# On Differentiating Symmetric Functions

Shaul Zemel

## Introduction

Consider a function $\phi$ of $N$ variables $x_{i}$, $1\leq i\leq N$, say from
a field $\mathbb{F}$, and assume that $\phi$ is symmetric. It is well-known
that the elementary symmetric functions $e_{r}:=\sum_{|I|=r}\prod_{i\in
I}x_{i}$ (where the sum is over the subsets $I$ of size $r$ of the indices
between 1 and $N$) generate the ring of symmetric polynomials in the
$x_{i}$’s, and therefore $\phi$ can be presented as a function $\psi$ of the
expressions $e_{r}$, $1\leq r\leq N$, viewed as coordinates on the $N$th
symmetric power $\operatorname{Sym}^{N}\mathbb{F}$, which is the quotient of
$\mathbb{F}^{N}$ under the action of the symmetric group $S_{N}$ by
interchanging the coordinates.

Assume now that $\mathbb{F}$ is the real field $\mathbb{R}$ and $\phi$ is
continuously differentiable, or that $\mathbb{F}$ is $\mathbb{C}$ and $\phi$
is holomorphic. Then so will the function $\psi$ of the symmetric expressions
$e_{r}$ be, and one may ask how to write the derivatives of the latter
function with respect to these coordinates, in terms of the derivatives of
$\phi$. Of course, the derivatives of $\phi$ are obtained from those of $\psi$
by the chain rule, but we are interested in inverting this relation, which
will, in particular, give a symmetric function of the $x_{i}$’s.

Such questions do show up in some applications, for example in the theory of
Riemann surfaces, Abel–Jacobi maps, and Thomae formulae. In proving Thomae’s
derivative formula (see, e.g., [EKZ] for a rather recent case), one has to
differentiate the Abel–Jacobi map as a map on positive divisors of fixed
degree, which is the same as a symmetric power of the Riemann surface (hence
locally a symmetric power of $\mathbb{C}$). The Jacobian of this map involves,
at the double points from [EKZ], derivatives of second order (this is already
hinted in the proof of the Riemann–Roch theorem in [FK], but without any
details). Now, the Abel–Jacobi map is the sum of the values of a single-
variable function on the coordinates (such functions are known as _trace
functions_ in [B] and others), which makes a few statements simpler. We will,
however, analyze the actions on general functions in this paper.

Since there will be many derivatives in this paper, we shall denote, when $f$
is a function of many variables including $v$, the derivative $\frac{\partial
f}{\partial v}$ by simply $f_{v}$. When the variables of $f$ are numbered, say
$v_{l}$, $1\leq l\leq M$, we shall shorthand $f_{v_{l}}$ to simply $f_{l}$.
Similarly, we shall write $\partial_{v}$ for the operator
$\frac{\partial}{\partial v}$, and with numbered variables we shall use the
shortened notation $\partial_{l}$ for $\partial_{v_{l}}$ in case no confusion
can arise. We emphasize that in every such derivative the other variables
remain constant when we differentiate, and when we express the same function
in terms of different coordinates, we will change the name of the function,
like the distinction we already made between $\phi$ and $\psi$ above already
exemplifies.

Now, as when $N=1$ there is nothing to consider, we examine the case of $N=2$,
with the variables $x=x_{1}$ and $y=x_{2}$ and the symmetric functions
$s=e_{1}=x+y$ and $p=e_{2}=xy$. Writing the symmetric function
$\phi=\phi(x,y)$ as $\psi=\psi(s,p)$, we obtain

$\phi_{x}=\psi_{s}+y\psi_{p}\qquad\mathrm{and}\qquad\phi_{y}=\psi_{s}+x\psi_{p},$

from which we get, when $y\neq x$, the expressions

$\psi_{s}=\frac{y\phi_{y}-x\phi_{x}}{y-x}\qquad\mathrm{and}\qquad\psi_{p}=\frac{\phi_{x}-\phi_{y}}{y-x}.$

Our first observation is that the differential operators that we have to apply
on $\psi$ for obtaining these derivatives are, in general, _not_ in the Weyl
algebra $\mathbb{C}[x,y]\langle\partial_{x},\partial_{y}\rangle^{S_{2}}$ from
[B] and others, as it involves division by $y-x$. The reason for this is that
while for general functions $f$ there is no relation between the derivatives
$f_{x}$ and $f_{y}$, for a symmetric function $f$ the difference $f_{x}-f_{y}$
is anti-symmetric, thus vanishes when $x=y$, which means that if $f$ is a
polynomial then this difference is divisible by $y-x$. This means that the
algebra acting on symmetric functions should be larger than the
$S_{N}$-invariants of the ordinary Weyl algebra, and contain also elements
like $\frac{\partial_{x}-\partial_{y}}{y-x}$ (which also equals
$\frac{\partial_{y}-\partial_{x}}{x-y}$ and is thus symmetric) in two
variables. The full Weyl algebra, for any (finite) number of variables, is
determined in Corollary 11 below.

