# Diagonal Harmonic Polynomials

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

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

with one equation for each pair of integers $a$ and $b$, such that $a+b>0$. It has been shown by Haiman that the linear span of these polynomials has dimension $(n+1)^{n-1}$. The bigraded Frobenius characteristic of the resulting $\mathbb{S}_n$-module (under the diagonal action) is given by $\nabla(e_n)$ (See the $\nabla$-operator page.). It is also noteworthy that the bigraded enumeration of the alternating component of this module gives rise to the famous $q,t$-Catalan polynomials.

One may generalize to $k$ sets of variables. For this, one considers a $k\times n$ matrix $X=(x_{ij})$ of variables. The symmetric group $\mathbb{S}_n$ (of $n\times n$ permutation matrices) acts on these variables by multiplication on the right, whereas the general linear group $GL_k$ acts by multiplication on the left. The module of Diagonal Harmonic Polynomials is stable under both of the respective extensions of these action to polynomials in the variables $X$. Since the two actions commute, one may decompose the concerned module under joint irreducibles. This may be globally encoded in the “universal” format

$\mathcal{D}_{n}(\mathbf{q};\mathbf{z})=\sum_{\mu\vdash n}\sum_\nu a_{\mu\nu} s_\nu(\mathbf{q})\,S_\mu(\mathbf{z}),$

with $\mathbf{q}=q_1,q_2,\ldots,q_k$ and $a_{\mu\nu}\in\mathbb{N}$ independent of $k$. Here the Schur functions $s_\nu(\mathbf{q})$ encode irreducibles for $GL_k$, while $S_\mu(\mathbf{z})$ encodes irreducibles for $\mathbb{S}_n$.

For small values of $n$, the resulting decompositions are

$\mathcal{D}_{1}(\mathbf{q};\mathbf{z})=S_{1}(\mathbf{z})$
$\mathcal{D}_{2}(\mathbf{q};\mathbf{z})=S_{2}(\mathbf{z})+s_{1}(\mathbf{q})\,S_{11}(\mathbf{z})$
$\mathcal{D}_{3}(\mathbf{q};\mathbf{z})=S_{3}(\mathbf{z})+(s_{1}(\mathbf{q})+s_{2}(\mathbf{q}))\,\,S_{21}(\mathbf{z})+(s_{3}(\mathbf{q})+s_{11}(\mathbf{q}))\,\,S_{111}(\mathbf{z})$

There are higher versions of these modules, for all $m\in\mathbb{N}$, denoted by $\mathcal{D}_n^m$. When $k=1$ all these modules have dimension $n!$. For $k=2$, it has been established by Haiman that they have dimension $(mn+1)^{n-1}$; and I have conjectured that they have dimension $(m+1)^n(mn+1)^{n-2}$ when $k=3$. One also has universal expressions for the associated characters, such as

$\mathcal{D}_{3}^2(\mathbf{q};\mathbf{z})=(s_{3}(\mathbf{q})+s_{11}(\mathbf{q}))\,\,S_{3}(\mathbf{z})$
$\qquad\qquad+(s_{4}(\mathbf{q})+s_{5}(\mathbf{q})+s_{21}(\mathbf{q})+s_{31}(\mathbf{q}))\,\,S_{21}(\mathbf{z})$
$\qquad\qquad+(s_{22}(\mathbf{q})+s_{41}(\mathbf{q})+s_{6}(\mathbf{q}))\,\,S_{111}(\mathbf{z})$

In the case of two sets of variables, these space originally arose around the study of Garsia-Haiman modules, themselves introduced for the study of positivity properties of the combinatorial Macdonald polynomials.

For more on all this, see