#  Gelfand-Kirillov Dimensions of the $\mathbb{Z}^{2}$-graded Oscillator
Representations of $\mathfrak{sl}(n)$

Zhanqiang Bai  Institute of Mathematics, Academy of Mathematics and System
Sciences, Chinese Academy of Sciences, Beijing 100080, China<EMAIL_ADDRESS>
###### Abstract.

We find an exact formula of Gelfand-Kirillov dimensions for the infinite-
dimensional explicit irreducible $\mathfrak{sl}(n,\mathbb{F})$-modules that
appeared in the $\mathbb{Z}^{2}$-graded oscillator generalizations of the
classical theorem on harmonic polynomials established by Luo and Xu. Three
infinite subfamilies of these modules have the minimal Gelfand-Kirillov
dimension. They contain weight modules with unbounded weight multiplicities
and completely pointed modules.

Key Words: Gelfand-Kirillov dimension; Highest-weight module; Oscillator
representation; Universal enveloping algebra.

2010 Mathematics Subject Classification. Primary 22E47; Secondary 17B10

##  1\. Introduction

In 1960s, Gelfand-Kirillov [8] introduced a quantity to measure the rate of
growth of an algebra in terms of any generating set, which is now known as
Gelfand-Kirillov dimension. In 1970s, Vogan [23] and Joseph [13] used the
Gelfand-Kirillov dimension to measure the size of the infinite-dimensional
modules. Gelfand-Kirillov dimension has been an important invariant in the
theory of algebras over a field for the past forty years. Although it is
rarely exact, it has the advantage over Krull dimension of being both
symmetric and ideal invariant. It has been applied successfully to enveloping
algebras, Weyl algebras, and more generally to filtered and graded algebras.
But in general, the Gelfand-Kirillov dimension of an infinite-dimensional
module is not easy to compute.

A module of a finite-dimensional simple Lie algebra is called a weight module
if it is a direct sum of its weight subspaces. Let $M$ be an irreducible
highest-weight module for a finite-dimensional simple Lie algebra
$\mathfrak{g}$. Then $M$ is naturally a weight module with finite-dimensional
weight subspaces. Denote by $d_{M}$ its Gelfand-Kirillov dimension. We fix a
Cartan subalgebra $\mathfrak{h}$, a root system
$\Delta\subset\mathfrak{h}^{*}$ and a set of positive roots
$\Delta_{+}\subset\Delta$. Let $\rho$ be half the sum of all positive roots.
Suppose that $\beta$ is the highest root. It is well known that $d_{M}=0$ if
and only if $M$ is finite-dimensional, in which case irreducible modules are
classified by the highest-weight theory. From Vogan [24] and Wang [26], we
know that the next smallest integer occurring is $d_{M}=(\rho,\beta^{\vee})$.
We call them the _minimal Gelfand-Kirillov dimension module_. These small
modules are of great interest in representation theory. A general introduction
can be found in Vogan [24]. Fernando [7] showed that the only simple Lie
algebras which have simple torsion-free modules (or cuspidal modules,
i.e.,weight modules on which all root vectors of the Lie algebra act
bijectively) are those of types $A_{n}$ or $C_{n}$. He also showed that these
finitely generated torsion-free $\mathcal{U}(\mathfrak{g})$-modules have the
minimal Gelfand-Kirillov dimension. A similar result was independently
obtained by Futorny [6]. In the last 20 years cuspidal modules have been
extensively studied by Benkart, Britten, Grantcharov, Hooper, Khomenko,
Lemire, Mathieu, Mazorchuk, Serganova and others (see e.g. [3],[4], [9],[10],
[18]).

Benkart, Britten and Lemire [1] generalized Fernando’s work. They proved that
infinite-dimensional weight modules, whose weight-subspace dimensions are
bounded, have the minimal Gelfand-Kirillov dimension and exist only for
finite-dimensional simple Lie algebras of types $A_{n}$ and $C_{n}$. Britten
and Lemire [5] described all weights $\omega\in\mathfrak{h}^{*}$ such that the
corresponding irreducible highest-weight $\mathfrak{sp}(2n,\mathbb{C})$-module
$L(\omega)$ has bounded multiplicities. Sun [22] classified genuine
irreducible lowest-weight modules of $\mathfrak{sp}(2n,\mathbb{C})$ with
minimal Gelfand-Kirillov dimension. But other examples of minimal Gelfand-
Kirillov dimension modules are not well-known.

In classical harmonic analysis, a fundamental theorem says that the spaces of
homogeneous harmonic polynomials are irreducible modules of the corresponding
orthogonal Lie group (algebra) and the whole polynomial algebra is a free
module over the invariant polynomials generated by harmonic polynomials. Bases
of these irreducible modules can be obtained easily (e.g., cf. [27]). The
algebraic beauty of the above theorem is that Laplace equation characterizes
the irreducible submodules of the polynomial algebra and the corresponding
quadratic invariant gives a decomposition of the polynomial algebra into a
direct sum of irreducible submodules, namely, the complete reducibility.
Algebraically, this can be interpreted as an
$(\mathfrak{sl}(2,\mathbb{R}),\mathfrak{o}(n,\mathbb{R}))$ Howe duality.
Recently Luo and Xu [17] established the $\mathbb{Z}^{2}$-graded oscillator
generalizations of the above theorem for $\mathfrak{sl}(n,\mathbb{F})$, where
the irreducible submodules are $\mathbb{Z}^{2}$-graded homogeneous polynomial
solutions of deformed Laplace equations. An important feature of the
irreducible modules in [17] is that the corresponding representation formulas
are simple and bases are easily given. In fact, these are explicit infinite-
dimensional highest-weight irreducible $\mathfrak{sl}(n,\mathbb{F})$-modules.
The goal of this paper is to find an exact formula of Gelfand-Kirillov
dimension for them. It turns out that these Gelfand-Kirillov dimensions are
independent of the double grading and four infinite subfamilies of these
modules have the minimal Gelfand-Kirillov dimension. In general, our result
will be useful in studying the $\mathfrak{sl}(n,\mathbb{F})$-modules with a
given Gelfand-Kirillov dimension. Below we give more detailed technical
introduction.

Throughout this paper, the base field $\mathbb{F}$ has characteristic $0$. For
convenience, we will use the notion
$\overline{{i,i+j}}=\\{i,i+1,i+2,...,i+j\\}$ for integers $i$ and $j$ with
$i\leq j$. Denote by $\mathbb{N}$ the additive semigroup of nonnegative
integers. Let $E_{r,s}$ be the square matrix with $1$ as its $(r,s)$-entry and
$0$ as the others. Moreover, we always assume that $n\geq 2$ is an integer.
Denote ${\mathscr{B}}=\mathbb{F}[x_{1},...,x_{n},y_{1},...,y_{n}]$. Fix
$n_{1},n_{2}\in\overline{1,n}$ with $n_{1}\leq n_{2}$. Changing operators
$\partial_{x_{r}}\mapsto-x_{r},\;x_{r}\mapsto\partial_{x_{r}}$ for
$r\in\overline{1,n_{1}}$ and $\partial_{y_{s}}\mapsto-
y_{s},\;y_{s}\mapsto\partial_{y_{s}}$ for $s\in\overline{n_{2}+1,n}$ in the
canonical oscillator representation
$E_{i,j}|_{\mathscr{B}}=x_{i}\partial_{x_{j}}-y_{j}\partial_{y_{i}}$ for
$i,j\in\overline{1,n}$, we get the following non-canonical oscillator
representation of $\mathfrak{sl}(n,\mathbb{F})$ on ${\mathscr{B}}$ determined
by

(1.1)
$E_{i,j}|_{\mathscr{B}}=E_{i,j}^{x}-E_{j,i}^{y}\qquad\text{for}~{}i,j\in\overline{1,n}$

with

(1.2)
$E_{i,j}^{x}|_{\mathscr{B}}=\left\\{\begin{array}[]{ll}-x_{j}\partial_{x_{i}}-\delta_{i,j}&\mbox{if}\;i,j\in\overline{1,n_{1}};\\\
\partial_{x_{i}}\partial_{x_{j}}&\mbox{if}\;i\in\overline{1,n_{1}},\;j\in\overline{n_{1}+1,n};\\\
-x_{i}x_{j}&\mbox{if}\;i\in\overline{n_{1}+1,n},\;j\in\overline{1,n_{1}};\\\
x_{i}\partial_{x_{j}}&\mbox{if}\;i,j\in\overline{n_{1}+1,n}\end{array}\right.$

and

(1.3)
$E_{i,j}^{y}|_{\mathscr{B}}=\left\\{\begin{array}[]{ll}y_{i}\partial_{y_{j}}&\mbox{if}\;i,j\in\overline{1,n_{2}};\\\
-y_{i}y_{j}&\mbox{if}\;i\in\overline{1,n_{2}},\;j\in\overline{n_{2}+1,n};\\\
\partial_{y_{i}}\partial_{y_{j}}&\mbox{if}\;i\in\overline{n_{2}+1,n},\;j\in\overline{1,n_{2}};\\\
-y_{j}\partial_{y_{i}}-\delta_{i,j}&\mbox{if}\;i,j\in\overline{n_{2}+1,n}.\end{array}\right.$

The related variated Laplace operator becomes

(1.4)
${\mathscr{D}}=-\sum_{i=1}^{n_{1}}x_{i}\partial_{y_{i}}+\sum_{r=n_{1}+1}^{n_{2}}\partial_{x_{r}}\partial_{y_{r}}-\sum_{s=n_{2}+1}^{n}y_{s}\partial_{x_{s}}$

and its dual

(1.5)
$\eta=\sum_{i=1}^{n_{1}}y_{i}\partial_{x_{i}}+\sum_{r=n_{1}+1}^{n_{2}}x_{r}y_{r}+\sum_{s=n_{2}+1}^{n}x_{s}\partial_{y_{s}}.$

Set

(1.6)
${\mathscr{B}}_{\langle\ell_{1},\ell_{2}\rangle}=\mbox{Span}\\{x^{\alpha}y^{\beta}\mid\alpha,\beta\in\mathbb{N}\>^{n},\sum_{r=n_{1}+1}^{n}\alpha_{r}-\sum_{i=1}^{n_{1}}\alpha_{i}=\ell_{1},\sum_{i=1}^{n_{2}}\beta_{i}-\sum_{r=n_{2}+1}^{n}\beta_{r}=\ell_{2}\\}$

for $\ell_{1},\ell_{2}\in\mathbb{Z}$. Define

(1.7)
${\mathscr{H}}_{\langle\ell_{1},\ell_{2}\rangle}=\\{f\in{\mathscr{B}}_{\langle\ell_{1},\ell_{2}\rangle}\mid{\mathscr{D}}(f)=0\\}.$

Luo and Xu [17] proved that for any $\ell_{1},\ell_{2}\in\mathbb{Z}$ such that
$\ell_{1}+\ell_{2}\leq n_{1}-n_{2}+1-\delta_{n_{1},n_{2}}$,
${\mathscr{H}}_{\langle\ell_{1},\ell_{2}\rangle}$ is an irreducible highest-
weight $\mathfrak{sl}(n,\mathbb{F})$-module. Moreover, the homogeneous
subspace
$\mathscr{B}_{\langle\ell_{1},\ell_{2}\rangle}=\bigoplus_{m=0}^{\infty}\eta^{m}(\mathscr{H}_{\langle\ell_{1}-m,\ell_{2}-m\rangle})$
is a direct sum of irreducible submodules. In some special cases, they
obtained more general results. The following is the main theorem of this
paper.

######  Theorem 1.1.

For any $\ell_{1},\ell_{2}\in\mathbb{Z}$, if
$\mathfrak{sl}(n,\mathbb{F})$-module
${\mathscr{H}}_{\langle\ell_{1},\ell_{2}\rangle}$ is irreducible, then it has
the Gelfand-Kirillov dimension

(1.8) $d=\left\\{\begin{array}[]{ll}2n-2,&\hbox{\text{\emph{if}}
$1<n_{1}<n_{2}<n-1$ \emph{or} $3\leq n_{1}=n_{2}\leq n-3,n\geq 7$;}\\\
2n-3,&\hbox{\emph{if} $1=n_{1}<n_{2}<n$ \emph{or} $n_{1}<n_{2}=n-1$ \emph{or}
$n_{1}=n_{2}=3<n=6$;}\\\ 2n-4,&\hbox{if
$n_{1}=n_{2}=2<n-1\mathrm{~{}or~{}}1<n_{1}=n_{2}=n-2$;}\\\ n,&\hbox{\emph{if}
$1<n_{1}=n_{2}<n-1$;}\\\ n-1,&\hbox{\emph{otherwise}.}\end{array}\right.$

We find that some of these irreducible highest-weight
$\mathfrak{sl}(n,\mathbb{F})$-modules have the minimal Gelfand-Kirillov
dimension. The result is as follows.