Another observation is exemplified by considering the points with $y=x$, for
which we take the limit $y\to x$ of our expressions for $\psi_{p}$ and
$\psi_{s}$. Since $\phi$ is symmetric, we know that at diagonal points we have
$\phi_{x}(x,x)=\phi_{y}(x,x)$, and we find that

$\lim_{y\to x}\psi_{p}=\lim_{y\to
x}\frac{\phi_{x}(x,y)-\phi_{x}(x,x)-\phi_{y}(x,y)+\phi_{y}(x,x)}{y-x}=\phi_{xy}-\phi_{yy}$

(the latter summand also equals $\phi_{xx}$ by symmetry, so that $\psi_{p}$ at
a diagonal point can be described as the difference between the mixed second
derivative of $\phi$ and the pure one, with _the_ pure one being independent
of the choice of variable). Since $\psi_{s}$ equals $\phi_{y}-x\psi_{p}$ and
$\phi_{x}-y\psi_{p}$ (or the more symmetric expression
$\frac{\phi_{x}+\phi_{y}}{2}-\frac{x+y}{2}\psi_{p}$), its value at the
diagonal point $(x,x)$, where $\phi_{x}=\phi_{y}$, is
$\phi_{y}-x(\phi_{xy}-\phi_{yy})=\phi_{x}-x(\phi_{xy}-\phi_{xx})$. Note that
these expressions contain the value of the variable itself ($x$ or $y$), and
mix derivatives of $\phi$ of different orders.

One might then pose the question, whether we can replace the $e_{r}$’s by a
set of coordinates in which the derivatives of $\psi$ will be more natural
expressions in the derivatives of $\phi$, in particular involving derivatives
of $\phi$ in as uniform a degree as possible when the values of the variables
coincide, and without having to multiply by these values themselves. We would
like the $r$th coordinates to be homogenous of degree $r$ in the $x_{i}$’s (so
that in particular the first one will be $e_{1}$, up to constant multiples),
thus expressible in terms of partitions of $r$.

The paper [B] considers some differential operators and coordinates, and the
ones chosen there are the Newton polynomials of the elementary symmetric
functions, with which $e_{r}$ is, up to a multiplicative constant perhaps, the
$r$th power sum function $p_{r}:=\sum_{i=1}^{N}x_{i}^{r}$. However, consider
the case $N=2$ again, choose again $s$ as the first coordinate but the power
sum $q=x^{2}+y^{2}$ as the second one, and write $\phi(x,y)$ as $\eta(s,q)$.
Then differentiation yields

$\phi_{x}=\eta_{s}+2x\eta_{q}\qquad\mathrm{and}\qquad\phi_{y}=\eta_{s}+2y\eta_{q},$

from which for $x\neq y$ one obtains

$\eta_{s}=\frac{y\phi_{x}-x\phi_{y}}{y-x}\qquad\mathrm{and}\qquad\eta_{q}=\frac{\phi_{y}-\phi_{x}}{2(y-x)},$

the limiting process at the diagonal is established by noticing that
$\eta_{q}$ is $-\frac{1}{2}\psi_{p}$ from before, and $\eta_{s}$ is
$\frac{\phi_{x}+\phi_{y}}{2}+(x+y)\eta_{q}$, with again a limit involving
derivatives of both orders 1 and 2 at diagonal points.

The right coordinates for $N=2$ are given in Lemma 2.3 of [EKZ]. With $s=x+y$
as before but $u=\frac{(y-x)^{2}}{2}$, and with $\phi(x,y)=\varphi(s,u)$, we
get

$\phi_{x}=\varphi_{s}-(y-x)\varphi_{u}\qquad\mathrm{and}\qquad\phi_{y}=\varphi_{s}+(y-x)\varphi_{u},$

relations whose inversion yields, when $x\neq y$, the expressions

$\varphi_{s}=\frac{\phi_{x}+\phi_{y}}{2}\qquad\mathrm{and}\qquad\varphi_{u}=\frac{\phi_{y}-\phi_{x}}{2(y-x)}.$

With these coordinates we now see that $\varphi_{s}$ no longer involves the
denominator $y-x$, so that it keeps its form also when $y=x$ (and then it
equals $\phi_{x}$ and $\phi_{y}$ as well), and involves only first derivatives
of $\phi$ also at such points (the form of the other derivative,
$\varphi_{u}$, as a difference of the pure and mixed second derivatives of
$\phi$ at the diagonal is preserved, up to a constant multiplier). These are
the type of coordinates that we look for in general.

