# Trivariate Diagonal Harmonics

Let $\mathbf{x}=x_1,x_2,\ldots,x_n$,  $\mathbf{y}=y_1,y_2,\ldots,y_n$ and $\mathbf{z}=z_1,z_2,\ldots,z_n$ be three sets of $n$ variables. Trivariate Diagonal harmonic polynomials are the solutions, in the polynomial ring $\mathbb{Q}[\mathbf{x},\mathbf{y},\mathbf{z}]$, of the system of partial differential equations

$\partial_{x_1}^a\partial_{y_1}^b\partial_{z_1}^c\,f(\mathbf{x},\mathbf{y},\mathbf{z})+\partial_{x_2}^a\partial_{y_2}^b\partial_{z_2}^c\,f(\mathbf{x},\mathbf{y},\mathbf{z})+\ldots+\partial_{x_n}^a\partial_{y_n}^b\partial_{z_n}^c\,f(\mathbf{x},\mathbf{y},\mathbf{z})=0,$

with one equation for each triple of integers $a$, $b$ and $c$, such that $a+b+c>0$. It has been conjectured by Haiman that the linear span of these polynomials has dimension $2^n(n+1)^{n-2}$. There are higher versions of these modules, for all $m\in\mathbb{N}$,  and I have conjectured that they have dimension

$(m+1)^n(mn+1)^{n-2}.$

The alternating component of these spaces are also conjectured to be of dimension

$\large{\frac{(m+1)}{n\,(mn+1)}\binom{(m+1)^2n+m}{n-1}}.$

For $m=1$ the sequence is as follows:

$1,\ 3,\ 13,\ 68,\ 399,\ 2530,\ 16965,\ 118668,\ 857956,\ 6369883\ \ldots$

Consulting the online encyclopedia of integer sequences, one finds that Chapoton identified this as being the number of intervals in the Tamari lattice. This led me to to introduce the notion $m$-Tamari lattice, expecting that the formula above would give the number of its intervals. This was shown to be the case by Bousquet-Mélou, Fusy and Préville-Ratelle. It was natural to wonder if the whole dimension of the space would then be obtained by “decorating” these intervals, and thus I found an explicit conjecture for the character of the action of $\mathbb{S}_n$ on such decorated intervals. This combinatorial question was settled (also by Bousquet-Mélou, Chapuy and Préville-Ratelle), first for the dimension, and then for the whole character of the permutation action of $\mathbb{S}_n$ on these decorated intervals. Still open however is to show that all this relates correctly to the $\mathbb{S}_n$-module of trivariate diagonal harmonics. Indeed, experiments suggest that this holds at the level of characters (up to a sign twist), including parameters that account for the trivariate grading. For example, the trigraded enumeration of the alternating component of that module is

$tr+qr+qt+{r}^{3}+t{r}^{2}+{t}^{2}r+{t}^{3}+q{r}^{2}+qtr+q{t}^{2}+{q}^{2}r+{q}^{2}t+{q}^{3}$

when $n=3$ and $m=1$.

The next step is to study multivariate diagonal harmonic polynomials.

For more details on all this, see