######  Corollary 1.1.

An irreducible $\mathfrak{sl}(n,\mathbb{F})$-module
${\mathscr{H}}_{\langle\ell_{1},\ell_{2}\rangle}$ has the minimal Gelfand-
Kirillov dimension $n-1$ if and only if

  1. (1)

$n_{1}<n_{2}=n$.

  2. (2)

$n_{1}=n_{2}=1$.

  3. (3)

$n_{1}=n_{2}=n-1$.

From Benkart-Britten-Lemire [1], an infinite-dimensional weight module is
called _completely pointed_ if its weight spaces are all one-dimensional.
Using the results on modules for Weyl algebras, they provided the whole list
of simple infinite-dimensional completely pointed modules. These modules are
described in terms of multiplication and differentiation operators on
“polynomials”. Comparing with their result [1], we can give different
realizations for some of these simple infinite-dimensional completely pointed
modules.

######  Corollary 1.2.

An irreducible $\mathfrak{sl}(n,\mathbb{F})$-module
${\mathscr{H}}_{\langle\ell_{1},\ell_{2}\rangle}$ (with highest weight vector
$v_{\lambda}$ of weight $\lambda$) is completely pointed if and only if

  1. (1)

$n_{1}+1<n_{2}=n$,
${\mathscr{H}}_{\langle\ell_{1},\ell_{2}\rangle}={\mathscr{H}}_{\langle
m_{1},0\rangle}$, with $v_{\lambda}=x_{n_{1}+1}^{m_{1}}$,
$\lambda=-(m_{1}+1)\lambda_{n_{1}}+m_{1}\lambda_{n_{1}+1}$,
$m_{1}\in\mathbb{N}$
or ${\mathscr{H}}_{\langle\ell_{1},\ell_{2}\rangle}={\mathscr{H}}_{\langle-
m_{1},0\rangle}$, with $v_{\lambda}=x_{n_{1}}^{m_{1}}$,
$\lambda=m_{1}\lambda_{n_{1}-1}-(m_{1}+1)\lambda_{n_{1}}$, $m_{1}\leq
n-n_{1}-2$ or $m_{1}\geq n-n_{1}-1$.

  2. (2)

$n_{1}+1=n_{2}=n$,
${\mathscr{H}}_{\langle\ell_{1},\ell_{2}\rangle}={\mathscr{H}}_{\langle-
m_{1},m_{2}\rangle}$, with $v_{\lambda}=x_{1}^{m_{1}}y_{2}^{m_{2}}$,
$\lambda=(m_{2}-m_{1}-1)\lambda_{1}$, $m_{i}\in\mathbb{N}$ and $m_{2}\leq
m_{1}$, $n=2$
or ${\mathscr{H}}_{\langle\ell_{1},\ell_{2}\rangle}={\mathscr{H}}_{\langle-
m_{1},0\rangle}$, with $v_{\lambda}=x_{n-1}^{m_{1}}$,
$\lambda=m_{1}\lambda_{n-2}-(m_{1}+1)\lambda_{n-1}$, $m_{1}\in\mathbb{N}$
or ${\mathscr{H}}_{\langle\ell_{1},\ell_{2}\rangle}={\mathscr{H}}_{\langle
m,0\rangle}$, with $v_{\lambda}=x_{n-1}^{m}$,
$\lambda=m\lambda_{n-2}-(m+1)\lambda_{n-1}$, $m\in\mathbb{Z}$.

  3. (3)

$n_{1}=n_{2}=1$,
${\mathscr{H}}_{\langle\ell_{1},\ell_{2}\rangle}={\mathscr{H}}_{\langle-
m_{1},0\rangle}$, with $v_{\lambda}=x_{1}^{m_{1}}$,
$\lambda=-(m_{1}+2)\lambda_{1}$, $m_{1}\in\mathbb{N}$
or ${\mathscr{H}}_{\langle\ell_{1},\ell_{2}\rangle}={\mathscr{H}}_{\langle
m_{1}+1,-m_{1}-1\rangle}$, with $v_{\lambda}=\zeta_{2}^{m_{1}+1}$,
$\lambda=-(m_{1}+3)\lambda_{1}$, $m_{1}\in\mathbb{N}$, $n=3$.

  4. (4)

$n_{1}=n_{2}=n-1$,
${\mathscr{H}}_{\langle\ell_{1},\ell_{2}\rangle}={\mathscr{H}}_{\langle
0,-m_{2}\rangle}$, with $v_{\lambda}=y_{n}^{m_{2}}$,
$\lambda=-(m_{2}+2)\lambda_{n-1}$, $m_{2}\in\mathbb{N}$
or ${\mathscr{H}}_{\langle\ell_{1},\ell_{2}\rangle}={\mathscr{H}}_{\langle-
m_{2}-1,m_{2}+1\rangle}$, with $v_{\lambda}=\zeta_{1}^{m_{2}+1}$,
$\lambda=-(m_{2}+3)\lambda_{2}$, $m_{2}\in\mathbb{N}$, $n=3$.

Benkart-Britten-Lemire [1] showed that when infinite-dimensional simple
$\mathfrak{sl}(n,\mathbb{F})$-modules have bounded weight multiplicities, they
must have the minimal GK-dimension $n-1$. From our formula and Luo-Xu [17], we
can easily find that the converse doesn’t hold. For example, when
$n_{1}=n_{2}=1<n$, the irreducible highest-weight module
${\mathscr{H}}_{\langle-1,-1\rangle}$ has a highest weight vector $x_{1}y_{2}$
of weight $-4\lambda_{1}+\lambda_{2}$. But it doesn’t have bounded weight
multiplicities.

In Section 2 we recall the basic definition of the Gelfand-Kirillov dimension.
In Section 3, we give the formula on the Gelfand-Kirillov dimensions of these
highest-weight modules ${\mathscr{H}}_{\langle\ell_{1},\ell_{2}\rangle}$. In
Section 4, we give a proof for our formula. The proof is given in a case-by-
case way. In Section 5, we give a corollary about the
$\mathfrak{sl}(n,\mathbb{F})$-modules which have the minimal Gelfand-Kirillov
dimension.

##  2\. Preliminaries on Gelfand-Kirillov Dimension

In this section we recall the definition and some properties of the Gelfand-
Kirillov dimension. The details can be found in Refs.[2, 12, 15, 20, 23, 25].

######  Definition 2.1.

Let $A$ be an algebra (not necessarily associative) generated by a finite-
dimensional subspace $V$. Let $V^{n}$ denote the linear span of all products
of length at most $n$ in elements of $V$. The _Gelfand-Kirillov dimension_ of
$A$ is defined by:

$GKdim(A)=\limsup_{n\rightarrow\infty}\frac{\log\mathrm{dim}(V^{n})}{\log n}$

######  Remark 2.1.

It is well-known that the above definition is independent of the choice of the
finite dimensional generating subspace $V$ (see Ref.[2, 15]). Clearly
$GKdim(A)=0$ if and only if $\mathrm{dim}(A)<\infty$.

The notion of Gelfand-Kirillov dimension can be extended for left $A$-modules.
In fact, we have the following definition.

######  Definition 2.2.

Let $A$ be an algebra (not necessarily associative) generated by a finite-
dimensional subspace $V$. Let $M$ be a left $A$-module generated by a finite-
dimensional subspace $M_{0}$. Let $V^{n}$ denote the linear span of all
products of length at most $n$ in elements of $V$. The _Gelfand-Kirillov
dimension_ $GKdim(M)$ of $M$ is defined by

$GKdim(M)=\limsup_{n\rightarrow\infty}\frac{\log\mathrm{dim}(V^{n}M_{0})}{\log
n}.$

In particular, let $\mathfrak{g}$ be a complex Lie algebra. Let
$A=\mathcal{U}(\mathfrak{g})$ be the enveloping algebra of $\mathfrak{g}$,
with the standard filtration given by $A_{n}=\mathcal{U}_{n}(\mathfrak{g})$,
the subspace of $\mathcal{U}(\mathfrak{g})$ spanned by products of at most
$n$-elements of $\mathfrak{g}$. By the Poincaré-Birkhoff-Witt theorem (see
Knapp [14, Prop. 3.16]), the graded algebra
$\text{gr}(\mathcal{U}(\mathfrak{g}))$ is canonically isomorphic to the
symmetric algebra $S(\mathfrak{g})$. Suppose $M$ is a
$\mathcal{U}(\mathfrak{g})$-module generated by a finite-dimensional subspace
$M_{0}$. We set $M_{n}=\mathcal{U}_{n}(\mathfrak{g})M_{0}$. Denote
$\text{gr}M=\bigoplus\limits_{n=0}^{\infty}\text{gr}_{n}M$, where
$\text{gr}_{n}M=M_{n}/M_{n-1}$. Then $\text{gr}M$ becomes a graded
$S(\mathfrak{g})$-module. We denote $\dim(M_{n})$ by $\varphi_{M}(n)$. Then we
have the following lemma.

######  Lemma 2.1.

_(Hilbert-Serre[29, Chapter VII. Th.41] and [28])_

  1. (1)

With the notations as above, there exists a unique polynomial
$\tilde{\varphi}_{M}(n)$ such that $\varphi_{M}(n)=\tilde{\varphi}_{M}(n)$ for
large $n$. The leading term of $\tilde{\varphi}_{M}(n)$ is

$\frac{c(M)}{(d_{M})!}n^{d_{M}},$

where $c(M)$ is an integer.

  2. (2)

The degree $d_{M}$ of this polynomial $\tilde{\varphi}_{M}(n)$ is equal to the
dimension of the _associated variety_

$\mathscr{V}(M)=\\{X\in\mathfrak{g}^{*}|p(X)=0,\forall
p\in\mathrm{Ann}_{S(\mathfrak{g})}(\text{\emph{gr}}M)\\},$

where $\mathrm{Ann}_{S(\mathfrak{g})}(\text{\emph{gr}}M)=\\{D\in
S(\mathfrak{g})|Dv=0,\forall v\in\text{\emph{gr}}M\\}$ is the annihilator
ideal of $\mathrm{gr}M$ in $S(\mathfrak{g})$, and $S(\mathfrak{g})$ is
identified with the polynomial ring over $\mathfrak{g}^{*}$ through the
Killing form of $\mathfrak{g}$.