With the coordinates $s$ and $u$ from above, the first derivative that we
obtained was a multiple of $\partial_{x}+\partial_{y}$, so that for general
$N$ we would like our first derivative to be $\sum_{i=1}^{N}\partial_{i}$. The
other derivative (in all three systems of coordinates) was a multiple of
$\frac{\partial_{x}-\partial_{y}}{y-x}$, which is symmetric with respect to
interchanging $x$ and $y$, and for obtaining a symmetric operator in $N$
variables we take the sum
$\sum_{i<j}\frac{\partial_{i}-\partial_{j}}{x_{j}-x_{i}}$ (when some of the
values of the $x_{i}$’s coincide, the corresponding expressions can be
replaced by the associated combination
$\partial_{i}\partial_{j}-\partial_{i}^{2}=\partial_{i}\partial_{j}-\partial_{j}^{2}$
of second derivatives). In three variables $x$, $y$ and $z$, the operator
$(y-z)\partial_{x}+(z-x)\partial_{y}+(x-y)\partial_{z}$ is anti-symmetric with
respect to interchanging any pair from the these variables. Therefore a
symmetric operator is obtained by dividing it by the product
$(x-y)(x-z)(y-z)$, which yields
$\frac{\partial_{x}}{(y-x)(z-x)}+\frac{\partial_{y}}{(x-y)(z-y)}+\frac{\partial_{z}}{(x-z)(y-z)}$.
Summing over all triples with $i<j<k$ of that operator obtained from $x_{i}$,
$x_{j}$, and $x_{k}$ then yields a symmetric operator in $N$ variables, which
only involves first partial derivatives and division by two differences of
variables. When $x_{i}=x_{j}=x_{k}$ the corresponding operator produces a
combination of third derivatives, which can be written as
$\partial_{i}\partial_{j}\partial_{k}-\frac{3}{2}\partial_{i}^{2}\partial_{j}+\frac{1}{2}\partial_{i}^{3}$
(or any expression obtained by the action of the symmetric group $S_{3}$ on
some of the summands).

We can now state our result (Theorem 8 below).

######  Theorem.

For every $1\leq r\leq N$ there is a symmetric polynomial $u_{r}$ in the
variables $x_{i}$, $1\leq i\leq N$, which is homogenous degree $r$, such that
$\\{u_{r}\\}_{r=1}^{N}$ is the unique system of coordinates with the following
property: If we consider a symmetric function $\phi$ of the variables $x_{i}$,
$1\leq i\leq N$, then the derivative $\varphi_{r}=\varphi_{u_{r}}$ equals
$\sum_{\\{i_{h}\\}_{h=1}^{r}\in\mathbb{N}_{N}^{r}\mathrm{\
distinct}}\sum_{h=1}^{r}\phi_{i_{h}}\big{/}\prod_{g\neq
h}(x_{i_{g}}-x_{i_{h}})$.

Note that for $N=2$ the variables from our theorem are $u_{1}=\frac{s}{2}$ and
$u_{2}=-\frac{u}{4}$, for $\varphi_{u_{1}}$ to be $\phi_{x}+\phi_{y}$ (without
the denominator 2) and $\varphi_{u_{2}}$ to be
$\frac{\phi_{x}-\phi_{y}}{y-x}+\frac{\phi_{y}-\phi_{x}}{x-y}$, namely
$2\frac{\phi_{x}-\phi_{y}}{y-x}$, with the correct sign and the extra multiple
(coming from the indices in the operator being distinct, and not necessarily
in increasing order). This will be the normalization with which our argument
will work in the optimal way (see Remark 29 below). Note that our operators do
not involve multiplications by non-constant functions the values of the
variables, as desired. We shall also deduce the formulae for the derivatives
at points where the values of some (or all) variables coincide, involving
higher order derivatives of $\phi$—see Theorems 21, Proposition 27, and
Corollary 28 below.

The coordinates that our main theorem produces have many interesting
properties: They can be expressed in terms of the ordinary Bell polynomials
(Remark 9—see also Remark 22 for a similar statement, with the exponential
Bell polynomials, for the forms of our derivatives along the total diagonal);
For $r\geq 2$ they vanish to a high order along the total diagonal (Theorem
13); They are characterized up to scalar multiples by not having non-trivial
numerators in the derivatives with respect to them (Theorem 12); And as
functions of the number $N$ of variables, they are characterized by all of
their mixed derivatives decaying faster than usual (Theorem 18). In fact, we
conjecture that the more mixed the derivative is, the faster it decays with
$N$—see Conjecture 19 below for the precise statement. Note that the Newton
polynomials and power sum functions do show up as the limits of these
symmetric functions as $N\to\infty$ (see Proposition 17), but not for the
finite values of $N$.

In fact, for evaluating the derivatives at specific points in the symmetric
space, we merge our coordinates with the simpler approach of differentiating
variables according to their values, and this produces the simplest form of
the derivatives around a given point—see Theorem 30 below.

The paper is divided into three sections. Section 1 defines the differential
operators and symmetric functions that we need at points where all the
variables take distinct values, and Section 2 investigates some of their
properties. Then Section 3 considers the limits of the resulting derivatives
at points where variables coincide (for obtaining higher order derivatives of
$\phi$), and presents the form of our differential operators at any point.

I am grateful to D. Barlet and Y. Kopeliovich for stimulating discussions on
this topic, as well as to A. Zemel for suggesting the use of L’Hôpital’s Rule
instead of the Taylor expansion for examining cases in an earlier draft, which
helped obtain the expressions with higher derivatives in general.

