lambda(q)$ denoting the generating functions for plane partitions, RPPs of shape $\lambda$, and plane partitions asymptotic to lambda, respectively,

$\displaystyle \operatorname{APP}_\lambda(q) = \operatorname{RPP}_\lambda(q)M(q)$.

This relation was formerly known only algebraically, but showing it bijectively was possible using $\operatorname{HG}$. A generalization of the relation was also bijectivized using Pak, and a bijectivizing a further generalization, which was shown algebraically by Jenne, Webb, and Young in 2020, remains an open problem.

That’s all for now, folks!