######  Remark 2.2.

From the definition of Gelfand-Kirillov dimension, we know

$GKdim(M)=\limsup_{n\rightarrow\infty}\frac{\log\dim(U_{n}(\mathfrak{g})M_{0})}{\log
n}=\limsup_{n\rightarrow\infty}\frac{\log\tilde{\varphi}_{M}(n)}{\log
n}=d_{M}=\dim\mathscr{V}(M).$

######  Example 2.1.

Let $M=\mathbb{C}[x_{1},...,x_{k}]$. Then $M$ is an algebra generated by the
finite-dimensional subspace $V=Span_{\mathbb{C}}\\{x_{1},...,x_{k}\\}$. So
$M_{n}=V^{n}=\bigoplus\limits_{0\leq q\leq n}P_{q}[x_{1},...,x_{k}]$ is the
subset of homogeneous polynomials of degree $\leq n$. Then

$\displaystyle\varphi_{M}(n)=$ $\displaystyle\sum\limits_{0\leq q\leq
n}\dim_{\mathbb{C}}(P_{q}[x_{1},...,x_{k}])$ $\displaystyle=$
$\displaystyle\sum\limits_{0\leq q\leq n}\binom{k+q-1}{q}$ $\displaystyle=$
$\displaystyle\binom{k+n}{n}$ $\displaystyle=$
$\displaystyle\frac{n^{k}}{k!}+O(n^{k-1}).$

Then we have $GKdim(M)=k.$

##  3\. A formula on the Gelfand-Kirillov dimension

We keep the same notations with the introduction. Write

(3.1)
$\zeta_{1}=x_{n_{1}-1}y_{n_{1}}-x_{n_{1}}y_{n_{1}-1},\;\;\zeta_{2}=x_{n_{1}+1}y_{n_{1}+2}-x_{n_{1}+2}y_{n_{1}+1}.$

From Luo-Xu [17], the followings are detailed irreducible highest weight
$\mathfrak{sl}(n,\mathbb{F})$-modules:

  1. (1)

$n_{1}+1<n_{2}<n$. For $m_{1},m_{2}\in\mathbb{N}$ with $m_{1}+m_{2}\geq
n_{2}-n_{1}-1$, ${\mathscr{H}}_{\langle-m_{1},-m_{2}\rangle}$ is an infinite-
dimensional irreducible module with a highest-weight vector
$x_{n_{1}}^{m_{1}}y_{n_{2}+1}^{m_{2}}$. For $m_{1},m_{2}\in\mathbb{N}$ with
$m_{2}-m_{1}\geq n_{2}-n_{1}-1$, ${\mathscr{H}}_{\langle m_{1},-m_{2}\rangle}$
is an infinite-dimensional irreducible module with a highest-weight vector
$x_{n_{1}+1}^{m_{1}}y_{n_{2}+1}^{m_{2}}$. For $m_{1},m_{2}\in\mathbb{N}$ with
$m_{1}-m_{2}\geq n_{2}-n_{1}-1$, ${\mathscr{H}}_{\langle-m_{1},m_{2}\rangle}$
is an infinite-dimensional irreducible module with a highest-weight vector
$x_{n_{1}}^{m_{1}}y_{n_{2}}^{m_{2}}$.

  2. (2)

$n_{1}+1<n_{2}=n$. For $m_{1},m_{2}\in\mathbb{N}$, ${\mathscr{H}}_{\langle
m_{1},m_{2}\rangle}$ is an infinite-dimensional irreducible module with a
highest-weight vector $x_{n_{1}+1}^{m_{1}}y_{n}^{m_{2}}$. For
$m_{1},m_{2}\in\mathbb{N}$ with $m_{1}\leq n-n_{1}-2$ or $m_{2}-m_{1}\leq
n_{1}-n+1$, ${\mathscr{H}}_{\langle-m_{1},m_{2}\rangle}$ is an infinite-
dimensional irreducible module with a highest-weight vector
$x_{n_{1}}^{m_{1}}y_{n}^{m_{2}}$.

  3. (3)

$n_{1}+1=n_{2}<n$. For $m_{1},m_{2}\in\mathbb{N}$, ${\mathscr{H}}_{\langle-
m_{1},-m_{2}\rangle}$ is an infinite-dimensional irreducible module with a
highest-weight vector $x_{n_{1}}^{m_{1}}y_{n_{1}+2}^{m_{2}}$. For
$m_{1},m_{2}\in\mathbb{N}$ with $m_{2}-m_{1}\geq 0$, ${\mathscr{H}}_{\langle
m_{1},-m_{2}\rangle}$ is an infinite-dimensional irreducible module with a
highest-weight vector $x_{n_{1}+1}^{m_{1}}y_{n_{1}+2}^{m_{2}}$. For
$m_{1},m_{2}\in\mathbb{N}$ with $m_{1}-m_{2}\geq 0$, ${\mathscr{H}}_{\langle-
m_{1},m_{2}\rangle}$ is an infinite-dimensional irreducible module with a
highest-weight vector $x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}}$.

  4. (4)

$n_{1}+1=n_{2}=n$. For $m_{1},m_{2}\in\mathbb{N}$ with $m_{2}\leq m_{1}$,
${\mathscr{H}}_{\langle-m_{1},m_{2}\rangle}$ is an infinite-dimensional
irreducible module with a highest-weight vector
$x_{n-1}^{m_{1}}y_{n}^{m_{2}}$. Moreover, ${\mathscr{H}}_{\langle m,0\rangle}$
is an infinite-dimensional irreducible module with a highest-weight vector
$x_{n-1}^{m}$. For $m_{1},m_{2}\in\mathbb{N}+1$, ${\mathscr{H}}_{\langle
m_{1},m_{2}\rangle}$ is an infinite-dimensional irreducible module with a
highest-weight vector
$\eta^{m_{1}+m_{2}}(x_{n-1}^{m_{2}}y_{n}^{-m_{1}})$.

  5. (5)

$n_{1}=n_{2}<n-1$. Let $m_{1},m_{2}\in\mathbb{N}$. The subspace
${\mathscr{H}}_{\langle-m_{1},-m_{2}\rangle}$ is an infinite-dimensional
irreducible module with a highest-weight vector
$x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}}$. The subspace ${\mathscr{H}}_{\langle
m_{1}+1,-m_{2}-m_{1}-1\rangle}$ is an infinite-dimensional irreducible module
with a highest-weight vector $y_{n_{1}+1}^{m_{2}}\zeta_{2}^{m_{1}+1}$. If
$n_{1}\geq 2$, the subspace ${\mathscr{H}}_{\langle-
m_{1}-m_{2}-1,m_{2}+1\rangle}$ is an infinite-dimensional irreducible module
with a highest-weight vector $x_{n_{1}}^{m_{1}}\zeta_{1}^{m_{2}+1}$.

  6. (6)

$n_{1}=n_{2}=n-1$. Let $m_{1},m_{2}\in\mathbb{N}$. The subspace
${\mathscr{H}}_{\langle-m_{1},-m_{2}\rangle}$ is an infinite-dimensional
irreducible module with a highest-weight vector
$x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}}$. If $n\geq 3$, the subspace
${\mathscr{H}}_{\langle-m_{1}-m_{2}-1,m_{2}+1\rangle}$ is an infinite-
dimensional irreducible module with a highest-weight vector
$x_{n_{1}}^{m_{1}}\zeta_{1}^{m_{2}+1}$.

  7. (7)

$n_{1}=n_{2}=n$. Let $m_{1},m_{2}\in\mathbb{N}$. The subspace
${\mathscr{H}}_{\langle-m_{1}-m_{2},m_{2}\rangle}$ is finite-dimensional and
is an irreducible module with a highest-weight vector
$x_{n}^{m_{1}}\zeta_{1}^{m_{2}}$.

Then we have our main theorem, i.e. Theorem 1.1.

##  4\. Proof of the main theorem

Now we want to compute the Gelfand-Kirillov dimension of the
$\mathfrak{sl}(n,\mathbb{F})$-module
${\mathscr{H}}_{\langle\ell_{1},\ell_{2}\rangle}$ for all cases. Suppose its
highest weight vector is $v_{\lambda}$ with highest weight $\lambda$.

We simply write $E_{i,j}|_{\mathscr{B}}$ as $E_{i,j}$. Take

(4.1) $\mathfrak{h}=\sum_{i=1}^{n-1}\mathbb{F}(E_{i,i}-E_{i+1,i+1})$

as a Cartan subalgebra of $\mathfrak{sl}(n,\mathbb{F})$ and the subspace
spanned by positive root vectors:

(4.2) $\mathfrak{sl}(n,\mathbb{F})_{+}=\sum_{1\leq i<j\leq
n}\mathbb{F}E_{i,j}.$

Correspondingly, we have

(4.3) $\mathfrak{sl}(n,\mathbb{F})_{-}=\sum_{1\leq i<j\leq
n}\mathbb{F}E_{j,i}.$

From the highest-weight module theorem we know that
${\mathscr{H}}_{\langle\ell_{1},\ell_{2}\rangle}=\mathcal{U}(\mathfrak{g})v_{\lambda}=\mathcal{U}(\mathfrak{g_{-}})v_{\lambda}$.
In the following we will compute the Gelfand-Kirillov dimension of
$\mathcal{U}(\mathfrak{g_{-}})v_{\lambda}$ in a case-by-case way.

Firstly we need the following two well-known lemmas.

######  Lemma 4.1.

_(Multinomial theorem)_
Let $n,m$ be two positive integers, then

(4.4)
$\left|\\{(k_{1},k_{2},...,k_{m})\in\mathbb{N}^{m}|\sum\limits_{i=1}^{m}k_{i}=n\\}\right|={n+m-1\choose
m-1}.$

######  Lemma 4.2.

Let $p,n$ be two positive integers, then

(4.5)
$\sum\limits_{i=0}^{n}i^{p}=\frac{(n+1)^{p+1}}{p+1}+\sum\limits_{k=1}^{p}\frac{B_{k}}{p-k+1}{p\choose
k}(n+1)^{p-k+1}\approx\frac{n^{p+1}}{p+1},$

where $B_{k}$ denotes a Bernoulli number.

From these two lemmas, we can get the following several propositions.

######  Proposition 4.1.

Let $k\in\mathbb{N}$ and we denote
$M_{k}=\left\\{\prod\limits_{\begin{subarray}{c}1\leq i\leq n_{1}\\\
n_{1}+1\leq t\leq
n\end{subarray}}({x_{i}x_{t}})^{p_{it}}|\sum\limits_{\begin{subarray}{c}1\leq
i\leq n_{1}\\\ n_{1}+1\leq t\leq
n\end{subarray}}p_{it}=k,p_{it}\in\mathbb{N}\right\\}$. Then

(4.6) $a_{k}=\dim Span_{\mathbb{R}}M_{k}={n_{1}+k-1\choose
k}{n-n_{1}+k-1\choose k}\approx ak^{n-2},$

for some constant _a_.

###### Proof.

From the definition of $M_{k}$, we know that all the elements in $M_{k}$ are
monomials and they must form a basis for $Span_{\mathbb{R}}M_{k}$. Thus

$\displaystyle a_{k}=$ $\displaystyle\dim
Span_{\mathbb{R}}M_{k}=\\#\left\\{\prod\limits_{\begin{subarray}{c}1\leq i\leq
n_{1}\\\ n_{1}+1\leq t\leq
n\end{subarray}}({x_{i}x_{t}})^{p_{it}}|\sum\limits_{\begin{subarray}{c}1\leq
i\leq n_{1}\\\ n_{1}+1\leq t\leq
n\end{subarray}}p_{it}=k,p_{it}\in\mathbb{N}\right\\}$ $\displaystyle=$
$\displaystyle\\#\left\\{\prod\limits_{\begin{subarray}{c}1\leq i\leq
n_{1}\end{subarray}}(x_{i})^{\sum_{n_{1}+1\leq t\leq
n}p_{it}}\prod\limits_{\begin{subarray}{c}n_{1}+1\leq t\leq
n\end{subarray}}(x_{t})^{\sum_{1\leq i\leq
n_{1}}p_{it}}|\sum\limits_{\begin{subarray}{c}1\leq i\leq n_{1}\\\ n_{1}+1\leq
t\leq n\end{subarray}}p_{it}=k,p_{it}\in\mathbb{N}\right\\}$ $\displaystyle=$
$\displaystyle{n_{1}+k-1\choose k}{n-n_{1}+k-1\choose k}\approx
ak^{n-2},\emph{~{}~{}for some constant~{}}\emph{a}.$

∎

######  Proposition 4.2.

Let $k\in\mathbb{N}$ and we denote
$N_{k}=\left\\{\prod\limits_{\begin{subarray}{c}1\leq i\leq n_{1}\\\
n_{1}+1\leq t\leq
n\end{subarray}}({x_{i}x_{t}}-y_{i}y_{t})^{h_{it}}|\sum\limits_{\begin{subarray}{c}1\leq
i\leq n_{1}\\\ n_{1}+1\leq t\leq
n\end{subarray}}h_{it}=k,h_{it}\in\mathbb{N}\right\\}$. Then we have

$\displaystyle d_{k}=\dim
Span_{\mathbb{R}}N_{k}\approx\left\\{\begin{array}[]{ll}c_{0}k^{n-2},&\text{if~{}}{n_{1}=1\mathrm{~{}or~{}}n_{1}=n-1;}\\\
c_{1}k^{2n-5},&\text{if~{}}{n_{1}=2<n-1\mathrm{~{}or~{}}1<n_{1}=n-2;}\\\
c_{2}k^{2n-4},&\text{if~{}}{n_{1}=3<n=6;}\\\ c_{3}k^{2n-3},&\text{if~{}}{3\leq
n_{1}\leq n-3,n\geq 7.}\end{array}\right.$

Here $c_{0},c_{1},c_{2}$ and $c_{3}$ are some positive constants which are
independent of $k$.