##  1 Normalized Symmetric Functions and Derivatives

Let $\mathbb{N}_{N}$ denote the set of integers $i$ with $1\leq i\leq N$. We
recall again that for $1\leq h\leq N$, the expression
$e_{h}:=\sum_{J\subseteq\mathbb{N}_{N},\ |J|=h}\prod_{i\in J}x_{i}$ is the
$h$th elementary symmetric function in the variables $x_{i}$,
$i\in\mathbb{N}_{N}$ (with $e_{0}$ being the constant function 1), which is
homogenous of degree $h$. Thus, if $\lambda$ is a partition of an integer $r$
as the sum of the positive integers $h_{q}$, $1\leq q\leq l$, with $h_{q}\leq
N$ for every $q$, then
$e_{\lambda}:=\prod_{q=1}^{l}e_{h_{q}}=\prod_{h=1}^{N}e_{h}^{m_{h}}$ (where
$m_{h}$ is the multiplicity of $h$ in $\lambda$) is a symmetric polynomial in
the $x_{i}$’s which is homogenous of degree $r$. The number $l$ of (non-zero)
integers $h_{q}$ that participate in $\lambda$, which also equals
$\sum_{h}m_{h}$, is the _length_ $l(\lambda)$ of the partition, and if we
denote the _conjugate_ , or _transpose_ , partition of $\lambda$, obtained by
reflecting its Ferrers diagram along the diagonal, by $\lambda^{t}$, then our
assumption that the numbers $h_{q}$ participating in $\lambda$ do not exceed
$N$ can be written as the inequality $l(\lambda^{t})\leq N$. The fact that
$\lambda$ is a partition of $r$ will henceforth be written as $\lambda\vdash
r$. We then recall the following result from the theory of symmetric
functions.

######  Theorem 1.

The products $\\{e_{\lambda}\\}_{\lambda\vdash r,\ l(\lambda^{t})\leq N}$ form
a basis, over $\mathbb{Z}$, for the symmetric homogenous polynomials of degree
$r$ in the $x_{i}$’s having integral coefficients. The ring
$\mathbb{Z}[x_{1},\ldots,x_{N}]^{S_{N}}$ of symmetric polynomials in these
variables is generated, as a $\mathbb{Z}$-algebra, by $\\{e_{h}\\}_{h=1}^{N}$.

Of course, the more natural version of Theorem 1 is for symmetric functions of
infinitely many variables, where the parameter $N$ is omitted, but for our
purposes we shall need the version in finitely many variables. Our version is,
of course, a consequence of the version with infinitely many variables, when
one substitutes all the variables $x_{i}$ with $i>N$ to be 0. Then $e_{h}$
with $h\leq N$ gives the finite versions, while for $h>N$ we get $e_{h}=0$
(since there is no subset of size $h$ in a set of size $N$), and with it
$e_{\lambda}=0$ when $l(\lambda^{t})>N$ because of the existence of a
vanishing multiplier. We shall thus henceforth adopt the convention for the
elementary symmetric function $e_{l}$ in any number $d$ of variables to give
the expected expression when $l\leq d$, and to vanish in case $l>d$. For more
on partitions and symmetric functions, including a proof of Theorem 1, see,
e.g., [M].

We shall soon differentiate the elementary symmetric functions, as well as
their products from Theorem 1, in various ways. For doing so we shall need the
following presentation of these functions.

######  Lemma 2.

Let $I\subseteq\mathbb{N}_{N}$ be a subset of size $d$, and take some $1\leq
h\leq N$. Then we have the expansion

$e_{h}(x_{1},\ldots,x_{N})=\textstyle{\sum_{l=0}^{h}e_{l}\big{(}\\{x_{i}\\}_{i\in
I}\big{)}\cdot e_{h-l}\big{(}\\{x_{i}\\}_{i\not\in I}\big{)}}.$

For simplifying the notation, given a set of indices $J$ and an integer $r$,
we can write $e_{r}(x_{J})$ for $e_{r}\big{(}\\{x_{i}\\}_{i\in J}\big{)}$.
Then Lemma 2 can be written as the equality
$e_{h}(x_{\mathbb{N}_{N}})=\sum_{l=0}^{h}e_{l}(x_{I})e_{h-l}(x_{\mathbb{N}_{N}\setminus
I})=\sum_{l=0}^{h}e_{l}(x_{I})e_{h-l}(x_{I^{c}})$ (where $I^{c}$ is the
complement of $I$ in the natural ambient set $\mathbb{N}_{N}$). Moreover, if
$h>d$ then the summand $e_{l}(x_{I})$ vanishes for every $d<l\leq h$, so that
the sum is effectively taken over $0\leq l\leq\min\\{d,h\\}$. However, we
shall not be needing the latter fact, and allow for vanishing terms to show up
in our expressions.

###### Proof.

Every subset $J\subseteq\mathbb{N}_{N}$ of size $h$ is the disjoint union of
$J\cap I$, of some size $0\leq l\leq h$, and $J\cap I^{c}$, of size $h-l$.
Conversely, fix $0\leq l\leq h$, and then from every subset $K\subseteq I$ of
size $l$, and every subset $H\leq I^{c}$ of size $h-l$, their union $H\cup K$
is a subset of size $h$ of $\mathbb{N}_{N}$. Moreover, the product
$\prod_{i\in K}x_{i}$ shows up in $e_{l}(x_{I})$, the product $\prod_{i\in
H}x_{i}$ appears in $e_{h-l}(x_{I^{c}})$, and multiplying them gives the
summand $\prod_{i\in H\cup K}x_{i}$ from $e_{h}(x_{\mathbb{N}_{N}})$. Now,
summing over $H$ and $K$ independently produces all the summands in
$e_{h}(x_{\mathbb{N}_{N}})$ that are associated with subsets
$J\subseteq\mathbb{N}_{N}$ of size $h$ such that $J\cap I$ has size $l$, and
by summing over $0\leq l\leq h$ we obtain all the sets $J$ of size $h$, and
therefore the desired expression $e_{h}(x_{\mathbb{N}_{N}})$. This proves the
lemma. ∎

Note that the summand with $l=0$ in Lemma 2 is $e_{h}(x_{I^{c}})$ since
$e_{0}(x_{I})=1$ (this matches the summands with empty $H$ in the proof).

When we let a differential operator $D$ act on
$e_{h}=e_{h}(x_{\mathbb{N}_{N}})$, if $I$ is the set of variables whose
respective derivatives appear in $D$, we can use the decomposition from Lemma
2 for evaluating $De_{h}$, by letting $D$ only act on the summands
$e_{l}(x_{I})$. Thus, for example, if $D=\partial_{i}$ then we can take
$I=\\{i\\}$, write $e_{h}$ with $h\geq 1$ as
$x_{i}e_{h-1}(x_{\\{i\\}^{c}})+e_{h}(x_{\\{i\\}^{c}})$, and easily obtain that
$\partial_{i}e_{h}=e_{h-1}(x_{\\{i\\}^{c}})$. As another example, let now $D$
be the operator $\frac{\partial_{i}-\partial_{j}}{x_{j}-x_{i}}$, like the one
appearing in the Introduction. We then take $I$ to be $\\{i,j\\}$, and Lemma 2
expresses $e_{h}$ for $h\geq 2$ as
$x_{i}x_{j}e_{h-2}(x_{I^{c}})+(x_{i}+x_{j})e_{h-1}(x_{I^{c}})+e_{h}(x_{I^{c}})$,
since for our $I$ we have $e_{1}(x_{I})=x_{i}+x_{j}$ and
$e_{2}(x_{I})=x_{i}x_{j}$. Now, our operator $D$ annihilates the last summand,
and as its numerator $\partial_{i}-\partial_{j}$ takes $x_{i}+x_{j}$ to 0 and
$x_{i}x_{j}$ to $x_{j}-x_{i}$, we deduce that $De_{h}=e_{h-2}(x_{I^{c}})$.
While the algebra involved is more tedious, one can verify that in the three
variables $x$, $y$, and $z$, the operator
$\frac{\partial_{x}}{(y-x)(z-x)}+\frac{\partial_{y}}{(x-y)(z-y)}+\frac{\partial_{z}}{(x-z)(y-z)}$
takes the elementary symmetric functions $e_{0}(x,y,z)=1$,
$e_{1}(x,y,z)=x+y+z$, and $e_{2}(x,y,z)=xy+xz+yz$ to 0 and sends the product
$e_{3}(x,y,z)=xyz$ to 1, which means that if $I\subseteq\mathbb{N}_{N}$ is a
set of size 3 and $D$ is the operator constructed from $I$ in that manner,
then $De_{h}=e_{h-3}(x_{I^{c}})$ wherever $h\geq 3$. We remark that we at this
point we view these formulae as operations on polynomials, which produce
polynomials and therefore we need not worry about the vanishing of
denominators. The reader may, however, assume that we only consider points
where all the variables take distinct values, and extend the results by
continuity to the remaining points. This point becomes relevant in Corollary
10, and is expanded on significantly in Section 3 below.

In any case, we now wish to generalize these expressions.

######  Lemma 3.

Take a non-empty subset $I\subseteq\mathbb{N}_{N}$, and let $d>0$ be the size
of $I$. Then the operator $D_{I}:=\sum_{i\in I}\partial_{i}\big{/}\prod_{i\neq
j\in I}(x_{j}-x_{i})$ takes $e_{l}(x_{I})$ to 1 when $l=d$, and to 0
otherwise.

Using the Kronecker $\delta$-symbol $\delta_{l,d}$, which equals 1 when $l=d$
and 0 otherwise, we can write the result of Lemma 3 as
$D_{I}e_{l}(x_{I})=\delta_{l,d}$ for every $l$ and $I$ of size $d$. The
analogue of $D_{I}$ in case $I$ is the empty set should not be the
corresponding empty sum (which thus vanishes), but rather the identity
operator, for it to take the function $e_{0}(x_{\emptyset})=1$ to 1, and not
0.