###### Proof.

When $n_{1}=1$, we have

$d_{k}=\dim
Span_{\mathbb{R}}N_{k}=\left\\{\prod\limits_{\begin{subarray}{c}2\leq t\leq
n\end{subarray}}({x_{1}x_{t}}-y_{1}y_{t})^{h_{t}}|\sum\limits_{\begin{subarray}{c}2\leq
t\leq n\end{subarray}}h_{t}=k,h_{t}\in\mathbb{N}\right\\}={n-1+k-1\choose
k}\approx c_{0}k^{n-2},$

for some positive constant $c_{0}$. The case for $n_{1}=n-1$ is dual to the
previous case.

When $n_{1}=2<n-1$, we have

$\displaystyle d_{k}$ $\displaystyle=\dim Span_{\mathbb{R}}N_{k}$
$\displaystyle=\dim
Span_{\mathbb{R}}\left\\{\prod\limits_{\begin{subarray}{c}1\leq i\leq 2\\\
3\leq t\leq
n\end{subarray}}({x_{i}x_{t}}-y_{i}y_{t})^{h_{it}}|\sum\limits_{\begin{subarray}{c}1\leq
i\leq 2\\\ 3\leq t\leq n\end{subarray}}h_{it}=k,h_{it}\in\mathbb{N}\right\\}$
$\displaystyle\geq\dim Span_{\mathbb{R}}\left\\{\prod\limits_{3\leq t\leq
n}(x_{1}x_{t})^{h_{1t}}\prod\limits_{3\leq t\leq
n}(y_{2}y_{t})^{h_{2t}}|\sum\limits_{\begin{subarray}{c}1\leq i\leq 2\\\ 3\leq
t\leq n\end{subarray}}h_{it}=k,h_{it}\in\mathbb{N}\right\\}$
$\displaystyle\approx c_{11}k^{2n-5}.$

On the other hand, we have $d_{k}=\dim Span_{\mathbb{R}}N_{k}\leq
c_{12}k^{n_{1}(n-n_{1})-1}=c_{12}k^{2n-5},$ for some positive constant
$c_{12}$. So we must have $d_{k}=\dim Span_{\mathbb{R}}N_{k}\approx
c_{1}k^{2n-5},$ for some positive constant $c_{1}$. The case for $1<n_{1}=n-2$
is dual to the previous case.

When $n_{1}=3<n=6$, we have

$\displaystyle d_{k}$ $\displaystyle=\dim Span_{\mathbb{R}}N_{k}$
$\displaystyle=\dim
Span_{\mathbb{R}}\left\\{\prod\limits_{\begin{subarray}{c}1\leq i\leq 3\\\
4\leq t\leq
6\end{subarray}}({x_{i}x_{t}}-y_{i}y_{t})^{h_{it}}|\sum\limits_{\begin{subarray}{c}1\leq
i\leq 3\\\ 4\leq t\leq 6\end{subarray}}h_{it}=k,h_{it}\in\mathbb{N}\right\\}$
$\displaystyle\geq\dim
Span_{\mathbb{R}}\left\\{({x_{1}x_{4}})^{h_{14}}({x_{1}x_{5}})^{h_{15}}({x_{2}x_{6}})^{h_{26}}({x_{2}x_{5}})^{h_{25}}({x_{3}x_{6}})^{h_{36}}\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\left.\cdot({y_{1}y_{6}})^{h_{16}}({y_{2}y_{4}})^{h_{24}}({y_{3}y_{4}})^{h_{34}}({y_{3}y_{5}})^{h_{35}}|\sum\limits_{\begin{subarray}{c}1\leq
i\leq 3\\\ 4\leq t\leq n\end{subarray}}h_{it}=k,h_{it}\in\mathbb{N}\right\\}$
$\displaystyle=\dim
Span_{\mathbb{R}}\left\\{\left(({x_{1}})^{h_{14}+h_{15}}({x_{2}})^{h_{25}+h_{26}}(x_{3})^{h_{36}}y_{4}^{h_{24}+h_{34}}y_{5}^{h_{35}}y_{6}^{h_{16}}\right)\right.$
$\displaystyle\quad\quad\quad\quad\quad\cdot\left.\left(({x_{4}})^{h_{14}}({x_{5}})^{h_{15}+h_{25}}(x_{6})^{h_{26}+h_{36}}(y_{2})^{h_{24}}(y_{3})^{h_{34}+h_{35}}y_{1}^{h_{16}}\right)|\sum\limits_{\begin{subarray}{c}1\leq
i\leq 3\\\ 4\leq t\leq n\end{subarray}}h_{it}=k,h_{it}\in\mathbb{N}\right\\}$
$\displaystyle\approx c_{21}k^{8}.$

On the other hand, we have $d_{k}=\dim Span_{\mathbb{R}}N_{k}\leq
c_{22}k^{n_{1}(n-n_{1})-1}=c_{22}k^{8},$ for some positive constant $c_{22}$.
So we must have $d_{k}=\dim Span_{\mathbb{R}}N_{k}\approx
c_{2}k^{8}=c_{2}k^{2n-4},$ for some positive constant $c_{2}$.

When $3=n_{1}<n-3$, we have

$\displaystyle d_{k}$ $\displaystyle=\dim Span_{\mathbb{R}}N_{k}$
$\displaystyle=\dim
Span_{\mathbb{R}}\left\\{(\prod\limits_{\begin{subarray}{c}1\leq i\leq 3\\\
4\leq t\leq
6\end{subarray}}({x_{i}x_{t}}-y_{i}y_{t})^{h_{it}})(\prod\limits_{\begin{subarray}{c}1\leq
i\leq 3\\\ 7\leq t\leq
n\end{subarray}}(x_{i}x_{t}-y_{i}y_{t})^{h_{it}})|\sum\limits_{\begin{subarray}{c}1\leq
i\leq 3\\\ 4\leq t\leq n\end{subarray}}h_{it}=k,h_{it}\in\mathbb{N}\right\\}$
$\displaystyle\geq\dim
Span_{\mathbb{R}}\left\\{(\prod\limits_{\begin{subarray}{c}1\leq i\leq 3\\\
4\leq t\leq
6\end{subarray}}({x_{i}x_{t}}-y_{i}y_{t})^{h_{it}})(\prod\limits_{\begin{subarray}{c}7\leq
t\leq
n\end{subarray}}({x_{3}x_{t}})^{h_{3t}}({y_{1}y_{t}})^{h_{1t}}(y_{2}y_{t})^{h_{2t}})\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\left.|\sum\limits_{\begin{subarray}{c}1\leq
i\leq 3\\\ 4\leq t\leq n\end{subarray}}h_{it}=k,h_{it}\in\mathbb{N}\right\\}$
$\displaystyle\geq\dim
Span_{\mathbb{R}}\left\\{\left(({x_{1}})^{h_{14}+h_{15}}({x_{2}})^{h_{25}+h_{26}}(x_{3})^{h_{36}+\sum
h_{3t}}y_{4}^{h_{24}+h_{34}}y_{5}^{h_{35}}y_{6}^{h_{16}}\prod\limits_{7\leq
t\leq n}y_{t}^{h_{1t}+h_{2t}}\right)\right.$
$\displaystyle\quad\quad\quad\quad\quad\cdot\left.\left(({x_{4}})^{h_{14}}({x_{5}})^{h_{15}+h_{25}}(x_{6})^{h_{26}+h_{36}}(\prod\limits_{7\leq
t\leq n}x_{t}^{h_{3t}})(y_{2})^{h_{24}+\sum
h_{2t}}(y_{3})^{h_{34}+h_{35}}y_{1}^{h_{16}+\sum h_{it}}\right)\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\left.|\sum\limits_{\begin{subarray}{c}1\leq
i\leq 3\\\ 4\leq t\leq n\end{subarray}}h_{it}=k,h_{it}\in\mathbb{N}\right\\}$
$\displaystyle\approx c_{31}k^{2n-3}.$

On the other hand, we have

$\displaystyle d_{k}$ $\displaystyle=\dim Span_{\mathbb{R}}N_{k}$
$\displaystyle\leq\dim
Span_{\mathbb{R}}\left\\{\prod\limits_{\begin{subarray}{c}1\leq i\leq n_{1}\\\
n_{1}+1\leq t\leq
n\end{subarray}}({x_{i}x_{t}})^{p_{it}}\prod\limits_{\begin{subarray}{c}1\leq
i\leq n_{1}\\\ n_{1}+1\leq t\leq n\end{subarray}}({y_{i}y_{t}})^{q_{it}}|\sum
p_{it}+\sum q_{it}=k\right\\}$ $\displaystyle\approx c_{32}k^{2n-3},$

for some positive constant $c_{32}$. So we must have $d_{k}=\dim
Span_{\mathbb{R}}N_{k}\approx c_{3}k^{2n-3},$ for some positive constant
$c_{3}$. The case for $3<n_{1}=n-3$ is dual to the previous case.

When $3<n_{1}<n-3$, we can use the same inductive argument with the above case
and get

$d_{k}=\dim Span_{\mathbb{R}}N_{k}\approx c_{3}k^{2n-3},$

for some positive constant $c_{3}$. ∎

######  Proposition 4.3.

Let $k\in\mathbb{N}$. Suppose $2<n_{1}+1\leq n_{2}<n-1$ and we denote

$\displaystyle N^{\prime}_{k}$
$\displaystyle=\left\\{\prod\limits(x_{i}x_{s})^{p_{is}}\prod\limits(y_{s}y_{t})^{q_{st}}\prod\limits({x_{i}x_{t}}-y_{i}y_{t})^{h_{it}}|\right.$
$\displaystyle\quad\quad\left.\sum\limits_{\begin{subarray}{c}1\leq i\leq
n_{1}\\\ n_{1}+1\leq s\leq
n_{2}\end{subarray}}p_{is}+\sum\limits_{\begin{subarray}{c}n_{1}+1\leq s\leq
n_{2}\\\ n_{2}+1\leq t\leq
n\end{subarray}}q_{st}+\sum\limits_{\begin{subarray}{c}1\leq i\leq n_{1}\\\
n_{2}+1\leq t\leq n\end{subarray}}h_{it}=k\right\\},$

then

$d^{\prime}_{k}=\dim Span_{\mathbb{R}}N^{\prime}_{k}\approx ck^{2n-3},$

for some constant $c$.

###### Proof.

When $n_{1}=2<n_{2}<n-1$, then from Prop 4.2 we have

$\displaystyle d^{\prime}_{k}=\dim
Span_{\mathbb{R}}\left\\{\prod\limits(x_{i}x_{s})^{p_{is}}\prod\limits(y_{s}y_{t})^{q_{st}}\prod\limits({x_{i}x_{t}}-y_{i}y_{t})^{h_{it}}|\sum\limits
p_{is}+\sum\limits q_{st}+\sum\limits h_{it}=k\right\\}$
$\displaystyle\geq\dim
Span_{\mathbb{R}}\left\\{\left(({x_{2}})^{h_{2,n_{2}+1}+\sum
p_{2s}}(x_{1})^{\sum p_{1s}}(\prod\limits_{n_{2}+2\leq t\leq
n}(x_{1})^{h_{1t}})(\prod\limits_{n_{2}+2\leq t\leq
n}(y_{t})^{h_{2t}})(y_{t})^{\sum
q_{st}}({y_{n_{2}+1}})^{h_{1,n_{2}+1}}\right)\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\left.\cdot\left(\prod\limits(x_{s})^{p_{1s}+p_{2s}}(x_{n_{2}+1})^{h_{2,n_{2}+1}}(\prod\limits_{n_{2}+2\leq
t\leq
n}(x_{t})^{h_{1t}})(\prod\limits(y_{s})^{q_{st}})({y_{1}})^{h_{1,n_{2}+1}}\prod\limits_{n_{2}+2\leq
t\leq n}(y_{2})^{h_{2t}}\right)\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\left.|\sum\limits
p_{is}+\sum\limits q_{st}+\sum\limits h_{it}=k\right\\}$ $\displaystyle\approx
c_{0}k^{2n-3},\emph{~{}for some constant }c_{0}.$

On the other hand, we have

$\displaystyle d^{\prime}_{k}=$ $\displaystyle\dim
Span_{\mathbb{R}}N^{\prime}_{k}$ $\displaystyle\leq$ $\displaystyle\dim
Span_{\mathbb{R}}\left\\{\prod\limits({x_{i}x_{s}})^{p_{is}}\prod({y_{s}y_{t}})^{q_{st}}\prod({x_{i}x_{t}})^{l_{it}}\prod(y_{i}y_{t})^{f_{it}}|\right.$
$\displaystyle\quad\quad\quad\quad\quad\left.\sum h_{it}+\sum p_{it}+\sum
l_{it}+\sum f_{it}=k\right\\}$ (4.7) $\displaystyle=$ $\displaystyle\dim
Span_{\mathbb{R}}\left\\{\prod\limits_{\begin{subarray}{c}1\leq i\leq n_{1}\\\
n_{1}+1\leq s_{0}\leq
n\end{subarray}}({x_{i}x_{s_{0}}})^{p_{is_{0}}}\prod\limits_{\begin{subarray}{c}1\leq
i_{0}\leq n_{2}\\\ n_{2}+1\leq t\leq
n\end{subarray}}({y_{i_{0}}y_{t}})^{q_{i_{0}t}}|\sum p_{is_{0}}+\sum
q_{i_{0}t}=k\right\\}$ $\displaystyle\approx$ $\displaystyle
c_{00}k^{2n-3},\emph{~{}for some constant }c_{00}.$

So we must have $d^{\prime}_{k}=\dim Span_{\mathbb{R}}N^{\prime}_{k}\approx
ck^{2n-3},$ for some positive constant $c$.

From Prop 4.2, we can use the similar argument to compute the other cases. And
for these cases we still have $d^{\prime}_{k}=\dim
Span_{\mathbb{R}}N^{\prime}_{k}\approx ck^{2n-3},$ for some positive constant
$c$.

∎

Next, we will compute the Gelfand-Kirillov dimensions of our modules in a
case-by-case way.

###  4.1. Case 1. $n_{1}+1<n_{2}<n$.

In this case we have:

(4.8) $\displaystyle E_{r,i}|_{\mathscr{B}}$
$\displaystyle=-x_{i}\partial_{x_{r}}-y_{i}\partial_{y_{r}}\qquad$ for
$\displaystyle 1\leq i<r\leq n_{1},$ (4.9) $\displaystyle
E_{s,i}|_{\mathscr{B}}$
$\displaystyle=-{x_{i}x_{s}}-{y_{i}}\partial_{y_{s}}\qquad$ for $\displaystyle
i\in\overline{1,n_{1}},\;s\in\overline{n_{1}+1,n_{2}},$ (4.10) $\displaystyle
E_{t,i}|_{\mathscr{B}}$ $\displaystyle=-{x_{i}x_{t}}+y_{i}y_{t}\qquad$ for
$\displaystyle i\in\overline{1,n_{1}},\;t\in\overline{n_{2}+1,n},$ (4.11)
$\displaystyle E_{s,j}|_{\mathscr{B}}$
$\displaystyle=x_{s}\partial_{x_{j}}-y_{j}\partial_{y_{s}}\qquad$ for
$\displaystyle n_{1}<j<s\leq n_{2},$ (4.12) $\displaystyle
E_{t,s}|_{\mathscr{B}}$ $\displaystyle=x_{t}\partial_{x_{s}}+y_{s}y_{t}\qquad$
for $\displaystyle s\in\overline{n_{1}+1,n_{2}},\;t\in\overline{n_{2}+1,n},$
(4.13) $\displaystyle E_{t,p}|_{\mathscr{B}}$
$\displaystyle=x_{t}\partial_{x_{p}}+y_{t}\partial_{y_{p}}\qquad$ for
$\displaystyle n_{2}+1\leq p<t\leq n.$

Then the above $E_{j,i}$ forms a basis for the subalgebra
$\mathfrak{sl}(n,\mathbb{F})_{-}$.

From Luo-Xu [17] we know that for $m_{1},m_{2}\in\mathbb{N}$ with
$m_{1}+m_{2}\geq n_{2}-n_{1}-1$, ${\mathscr{H}}_{\langle-m_{1},-m_{2}\rangle}$
has a highest-weight vector $v_{\lambda}=x_{n_{1}}^{m_{1}}y_{n_{2}+1}^{m_{2}}$
of weight
$\lambda=m_{1}\lambda_{n_{1}-1}-(m_{1}+1)\lambda_{n_{1}}-(m_{2}+1)\lambda_{n_{2}}+m_{2}(1-\delta_{n_{2},n-1})\lambda_{n_{2}+1}$.
Then

(4.14) $\displaystyle E_{s,j}|_{\mathscr{B}}v_{\lambda}$
$\displaystyle=0\qquad$ for $\displaystyle n_{1}<j<s\leq n_{2},$ (4.15)
$\displaystyle E_{n_{1},i}|_{\mathscr{B}}v_{\lambda}$
$\displaystyle=-x_{i}\partial_{x_{n_{1}}}(x_{n_{1}}^{m_{1}}y_{n_{2}+1}^{m_{2}}),\qquad$
for $\displaystyle 1\leq i<n_{1},$ (4.16) $\displaystyle
E_{r,i}|_{\mathscr{B}}v_{\lambda}$ $\displaystyle=0\qquad$ for $\displaystyle
1\leq i<r<n_{1},$ (4.17) $\displaystyle E_{t,p}|_{\mathscr{B}}v_{\lambda}$
$\displaystyle=0\qquad$ for $\displaystyle n_{2}+1<p<t\leq n,$ (4.18)
$\displaystyle E_{t,n_{2}+1}|_{\mathscr{B}}v_{\lambda}$
$\displaystyle=y_{t}\partial_{y_{n_{2}+1}}(x_{n_{1}}^{m_{1}}y_{n_{2}+1}^{m_{2}})\qquad$
for $\displaystyle n_{2}+1<t\leq n.$

Let $\mathfrak{g}_{1}$ be the subalgebra of $\mathfrak{sl}(n,\mathbb{F})$
spanned by the following set:

$\\{E_{s,j},E_{r,i},E_{t,p}|n_{1}<j<s\leq n_{2},1\leq i<r\leq
n_{1},n_{2}+1\leq p<t\leq n\\}.$

Let $\mathfrak{g}_{2}$ be the subalgebra of $\mathfrak{sl}(n,\mathbb{F})$
spanned by the following set:

$\\{E_{s,i},E_{t,i},E_{t,s}|i\in\overline{1,n_{1}},s\in\overline{n_{1}+1,n_{2}},t\in\overline{n_{2}+1,n}\\}.$

So we get
$\mathcal{U}(\mathfrak{g}_{-})=\mathcal{U}(\mathfrak{g}_{2})\mathcal{U}(\mathfrak{g}_{1}).$

Observe that

$\displaystyle\mathcal{U}(\mathfrak{g}_{1})v_{\lambda}$ $\displaystyle=$
$\displaystyle
Span_{\mathbb{R}}\\{\prod\limits_{i=1}^{n_{1}-1}E_{n_{1},i}^{k_{i}}\prod\limits_{t=n_{2}+2}^{n}E_{t,n_{2}+1}^{l_{t}}v_{\lambda}|k_{i},l_{t}\in\mathbb{N}\\}$
$\displaystyle=$ $\displaystyle
Span_{\mathbb{R}}\\{\prod\limits_{i=1}^{n_{1}-1}x_{i}^{k_{i}}x_{n_{1}}^{(m_{1}-\sum\limits_{i=1}^{n_{1}-1}k_{i})}\prod\limits_{t=n_{2}+2}^{n}y_{t}^{l_{t}}y_{n_{2}+1}^{(m_{2}-\sum\limits_{t=n_{2}+2}^{n}l_{t})}|k_{i},l_{t}\in\mathbb{N}\\}$
$\displaystyle=$ $\displaystyle
Span_{\mathbb{R}}\\{\prod\limits_{i=1}^{n_{1}}x_{i}^{k_{i}}\prod\limits_{t=n_{2}+1}^{n}y_{t}^{l_{t}}|\sum\limits_{i=1}^{n_{1}}k_{i}=m_{1},\sum\limits_{t=n_{2}+1}^{n}l_{t}=m_{2}\\}$

We denote this space by $M_{0}$. Then $M_{0}$ is a subspace of the space
spanned by the homogeneous polynomials of degree $m_{1}+m_{2}$ in
$\mathbb{F}[x_{1},...,x_{n_{1}},y_{n_{2}+1},...,y_{n}]$. So $M_{0}$ is finite-
dimensional.

Thus

$\mathcal{U}(\mathfrak{g_{-}})v_{\lambda}=\mathcal{U}(\mathfrak{g}_{2})M_{0}.$

Now we take any base element
$u_{0}=\prod\limits_{i=1}^{n_{1}}x_{i}^{k_{i}}\prod\limits_{t=n_{2}+1}^{n}y_{t}^{l_{t}}\in
M_{0}$. Let $k$ be any positive integer. We want to compute
$\mathrm{dim}(\mathcal{U}_{k}(\mathfrak{g}_{2})M_{0})$, and get the Gelfand-
Kirillov dimension of $\mathcal{U}(\mathfrak{g}_{2})M_{0}$.

We denote

$\displaystyle N_{0}(k)=$ $\displaystyle\left\\{\left(\prod
E_{s,i}^{p_{si}}\prod E_{t,i}^{h_{ti}}\prod
E_{t,s}^{q_{ts}}\right)u_{0}|p_{si},h_{ti},q_{ts}\in\mathbb{N},\right.$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\left.\sum\limits_{\begin{subarray}{c}1\leq
i\leq n_{1}\\\ n_{1}+1\leq s\leq
n_{2}\end{subarray}}p_{si}+\sum\limits_{\begin{subarray}{c}1\leq i\leq
n_{1}\\\ n_{2}+1\leq t\leq
n\end{subarray}}h_{ti}+\sum\limits_{\begin{subarray}{c}n_{1}+1\leq s\leq
n_{2}\\\ n_{2}+1\leq t\leq n\end{subarray}}q_{ts}=k\right\\}.$

From the definition we know

$\displaystyle\left(\prod E_{s,i}^{p_{si}}\prod E_{t,i}^{h_{ti}}\prod
E_{t,s}^{q_{ts}}\right)u_{0}$ $\displaystyle=$
$\displaystyle\left(\prod(-{x_{i}x_{s}}-{y_{i}}\partial_{y_{s}})^{p_{si}}\prod(-{x_{i}x_{t}}+y_{i}y_{t})^{h_{ti}}\prod(y_{s}y_{t})^{q_{ts}}\right)\prod\limits_{i=1}^{n_{1}}x_{i}^{k_{i}}\prod\limits_{t=n_{2}+1}^{n}y_{t}^{l_{t}}$
$\displaystyle=$
$\displaystyle\left(\prod(-{x_{i}x_{s}})^{p_{si}}\prod(-{x_{i}x_{t}}+y_{i}y_{t})^{h_{ti}}\prod(y_{s}y_{t})^{q_{ts}}\right)\prod\limits_{i=1}^{n_{1}}x_{i}^{k_{i}}\prod\limits_{t=n_{2}+1}^{n}y_{t}^{l_{t}}$
$\displaystyle+\text{lower degree part of }~{}y_{s}.$

Then we must have

$\dim Span_{\mathbb{R}}N_{0}(m)\geq d^{\prime}_{m}.$

Using the same idea with inequality 4, we can also get

$\dim Span_{\mathbb{R}}N_{0}(m)\leq d^{\prime}_{m}.$

Thus $\dim Span_{\mathbb{R}}N_{0}(m)=d^{\prime}_{m}.$

Then using proposition 4.3, we can get

$\displaystyle\dim Span_{\mathbb{R}}(\bigcup\limits_{0\leq m\leq k}N_{0}(m))$
$\displaystyle=$ $\displaystyle\sum\limits_{0\leq m\leq k}d^{\prime}_{m}$
$\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}b_{1}\sum\limits_{0\leq m\leq
k}m^{2n-4},&\hbox{if $2=n_{1}+1<n_{2}<n$ ~{}or~{} $n_{1}+1<n_{2}=n-1$}\\\
c_{1}\sum\limits_{0\leq m\leq k}m^{2n-3},&\hbox{if
$2<n_{1}+1<n_{2}<n-1$}\end{array}\right.$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}bk^{2n-3},&\hbox{if
$2=n_{1}+1<n_{2}<n$ ~{}or~{} $n_{1}+1<n_{2}=n-1$}\\\ ck^{2n-2},&\hbox{if
$2<n_{1}+1<n_{2}<n-1$.}\end{array}\right.$