###### Proof.

If $d=1$ then $I=\\{i\\}$, $D_{I}=\partial_{i}$ since the product is empty,
and as we have $e_{0}(x_{I})=1$ and $e_{1}(x_{I})=x_{i}$, the result is clear.
We therefore assume that $d\geq 2$, where permuting the variables can be non-
trivial. We then observe that the operator $D_{I}$ is invariant under applying
a permutation to the elements of $I$, and is thus symmetric. Therefore if we
let it act on a symmetric polynomial in the variables $\\{x_{i}\\}_{i\in I}$,
and then multiply by the anti-symmetric expression $\prod_{i<j\in
I}(x_{i}-x_{j})$ (according to the order of elements in $I$ as a subset of
$\mathbb{N}_{N}$, say), then we obtain an anti-symmetric polynomial in these
variables.

Now, an anti-symmetric polynomial in $\\{x_{i}\\}_{i\in I}$ vanishes wherever
$x_{i}=x_{j}$, so that its zero set contains that of the irreducible
polynomial $x_{i}-x_{j}$, making it divisible by $x_{i}-x_{j}$ in the UFD
$\mathbb{C}[\\{x_{i}\\}_{i\in I}]$. Since this is valid for every pair
$\\{i,j\\}\subseteq I$ with $i<j$, such a polynomial will be divisible by
$\prod_{i<j\in I}(x_{i}-x_{j})$. In particular, if it does not vanish then its
degree is at least the degree $\binom{d}{2}$ of the latter multiplier.

Consider now the action of $\prod_{i<j\in I}(x_{i}-x_{j})\cdot D_{I}$ on the
function $e_{l}(x_{I})$. Since the product contains every denominator from the
definition of $D_{I}$, and every such denominator is homogenous of degree
$d-1$, we deduce that this differential operator is the sum of $d$ operators,
each of which is a partial derivative with respect to one variable multiplied
by a homogenous polynomial of degree $\binom{d-1}{2}$. Letting such an
operator act on $e_{l}$, of degree $l$, decreases the degree by 1 (via the
derivatives), and then multiplies by a polynomial of degree $\binom{d-1}{2}$.
The result is therefore an anti-symmetric polynomial of degree
$\binom{d-1}{2}+l-1=\binom{d}{2}+l-d$. As this degree is smaller than
$\binom{d}{2}$ when $l<d$, we deduce the asserted vanishing.

Finally, when $l=d$ the degree is $\binom{d}{2}$, meaning that $\prod_{i<j\in
I}(x_{i}-x_{j})\cdot D_{I}e_{d}$ is a constant multiple of $\prod_{i<j\in
I}(x_{i}-x_{j})$. For evaluating the constant, which is the value of
$D_{I}e_{d}$, assume that $I=\mathbb{N}_{d}\subseteq\mathbb{N}_{n}$ for easing
the notation, and then in the anti-symmetric product $\prod_{i<j\in
I}(x_{i}-x_{j})$, the monomial $\prod_{j=1}^{d}x_{j}^{d-j}$ shows up with a
coefficient of 1. The constant value of $D_{I}e_{d}$ is therefore the
coefficient with which this monomial shows up in $\prod_{i<j\in
I}(x_{i}-x_{j})\cdot D_{I}e_{d}$. But $x_{d}$ does not divide our monomial,
and as $e_{d}=\prod_{j=1}^{d}x_{j}$, letting each $\partial_{i}$ with $i<d$
act on it and multiplying by some polynomial only gives combinations of
monomials that are divisible by $x_{d}$. Thus our monomial only shows up in
the summand associated with $i=d$ in $D_{I}$, where the derivative is
$\prod_{j=1}^{d-1}x_{j}$ and the multiplying polynomial is
$\prod_{i<j<d}(x_{i}-x_{j})$ after canceling. As our monomial appears with the
coefficient 1 in the resulting product, the last value is also the desired
one. This proves the lemma. ∎

We can now obtain the following consequence.

######  Corollary 4.

The operator $D_{I}$ from Lemma 3 sends the symmetric function
$e_{h}=e_{h}(x_{\mathbb{N}_{N}})$ with $h\geq d$ to $e_{h-d}(x_{I^{c}})$.

###### Proof.

Write $e_{h}$ as in Lemma 2, and as $D_{I}$ contains no derivatives with
respect to variables $x_{i}$ with $i\not\in I$, it only acts on the
multipliers $e_{l}(x_{I})$ from that lemma. The result thus follows directly
from Lemma 3. This proves the corollary. ∎

Corollary 4 is also valid for $h<d$, if we define $e_{r}$ with negative $r$ to
be 0. This is also visible in the examples calculated explicitly above.
Moreover, the convention that $D_{I}=\operatorname{Id}$ (and not 0) when $I$
is empty is indeed the natural extension for this case, as this is the
operator taking $e_{h}$ to $e_{h-0}(x_{I^{c}})$ for empty $I$. However, we
shall not need this operator in what follows.

We are interested in symmetric differential operators in our variables
$x_{i}$, $i\in\mathbb{N}_{N}$. The operator $D_{I}$ associated via Lemma 3 to
a subset $I$ was seen to be symmetric, but only in the variables
$\\{x_{i}\\}_{i\in I}$, and a permutation on
$\\{x_{i}\\}_{i\in\mathbb{N}_{N}}$ that does not preserve $I$ clearly no
longer leaves the operator $D_{I}$ invariant, but takes it to the operator of
another set of size $d$. We shall use $|I|$ for the size of $I$ throughout
this paper, and we consider the following symmetric operators and their
action.

######  Proposition 5.

For every $d>0$, the operator $D_{d}:=d!\sum_{I\subseteq\mathbb{N}_{N},\
|I|=d}D_{I}$ is a symmetric differential operator, which takes $e_{h}$ with
$h\geq d$ to $\frac{(N-h+d)!}{(N-h)!}e_{h-d}$ and $e_{h}$ with $h<d$ to 0.

###### Proof.

The symmetry of $D_{d}$ is obvious from the fact that permutations on
$\\{x_{i}\\}_{i\in\mathbb{N}_{N}}$ permute the subsets of size $d$ in
$\mathbb{N}_{N}$. Now, we saw in Corollary 4 that each $D_{I}$ takes $e_{h}$
to $e_{h-d}(x_{I^{c}})$, which equals $\sum_{H\leq I^{c},\ |H|=h-d}\prod_{i\in
H}x_{i}$ in case $h\geq d$ and 0 otherwise. Thus $D_{d}e_{h}$ indeed vanishes
when $h<d$, and if $h\geq d$ then in the image of $e_{h}$ under $D_{d}$, the
expression $\prod_{i\in H}x_{i}$ for some subset $H\subseteq\mathbb{N}_{N}$ of
size $h-d$ appears with one multiple of $d!$ for every subset
$I\subseteq\mathbb{N}_{N}$ of size $d$ such that $H\subseteq I^{c}$. But as
the latter condition is equivalent to $I\subseteq H^{c}$, it means that the
number of such sets $I$ is obtained by choosing $d$ elements from the set
$H^{c}$, of size $N-h+d$. As $d!$ times the number $\binom{N-h+d}{d}$ of such
choices equals $\frac{(N-h+d)!}{(N-h)!}$ (independently of $H$, of course),
and $\sum_{H\subseteq\mathbb{N}_{N},\ |H|=h-d}\prod_{i\in H}x_{i}$ is
$e_{h-d}$ by definition, this proves the proposition. ∎

The reason for the factor $d!$ in the definition of $D_{d}$ in Proposition 5,
which can be considered as summing over $I$ as ordered $d$-tuples of distinct
indices rather than subsets, will soon be seen to be more convenient. The
convention of $e_{r}=0$ for $r<0$ makes the statement of Proposition 5 uniform
for all $h$.

The coefficients from Proposition 5 depend on the number $N$ of variables. We
thus renormalize each of our functions $e_{h}$ with
$h\in\mathbb{N}\cup\\{0\\}$ by setting
$\tilde{e}_{h}:=e_{h}\big{/}h!\binom{N}{h}=\frac{(N-h)!e_{h}}{N!}$, which have
the property that when all the $x_{i}$’s take the same value $a$, this
function attains $\frac{a^{h}}{h!}$, a value that no longer depends on $N$.
Note that this indeed holds also for $h=0$, and we wrote the union
$\mathbb{N}\cup\\{0\\}$ since it will be more convenient to later in this
paper to only include positive integers in $\mathbb{N}$. We immediately obtain
the following consequence.

######  Corollary 6.

The operator $D_{d}$ from Proposition 5 sends $\tilde{e}_{h}$ with $0\leq
h\leq N$ to $\tilde{e}_{h-d}$, which vanishes when $h<d$.

###### Proof.

By Proposition 5, $D_{d}$ takes $\tilde{e}_{h}=\frac{(N-h)!e_{h}}{N!}$ to
$\frac{(N-h)!}{N!}\cdot\frac{(N-h+d)!}{(N-h)!}e_{h-d}$ (and to 0 when $h<d$),
which indeed equals $\frac{(N-h+d)!e_{h-d}}{N!}=\tilde{e}_{h-d}$ as desired.
This proves the corollary. ∎

Note that the condition $h\leq N$ is important in Corollary 6, since for an
index $N<h\leq N+d$ there is no function $e_{h}$ that is mapped to $e_{h-d}$
via $D_{d}$. In Proposition 5 this issue did not show up, since for such $h$
the numerator $(N-h+d)!$ is finite (a factorial of a non-negative integer)
while the denominator $(N-h)!$ is infinite as the factorial of a negative
integer, meaning that $e_{h-d}$ is multiplied by 0 to give the correct value
of $D_{d}e_{h}=D_{d}0$ for $e_{h}$ with $h>N$. Moreover, without the
coefficient $d!$ in the definition of $D_{d}$, Corollary 6 would have
contained it, which is the reason for our normalization.