We know

$\dim Span_{\mathbb{R}}(\bigcup\limits_{0\leq m\leq
k}N_{0}(m))\leq\mathrm{dim}(\mathcal{U}_{k}(\mathfrak{g}_{2})M_{0})\leq\dim{M_{0}}\dim
Span_{\mathbb{R}}(\bigcup\limits_{0\leq m\leq k}N_{0}(m)).$

Then from the definition, we know that the Gelfand-Kirillov dimension of
$\mathcal{U}(\mathfrak{g}_{-})v_{\lambda}$ is

$d=\left\\{\begin{array}[]{ll}2n-3,&\hbox{if $2=n_{1}+1<n_{2}<n$ ~{}or~{}
$n_{1}+1<n_{2}=n-1$}\\\ 2n-2,&\hbox{if
$2<n_{1}+1<n_{2}<n-1$.}\end{array}\right.$

From Luo-Xu [17] we know that for $m_{1},m_{2}\in\mathbb{N}$ with
$m_{2}-m_{1}\geq n_{2}-n_{1}-1$, ${\mathscr{H}}_{\langle m_{1},-m_{2}\rangle}$
has a highest-weight vector $x_{n_{1}+1}^{m_{1}}y_{n_{2}+1}^{m_{2}}$ of weight
$-(m_{1}+1)\lambda_{n_{1}}+m_{1}\lambda_{n_{1}+1}-(m_{2}+1)\lambda_{n_{2}}+m_{2}(1-\delta_{n_{2},n-1})\lambda_{n_{2}+1}$.
For $m_{1},m_{2}\in\mathbb{N}$ with $m_{1}-m_{2}\geq n_{2}-n_{1}-1$,
${\mathscr{H}}_{\langle-m_{1},m_{2}\rangle}$ has a highest-weight vector
$x_{n_{1}}^{m_{1}}y_{n_{2}}^{m_{2}}$ of weight
$m_{1}\lambda_{n_{1}-1}-(m_{1}+1)\lambda_{n_{1}}+m_{2}\lambda_{n_{2}-1}-(m_{2}+1)\lambda_{n_{2}}$.
The arguments for these two cases are similar to the above case, and we have
the same Gelfand-Kirillov dimension

###  4.2. Case 2. $n_{1}+1<n_{2}=n$.

In this case we have:

(4.19) $\displaystyle E_{r,i}|_{\mathscr{B}}$
$\displaystyle=-x_{i}\partial_{x_{r}}-y_{i}\partial_{y_{r}}\qquad$ for
$\displaystyle 1\leq i<r\leq n_{1},$ (4.20) $\displaystyle
E_{s,i}|_{\mathscr{B}}$
$\displaystyle=-{x_{i}x_{s}}-{y_{i}}\partial_{y_{s}}\qquad$ for $\displaystyle
i\in\overline{1,n_{1}},\;s\in\overline{n_{1}+1,n},$ (4.21) $\displaystyle
E_{s,j}|_{\mathscr{B}}$
$\displaystyle=x_{s}\partial_{x_{j}}-y_{j}\partial_{y_{s}}\qquad$ for
$\displaystyle n_{1}<j<s\leq n.$

Then the above $E_{j,i}$ forms a basis for the subalgebra
$\mathfrak{sl}(n,\mathbb{F})_{-}$.

From Luo-Xu [17] we know that for $m_{1},m_{2}\in\mathbb{N}$,
${\mathscr{H}}_{\langle m_{1},m_{2}\rangle}$ has a highest-weight vector
$x_{n_{1}+1}^{m_{1}}y_{n}^{m_{2}}$ of weight
$-(m_{1}+1)\lambda_{n_{1}}+m_{1}\lambda_{n_{1}+1}+m_{2}\lambda_{n-1}$. For
$m_{1},m_{2}\in\mathbb{N}$ with $m_{1}\leq n-n_{1}-2$ or $m_{2}-m_{1}\leq
n_{1}-n+1$, ${\mathscr{H}}_{\langle-m_{1},m_{2}\rangle}$ has a highest-weight
vector $x_{n_{1}}^{m_{1}}y_{n}^{m_{2}}$ of weight
$m_{1}\lambda_{n_{1}-1}-(m_{1}+1)\lambda_{n_{1}}+m_{2}\lambda_{n-1}$.

The arguments for these two cases are similar to case 1, and from proposition
4.1 we have the Gelfand-Kirillov dimension equal to

$n-1.$

###  4.3. Case 3. $n_{1}+1=n_{2}<n$.

In this case we have:

(4.22) $\displaystyle E_{r,i}|_{\mathscr{B}}$
$\displaystyle=-x_{i}\partial_{x_{r}}-y_{i}\partial_{y_{r}}\qquad$ for
$\displaystyle 1\leq i<r\leq n_{1},$ (4.23) $\displaystyle
E_{n_{1}+1,i}|_{\mathscr{B}}$
$\displaystyle=-{x_{i}x_{n_{1}+1}}-{y_{i}}\partial_{y_{n_{1}+1}}\qquad$ for
$\displaystyle i\in\overline{1,n_{1}},$ (4.24) $\displaystyle
E_{t,i}|_{\mathscr{B}}$ $\displaystyle=-{x_{i}x_{t}}+y_{i}y_{t}\qquad$ for
$\displaystyle i\in\overline{1,n_{1}},\;t\in\overline{n_{1}+2,n},$ (4.25)
$\displaystyle E_{t,n_{1}+1}|_{\mathscr{B}}$
$\displaystyle=x_{t}\partial_{x_{n_{1}+1}}+y_{n_{1}+1}y_{t}\qquad$ for
$\displaystyle t\in\overline{n_{1}+2,n},$ (4.26) $\displaystyle
E_{t,p}|_{\mathscr{B}}$
$\displaystyle=x_{t}\partial_{x_{p}}+y_{t}\partial_{y_{p}}\qquad$ for
$\displaystyle n_{1}+2\leq p<t\leq n.$

Then the above $E_{j,i}$ forms a basis for the subalgebra
$\mathfrak{sl}(n,\mathbb{F})_{-}$.

From Luo-Xu [17] we know that for $m_{1},m_{2}\in\mathbb{N}$,
${\mathscr{H}}_{\langle-m_{1},-m_{2}\rangle}$ has a highest-weight vector
$x_{n_{1}}^{m_{1}}y_{n_{1}+2}^{m_{2}}$ of weight
$m_{1}\lambda_{n_{1}-1}-(m_{1}+1)\lambda_{n_{1}}-(m_{2}+1)\lambda_{n_{1}+1}+m_{2}(1-\delta_{n_{1},n-2})\lambda_{n_{1}+2}$.
For $m_{1},m_{2}\in\mathbb{N}$ with $m_{2}-m_{1}\geq 0$,
${\mathscr{H}}_{\langle m_{1},-m_{2}\rangle}$ has a highest-weight vector
$x_{n_{1}+1}^{m_{1}}y_{n_{1}+2}^{m_{2}}$ of weight
$-(m_{1}+1)\lambda_{n_{1}}+(m_{1}-m_{2}-1)\lambda_{n_{1}+1}+m_{2}(1-\delta_{n_{1},n-2})\lambda_{n_{1}+2}$.
For $m_{1},m_{2}\in\mathbb{N}$ with $m_{1}-m_{2}\geq 0$,
${\mathscr{H}}_{\langle-m_{1},m_{2}\rangle}$ has a highest-weight vector
$x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}}$ of weight
$m_{1}\lambda_{n_{1}-1}+(m_{2}-m_{1}-1)\lambda_{n_{1}}-(m_{2}+1)\lambda_{n_{1}+1}$.

The arguments for these three cases are similar to case 1, and we have the
Gelfand-Kirillov dimension equal to

$d=\left\\{\begin{array}[]{ll}2n-3,&\hbox{if $2=n_{1}+1=n_{2}<n$ ~{}or~{}
$n_{1}+1=n_{2}=n-1$}\\\ 2n-2,&\hbox{if
$2<n_{1}+1=n_{2}<n-1$.}\end{array}\right.$

###  4.4. Case 4. $n_{1}+1=n_{2}=n$.

In this case we have:

(4.27) $\displaystyle E_{r,i}|_{\mathscr{B}}$
$\displaystyle=-x_{i}\partial_{x_{r}}-y_{i}\partial_{y_{r}}\qquad$ for
$\displaystyle 1\leq i<r\leq n-1,$ (4.28) $\displaystyle
E_{n,i}|_{\mathscr{B}}$
$\displaystyle=-{x_{i}x_{n}}-{y_{i}}\partial_{y_{n}}\qquad$ for $\displaystyle
i\in\overline{1,n-1}.$

Then the above $E_{j,i}$ forms a basis for the subalgebra
$\mathfrak{sl}(n,\mathbb{F})_{-}$. In this case, we denote

(4.29) $\eta=\sum_{i=1}^{n-1}y_{i}\partial_{x_{i}}+x_{n}y_{n}.$

From Luo-Xu [17] we know that for $m_{1},m_{2}\in\mathbb{N}$ with $m_{2}\leq
m_{1}$, ${\mathscr{H}}_{\langle-m_{1},m_{2}\rangle}$ has a highest-weight
vector $x_{n-1}^{m_{1}}y_{n}^{m_{2}}$ of weight
$m_{1}\lambda_{n-2}+(m_{2}-m_{1}-1)\lambda_{n-1}$. Moreover,
${\mathscr{H}}_{\langle m,0\rangle}$ has a highest-weight vector $x_{n-1}^{m}$
of weight $m\lambda_{n-2}-(m+1)\lambda_{n-1}$ for $m\in\mathbb{Z}$.

The arguments for these two cases are similar to case 2, and we have the
Gelfand-Kirillov dimension equal to

$n-1.$

For $m_{1},m_{2}\in\mathbb{N}+1$, ${\mathscr{H}}_{\langle m_{1},m_{2}\rangle}$
has a highest-weight vector
$\eta^{m_{1}+m_{2}}(x_{n-1}^{m_{2}}y_{n}^{-m_{1}})$ of weight
$m_{2}\lambda_{n-2}+(m_{1}-m_{2}-1)\lambda_{n-1}$. Then similar to the the
arguments in case 2, we have the Gelfand-Kirillov dimension equal to

$n-1.$

###  4.5. Case 5. $n_{1}=n_{2}<n-1$.

In this case we have:

(4.30) $\displaystyle E_{r,i}|_{\mathscr{B}}$
$\displaystyle=-x_{i}\partial_{x_{r}}-y_{i}\partial_{y_{r}}\qquad$ for
$\displaystyle 1\leq i<r\leq n_{1},$ (4.31) $\displaystyle
E_{t,i}|_{\mathscr{B}}$ $\displaystyle=-{x_{i}x_{t}}+y_{i}y_{t}\qquad$ for
$\displaystyle i\in\overline{1,n_{1}},\;t\in\overline{n_{1}+1,n},$ (4.32)
$\displaystyle E_{t,p}|_{\mathscr{B}}$
$\displaystyle=x_{t}\partial_{x_{p}}+y_{t}\partial_{y_{p}}\qquad$ for
$\displaystyle n_{1}+1\leq p<t\leq n.$

Then the above $E_{j,i}$ forms a basis for the subalgebra
$\mathfrak{sl}(n,\mathbb{F})_{-}$.

In this case, recall

(4.33)
$\zeta_{1}=x_{n_{1}-1}y_{n_{1}}-x_{n_{1}}y_{n_{1}-1},\;\;\zeta_{2}=x_{n_{1}+1}y_{n_{1}+2}-x_{n_{1}+2}y_{n_{1}+1}.$

From Luo-Xu [17] we know that for $m_{1},m_{2}\in\mathbb{N}$, the subspace
${\mathscr{H}}_{\langle-m_{1},-m_{2}\rangle}$ has a highest-weight vector
$x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}}$ of weight
$m_{1}(1-\delta_{1,n_{1}})\lambda_{n_{1}-1}-(m_{1}+m_{2}+2)\lambda_{n_{1}}+m_{2}\lambda_{n_{1}+1}$.
The subspace ${\mathscr{H}}_{\langle m_{1}+1,-m_{2}-m_{1}-1\rangle}$ has a
highest-weight vector $y_{n_{1}+1}^{m_{2}}\zeta_{2}^{m_{1}+1}$ of weight
$-(m_{1}+m_{2}+3)\lambda_{n_{1}}+m_{2}\lambda_{n_{1}+1}-(m_{1}+1)(1-\delta_{n_{1},n-2})\lambda_{n_{1}+2}$.
If $n_{1}\geq 2$, the subspace ${\mathscr{H}}_{\langle-
m_{1}-m_{2}-1,m_{2}+1\rangle}$ has a highest-weight vector
$x_{n_{1}}^{m_{1}}\zeta_{1}^{m_{2}+1}$ of weight
$(m_{2}+1)\lambda_{n_{1}-2}-m_{1}\lambda_{n_{1}-1}-(m_{1}+m_{2}+3)\lambda_{n_{1}}$.

Then from Prop 4.2 we have the Gelfand-Kirillov dimension equal to

$d=\left\\{\begin{array}[]{ll}n-1,&\hbox{if $1=n_{1}=n_{2}<n-1$}\\\
2n-4,&\hbox{if $n_{1}=n_{2}=2<n-1\mathrm{~{}or~{}}1<n_{1}=n_{2}=n-2$}\\\
2n-3,&\hbox{if $n_{1}=n_{2}=3<n=6$}\\\ 2n-2,&\hbox{if $3\leq n_{1}=n_{2}\leq
n-3,n\geq 7.$}\end{array}\right.$

###  4.6. Case 6. $n_{1}=n_{2}=n-1$.

In this case we have:

(4.34) $\displaystyle E_{r,i}|_{\mathscr{B}}$
$\displaystyle=-x_{i}\partial_{x_{r}}-y_{i}\partial_{y_{r}}\qquad$ for
$\displaystyle 1\leq i<r\leq n-1,$ (4.35) $\displaystyle
E_{n,i}|_{\mathscr{B}}$ $\displaystyle=-{x_{i}x_{n}}+y_{i}y_{n}\qquad$ for
$\displaystyle i\in\overline{1,n-1}.$

Then the above $E_{j,i}$ forms a basis for the subalgebra
$\mathfrak{sl}(n,\mathbb{F})_{-}$.

In this case, we denote

(4.36) $\zeta_{1}=x_{n-2}y_{n-1}-x_{n-1}y_{n-2}.$

From Luo-Xu [17] we know that for $m_{1},m_{2}\in\mathbb{N}$, the subspace
${\mathscr{H}}_{\langle-m_{1},-m_{2}\rangle}$ has a highest-weight vector
$x_{n_{1}}^{m_{1}}y_{n_{1}+1}^{m_{2}}$ of weight
$m_{1}(1-\delta_{n,2})\lambda_{n-2}-(m_{1}+m_{2}+2)\lambda_{n-1}$. If $n\geq
3$, the subspace ${\mathscr{H}}_{\langle-m_{1}-m_{2}-1,m_{2}+1\rangle}$ has a
highest-weight vector $x_{n_{1}}^{m_{1}}\zeta_{1}^{m_{2}+1}$ of weight
$(m_{2}+1)(1-\delta_{n,3})\lambda_{n-3}-m_{1}\lambda_{n-2}-(m_{1}+m_{2}+3)\lambda_{n-1}$.

Then similar to the the arguments in case 2, we have the Gelfand-Kirillov
dimension equal to

$n-1.$

###  4.7. Case 7. $n_{1}=n_{2}=n$.

In this case we have:

(4.37) $\displaystyle E_{r,i}|_{\mathscr{B}}$
$\displaystyle=-x_{i}\partial_{x_{r}}-y_{i}\partial_{y_{r}}\qquad$ for
$\displaystyle 1\leq i<r\leq n$

Then the above $E_{j,i}$ forms a basis for the subalgebra
$\mathfrak{sl}(n,\mathbb{F})_{-}$.

From Luo-Xu [17] we know that the subspace ${\mathscr{H}}_{\langle-
m_{1}-m_{2},m_{2}\rangle}$ is finite dimensional and has a highest-weight
vector $x_{n}^{m_{1}}\zeta_{1}^{m_{2}}$ of weight
$m_{2}(1-\delta_{n,2})\lambda_{n-2}+m_{1}\lambda_{n-1}$. Obviously its
Gelfand-Kirillov dimension is equal to $0$.

This completes the proof of our formula.

##  5\. Minimal Gelfand-Kirillov dimension module

Let $M$ be an irreducible highest-weight module of
$\mathfrak{sl}(n,\mathbb{F})$. We denote its Gelfand-Kirillov dimension by
$d_{M}$. From Vogan [24] and Wang [26], we know that the minimal Gelfand-
Kirillov dimension is $d_{M}=n-1$. The corresponding modules are called the
minimal GK-dimension module.

From Luo-Xu [17] and our formula 1.8, we find the following result.

######  Corollary 5.1.

Let $n(\geq 2)$ be a positive integer. Fix $n_{1},n_{2}\in\overline{1,n}$ with
$n_{1}\leq n_{2}$. The irreducible highest-weight
$\mathfrak{sl}(n,\mathbb{F})$-module
${\mathscr{H}}_{\langle\ell_{1},\ell_{2}\rangle}$ has the minimal Gelfand-
Kirillov dimension if and only if

  1. (1)

$n_{1}<n_{2}=n$.

  2. (2)

$n_{1}=n_{2}=1$.

  3. (3)

$n_{1}=n_{2}=n-1$.

We denote $\mathscr{A}=\mathbb{F}[x_{1},...,x_{n}]$. Fix $1\leq n_{1}<n$.
Changing operators $\partial_{x_{r}}\mapsto-x_{r}$ and
$x_{r}\mapsto\partial_{x_{r}}$ in the canonical oscillator representation
$E_{i,j}|\mathscr{A}=x_{i}\partial_{j}$ for $r\in\overline{1,n_{1}}$, we
obtain the following non-canonical oscillator representation of
$\mathfrak{sl}(n,\mathbb{F})$ determined by:

(5.1)
$E_{i,j}|_{\mathscr{A}}=\left\\{\begin{array}[]{ll}-x_{j}\partial_{x_{i}}-\delta_{i,j}&\mbox{if}\;i,j\in\overline{1,n_{1}};\\\
\partial_{x_{i}}\partial_{x_{j}}&\mbox{if}\;i\in\overline{1,n_{1}},\;j\in\overline{n_{1}+1,n};\\\
-x_{i}x_{j}&\mbox{if}\;i\in\overline{n_{1}+1,n},\;j\in\overline{1,n_{1}};\\\
x_{i}\partial_{x_{j}}&\mbox{if}\;i,j\in\overline{n_{1}+1,n}.\end{array}\right.$

For any $k\in\mathbb{Z}$, we denote

${\mathscr{A}}_{\langle
k\rangle}=\mbox{Span}\>\\{x^{\alpha}\mid\alpha\in\mathbb{N}\>^{n};\sum_{r=n_{1}+1}^{n}\alpha_{r}-\sum_{i=1}^{n_{1}}\alpha_{i}=k\\}.$

It was presented by Howe [11] that for $m_{1},m_{2}\in\mathbb{N}$ with
$m_{1}>0$, ${\mathscr{A}}_{\langle-m_{1}\rangle}$ is an irreducible highest-
weight $\mathfrak{sl}(n,\mathbb{F})$-submodule with highest weight
$m_{1}\lambda_{n_{1}-1}-(m_{1}+1)\lambda_{n_{1}}$ and ${\mathscr{A}}_{\langle
m_{2}\rangle}$ is an irreducible highest-weight
$\mathfrak{sl}(n,\mathbb{F})$-submodule with highest weight
$-(m_{2}+1)\lambda_{n_{1}}+m_{2}(1-\delta_{n_{1},n-1})\lambda_{n_{1}+1}$.

By a similar argument with our formula, we have the following corollary.

######  Corollary 5.2.

Fix $1\leq n_{1}<n$. For $m_{1},m_{2}\in\mathbb{N}$ with $m_{1}>0$, the
irreducible highest-weight module ${\mathscr{A}}_{\langle-m_{1}\rangle}$ and
${\mathscr{A}}_{\langle m_{2}\rangle}$ have the minimal Gelfand-Kirillov
dimension $(n-1).$

Acknowledgment. The author would like to thank Prof. Xiaoping Xu for several
helpful discussions and his financial support from NSFC Grant 11171324. The
author is also very grateful to the referee for detailed comments on the
manuscript, which helped to improve it greatly.

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# A Hybrid Approach for Depression Classification: Random Forest-ANN Ensemble
on Motor Activity Signals

Anket Patil
Department of Information Technology
K. J. Somaiya Institute of Technology
Maharashtra, India
<EMAIL_ADDRESS>
&Dhairya Shah
Department of Information Technology
K. J. Somaiya Institute of Technology
Maharashtra, India
<EMAIL_ADDRESS>
&Abhishek Shah
Department of Information Technology
K. J. Somaiya Institute of Technology
Maharashtra, India
<EMAIL_ADDRESS>
&Mokshit Gala
Department of Information Technology
K. J. Somaiya Institute of Technology
Maharashtra, India
<EMAIL_ADDRESS>

###### Abstract

Regarding the rising number of people suffering from mental health illnesses
in today’s society, the importance of mental health cannot be overstated.
Wearable sensors, which are increasingly widely available, provide a potential
way to track and comprehend mental health issues. These gadgets not only
monitor everyday activities but also continuously record vital signs like
heart rate, perhaps providing information on a person’s mental state. Recent
research has used these sensors in conjunction with machine learning methods
to identify patterns relating to different mental health conditions,
highlighting the immense potential of this data beyond simple activity
monitoring. In this research, we present a novel algorithm called the Hybrid
Random forest - Neural network that has been tailored to evaluate sensor data
from depressed patients. Our method has a noteworthy accuracy of 80% when
evaluated on a special dataset that included both unipolar and bipolar
depressive patients as well as healthy controls. The findings highlight the
algorithm’s potential for reliably determining a person’s depression condition
using sensor data, making a substantial contribution to the area of mental
health diagnostics.

_K_ eywords Depression $\cdot$ Motor Activity $\cdot$ Machine Learning $\cdot$
Random Forest $\cdot$ Neural Network $\cdot$ Hybrid Model $\cdot$ Artificial
Intelligence

##  1 Introduction

Global health is greatly impacted by the widespread mental health problems of
depression and bipolar disorder. The World Health Organization (WHO) estimates
that over 264 million people worldwide suffer from depression, making it the
main cause of disability [1]. Although bipolar disorder only affects 1% to 2%
of the world’s population, it has a significant negative impact on the
affected population’s quality of life and functional impairment [2]. These
diseases have significant financial repercussions; depression and anxiety are
estimated to cost US$ 1 trillion annually in lost productivity [3]. With the
introduction of on-body sensors, personal health monitoring has undergone a
revolutionary change. Today’s people use enormous amounts of data every day
for a variety of goals, such as improving life quality, tracking their fitness
levels, and changing unhealthy habits. This information includes continuous
records of heart rate and activity levels, which have considerable promise in
the field of psychiatry and go beyond the simple metrics of daily steps taken
or calories burned. Growing emphasis has been paid to the complex association
between activity data and a variety of mental health problems such mood
swings, stress management, and social disengagement [4–5]. Since 2010, mental
health issues—with depression leading the list of most common illnesses
[6]–[8] have been the primary reason for years lived with disability
worldwide. Depression presents a variety of difficulties in the physical,
financial, and emotional spheres, which frequently result in problems at work
and sick days [9]. The underlying etiology of these illnesses involves a
complex combination of genetic, environmental, and social variables, with
biological rhythm disruptions—often sparked by environmental
disturbances—showing up in afflicted people as altered motor activity patterns
[10]. By examining actigraph data to find patterns of motor activity
suggestive of depressive and bipolar illnesses, this study aims to advance our
understanding of these disorders. The study intends to clarify the distinctive
motor activity patterns connected to various mood disorders using cutting-edge
statistical and machine learning technologies, potentially permitting better
diagnostic and curative approaches.

The main contributions of this paper are:

  1. 1.

Our study introduces a groundbreaking Hybrid Random Forest – Neural Network
model for depression classification, promising enhanced accuracy in mental
health diagnosis.

  2. 2.

Our findings directly benefit clinical practice by enabling early depression
detection, potentially improving patient outcomes in mental healthcare.

The paper is organized as follows: Section 2 provides a comprehensive
literature review, summarizing prior work and baseline algorithms. Section 3
presents the proposed approach, which focuses on the novel Algorithm. Section
4 discusses the implementation results, demonstrating the effectiveness of the
proposed approach. Section 5 concludes the work and suggests directions for
future research.

##  2 Related Work

The existing work on applications of mental healthcare field are presented in
this section, which has been the base for this study.

###  2.1 Baseline Algorithms

The use of classification algorithms is vital in the effort to identify trends
and draw conclusions from data, especially in the area of mental health.
Understanding the underlying structure of the data and creating accurate
predictions depend greatly on the classification method’s objective, which is
to group data into predetermined groups. There are countless categorization
algorithms in the literature, each with their own advantages, presumptions,
and application to various kinds of data.

####  2.1.1 Artificial Neural Network [11]

To convert neural networks into a classification or regression model in neural
networks, utilize the activation function of the node in the output layer.

Figure 1: Artificial Neural Network Model

A regression problem’s output node has a linear activation function (or none
at all). Continuous output from $-\infty$ to $+\infty$ is the range for a
linear function. As a result, the output layer will be based on regression and
be a function of the nodes in the layer before it, just as it was for testing
the proposed system. Figure 1 depicts the ANN Model’s system flow. In the
context of the proposed study, the artificial neural network (ANN) model takes
various activity statistics ($a_{1}$, $a_{2}$, $a_{3}$) as input and
classifies the binary variable "state" which denotes sad vs. non-depressed
states, by using linked layers of neurons.

####  2.1.2 Random Forest [12]

A random forest functions as a group of binary regression trees. An
independent subset of variables is used to create these enormous numbers of
binary regression trees. The variables for division are selected at random
using Random Forest, and the decision trees are constructed using bootstrapped
samples from the dataset.

Figure 2: Random Forest Model

Figure 2 depicts the Random Forest Regressor Model’s system flow. For the
system covered by the study, the random forest model applies activity
statistics, such as mean logarithmic activity, standard deviation, minimum and
maximum logarithmic activity, and proportion of zero activity, as input
features to categorize the binary variable ’state’ which denotes depressed or
non-depressed states, through an ensemble of decision trees.

###  2.2 Mental Health Monitoring Systems

O’brien et al. [13] using a wrist-worn activity monitoring device to track
persons with late-life depression, researchers discovered that their physical
activity was lower than that of healthy controls.

In another study of Faurholt Jepsen et al. [14], they used smartphone sensors
to track patients with bipolar illness and discovered that the more severe the
depression symptoms, the fewer incoming calls were answered and the fewer
outbound calls were recorded. Based on cell tower IDs, they also discovered
that sad patients traveled less. Our work varies from the previous two in that
we automatically categorize depressed vs. non-depressed individuals using
machine learning on statistical variables generated from the sensor data.

ActiGraphs were used in a research to examine the viability of community-based
remote biometric data gathering in people with behavioral signs of Alzheimer’s
disease [15]. The value of actigraphy baseline data (mean motor activity; MMA)
was evaluated in the study. This study, although being specifically focused on
Alzheimer’s disease, highlights the potential and broader usefulness of
actigraphy in monitoring motor activity across a range of mental and
neurological diseases. The research reported in this paper tries to close this
gap by developing a unique, diagnosis-focused ML model to determine whether or
not a person is depressed.

##  3 Proposed Approach

The proposed approach for classifying depression is presented in this section
with a comprehensive description of its architectural design illustrated in
Figure 3. Following this is a step-by-step breakdown of the various phases
involved in this approach.

Figure 3: Architectural Design of the Proposed Approach

###  3.1 Data Acquisition

The acquired dataset is divided into two separate files that represent the
control group and the condition group, respectively. Each participant’s data
is encapsulated, and the actigraph data amassed over a predetermined time
period is chronicled. Each entry includes a timestamp, the measurement date,
and the activity measurement obtained from the actigraph watch. The data is
collected at one-minute intervals.

###  3.2 Data Pre-Processing

The activity column underwent a log transformation to lessen the impact of
extreme values. The data was then divided into daily segments using a date-
based filtering strategy, allowing for a controlled and methodical study of
the motor activity patterns across time.

###  3.3 Feature Extraction

Several statistics, including the mean, standard deviation, minimum, maximum,
and zero count of the log-transformed activity values, are retrieved in the
code for each distinct date in the dataset. For each date, these retrieved
characteristics are kept in a dictionary.

###  3.4 Proposed Machine Learning Novel Algorithm: Hybrid Random Forest –
Neural Network

The Hybrid Random Forest – Neural Network combines the power of two distinct
machine learning techniques, Random Forests (RF) and Artificial Neural
Networks (ANN), to create a formidable ensemble model for predictive
analytics. The algorithm is designed as follows:

Step 1: Data Splitting

Split the dataset into training and testing sets using an 80-20 split ratio.

Step 2: Data Standardization

Standardize the feature data using a StandardScaler to ensure mean=0 and
standard deviation=1.

Step 3: Random Forest Classifier

  1. 1.

Initialize a Random Forest classifier with the following parameters:

     * •

Number of decision trees (n estimators): 100

     * •

Random seed (random state): 42

  2. 2.

Train the Random Forest classifier on the standardized training data.

  3. 3.

Predict labels for the testing data using the trained Random Forest
classifier.

Step 4: Neural Network Model

  1. 1.

Define a feedforward neural network model with the following layers:

     * •

Input layer: Shape matching the number of features.

     * •

Hidden layer 1: 64 neurons with ReLU activation.

     * •

Hidden layer 2: 32 neurons with ReLU activation.

     * •

Output layer: 1 neuron with sigmoid activation for binary classification.

  2. 2.

Compile the neural network with the following settings:

     * •

Optimizer: Adam

     * •

Loss function: Binary cross-entropy

     * •

Metrics: Accuracy

  3. 3.

Train the neural network on the standardized training data for 100 epochs with
a batch size of 16.

Step 5: Combining Predictions

Combine the binary predictions from the Random Forest classifier and the
Neural Network.

Step 6: Final Random Forest Classifier

  1. 1.

Initialize a final Random Forest classifier with the following parameters:

     * •

Number of decision trees (n estimators): 100

     * •

Random seed (random state): 42

  2. 2.

Train the final Random Forest classifier on the combined predictions obtained
in Step 5 using the testing labels.

Step 7: Evaluation

Evaluate the final classifier’s performance using a classification report,
which includes precision, recall, F1-score, and accuracy.

##  4 Implementation Details

In the ensuing section, we delineate the implementation intricacies and the
resultant outcomes of the analytical procedures undertaken in this study.

###  4.1 Dataset

The dataset [16] has been divided into two separate sections, one of which
represents the control group and the other of which represents the condition
group. The actigraph data collected over a certain period is housed in a
structured manner that contains each patient’s data. The actigraph data
includes activity measures obtained from an actigraph watch, timestamps at
one-minute intervals, and the measurement date. Additionally, the dataset
provides MADRS (Montgomery-Sberg Depression Rating Scale) scores, which
include a unique identifier, the number of days of measurements, gender, age
groups, the type of affective disorder (bipolar II, unipolar depressive, or
bipolar I), the presence of melancholia, patient status (inpatient or
outpatient), education level categorized in years, marital status, employment
status, and MADRS scores for each patient.

###  4.2 Algorithmic Parameters

The implementation of Machine algorithms in this work involved careful
selection and tuning of algorithmic hyperparameters. The algorithmic
parameters used are depicted in Table 1.

Table 1: Algorithmic Parameters for Random Forest and Neural Network Model | Description | Value
---|---|---
Random Forest | Number of decision trees | 100
| Maximum depth | None
| Minimum samples to split | 2
| Minimum samples at leaf node | 1
Neural Network | Learning Rate | 0.001
| Number of Epochs | 100
| Batch Size | 16
| Standardization | Yes
| Hidden Layer 1 Units | 64
| Hidden Layer 2 Units | 32

###  4.3 Evaluation Criteria

To analyze the performance of the Machine Learning models, the following
measures are used to evaluate the performance.

####  4.3.1 Accuracy [17]

In classification tasks, accuracy is a key parameter that measures the ratio
of accurately predicted instances to all occurrences in the dataset. It offers
a clear idea of the model’s overall performance.

$\text{Accuracy}=\frac{\text{True Positives}+\text{True
Negatives}}{\text{Total Instances}}$ (1)

####  4.3.2 Precision [18]

Precision is a measure of the accuracy of the positive predictions made by a
classification model. It calculates the ratio of accurately predicted positive
observations to all anticipated positive observations.

$\text{Precision}=\frac{\text{True Positives}}{\text{True
Positives}+\text{False Positives}}$ (2)

####  4.3.3 Recall [19]

Recall, often referred to as Sensitivity or True Positive Rate, measures a
model’s capacity to find all pertinent instances within the data.

$\text{Recall}=\frac{\text{True Positives}}{\text{True Positives}+\text{False
Negatives}}$ (3)

####  4.3.4 F1 Score [20]

The F1 Score provides a balance between precision and recall by calculating a
harmonic mean of accuracy and recall.

$\text{F1
Score}=\frac{2\times\text{Precision}\times\text{Recall}}{\text{Precision}+\text{Recall}}$
(4)

Figure 4: Performance Comparison of Machine Learning Algorithms for
Classification

###  4.4 Results and Discussions

The findings derived from the conducted research are presented in this
section.

Results from binary classification models, including the Hybrid Model, Neural
Model, and Random Forest Model, are displayed in Chart 4. In comparison to its
individual components, the hybrid model excels across all evaluation criteria.
With an accuracy rating of 0.81, the Hybrid Model outperforms both the Neural
Model (0.71) and the Random Forest Model (0.65) in terms of accuracy. This
demonstrates the noteworthy accuracy of the Hybrid Model’s predictions for
about 81% of the dataset. Additionally, the Hybrid Model performs admirably in
terms of weighted precision, recall, and F1-score with scores of 0.8, 0.81,
and 0.8, respectively, demonstrating its efficiency in categorizing cases
across both classes while taking class imbalance into account. The Hybrid
Model retains high performance in the macro-average (M) category, treating
each class equally, with precision, recall, and F1-score all at 0.78,
illustrating its constant prediction abilities without bias toward any
particular class. The Neural Model and Random Forest Model, on the other hand,
score significantly worse in both weighted and macro-average categories. The
Hybrid Model is the best option for this binary classification problem since
it can use the advantages of both models to produce a more reliable and
accurate classification strategy.

##  5 Conclusion

The study presents a novel method for categorizing depression that is based on
a Hybrid Random Forest-Neural Network model. Across all evaluation criteria,
our hybrid model performs better than both the standalone Neural Network and
Random Forest models. The Hybrid Model’s astounding 81% accuracy rate
demonstrates its outstanding capacity for making accurate forecasts.
Additionally, it showed excellent precision, recall, and F1-score, handling
class imbalance and ensuring consistent predictive performance. This study
highlights the ability of hybrid models to handle challenging tasks like
depression categorization. This hybrid model provides a viable path for
enhancing mental health diagnostics by integrating the benefits of both Random
Forest and Neural Network techniques. It may help medical personnel identify
patients earlier and intervene more effectively, thereby improving the quality
of life for those who are depressed. Future research should concentrate on
determining how well the model generalizes across other datasets and
populations. Key paths for additional research and practical application
include investigating cutting-edge deep learning methods, merging real-time
monitoring with wearable technologies, and performing clinical trials for
real-world validation.

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