We can also normalize the products showing up in Theorem 1, and set the
expression $\tilde{e}_{\lambda}$ associated with a partition $\lambda\vdash r$
with $l(\lambda^{t})\leq N$ to be the product
$\prod_{q=1}^{l}\tilde{e}_{h_{q}}=\prod_{h=1}^{N}\tilde{e}_{h}^{m_{h}}$. It is
a multiple of the original $\tilde{e}_{\lambda}$, which equals
$a^{r}\big{/}\prod_{q=1}^{l}h_{q}!=a^{r}\big{/}\prod_{h=1}^{N}h!^{m_{h}}$ when
$x_{i}=a$ for all $i\in\mathbb{N}_{N}$, and the spanning property and ring
generation property from that theorem hold for these expressions as well, when
we work over $\mathbb{Q}$. For describing the action of $D_{d}$ on these
expressions we consider a partition $\lambda$ of $r$ and some $d\leq h\leq N$
that shows up in $\lambda$ (i.e., with $m_{h}\geq 1$), and we define
$\lambda-d\varepsilon_{h}$ to be the partition of $r-d$ that is obtained by
replacing one instance of $h$ in $\lambda$ by $h-d$. We thus obtain the
following result.

######  Corollary 7.

For every $1\leq d\leq N$ and every partition $\lambda\vdash r$ with
$l(\lambda^{t})\leq N$, the operator $D_{d}$ takes $\tilde{e}_{\lambda}$ to
$\sum_{h=d}^{N}m_{h}\tilde{e}_{\lambda-d\varepsilon_{h}}$.

In fact, since the maximal number $h$ for which $m_{h}>0$ is $l(\lambda^{t})$,
we can take the sum to be only up to this value. In particular when
$l(\lambda^{t})<d$, i.e., when all the multipliers constructing
$\tilde{e}_{\lambda}$ are $\tilde{e}_{h}$ with $h<d$, Corollary 7 yields
$D_{d}\tilde{e}_{\lambda}=0$.

###### Proof.

Since $D_{d}$ is a differential operator of order 1, its action on the product
$\tilde{e}_{\lambda}$ is expressed via Leibniz’ rule, which yields
$\sum_{h=1}^{N}m_{h}\tilde{e}_{h}^{m_{h}-1}\prod_{g\neq
h}\tilde{e}_{g}^{m_{g}}\cdot D_{d}\tilde{e}_{h}$. The summands with $h<d$
vanish, and if $h\geq d$ then the last multiplier is $\tilde{e}_{h}$ by
Corollary 6. As for every $d\leq h\leq N$ this yields the multiplier $m_{h}$
times the product that defines $\tilde{e}_{\lambda-d\varepsilon_{h}}$, this
proves the corollary. ∎

Note that Corollary 7 is no longer valid for $d=0$ with the identity operator,
since the latter operator does not satisfy the Leibniz’ rule (and indeed, the
asserted value of $D_{d}\tilde{e}_{\lambda}$ would become
$\tilde{e}_{\lambda}$ times $\sum_{h=1}^{N}m_{h}=l(\lambda)$, rather than 1).

We remark that the if $I$ is a set of size $d$ as above then $\prod_{i\in
I}\partial_{i}$ also takes each $e_{h}$ to $e_{h-d}(x_{I^{c}})$, as in
Corollary 4, and can thus be used for establishing counterparts for
Proposition 5 and Corollary 6. However, the main difference shows up in
Corollary 7, where the proof uses the fact that $D_{I}$ or $D_{d}$ are
operators of degree 1 (at a generic point—the results in Section 3 consider
general points), which thus satisfies Leibniz’ Rule, a property that is not
shared by the product in question when $d\geq 2$. For example, when
$I=\\{i,j\\}$ of size 2, $D_{I}$ takes the product $e_{h}e_{g}$ to
$e_{h-2}(x_{I^{c}})e_{g}+e_{h}e_{g-2}(x_{I^{c}})$, while the action of
$\partial_{i}\partial_{j}$ will produce these two terms plus
$e_{h-1}(x_{\\{i\\}^{c}})e_{g-1}(x_{\\{j\\}^{c}})+e_{h-1}(x_{\\{j\\}^{c}})e_{g-1}(x_{\\{i\\}^{c}})$.
This is the reason why certain proofs along the total diagonal does not
simplify in an obvious manner—see Remark 31 below.

We can now define our symmetric coordinates, and prove that they are the only
ones with the property that we seek.

######  Theorem 8.

For every partition $\lambda\vdash r$ of some $1\leq r\leq N$, let $m_{h}$ be
the multiplicity with which the integer $1\leq h\leq r$ appears in $\lambda$,
and let $l(\lambda)$ be its length $\sum_{h=1}^{r}m_{h}$. We then set

$u_{r}:=\sum_{\lambda\vdash
r}(-1)^{l(\lambda)-1}\frac{\big{(}l(\lambda)-1\big{)}!}{\prod_{h=1}^{r}m_{h}!}\tilde{e}_{\lambda},$

where $\tilde{e}_{\lambda}$ is the normalized product appearing in Corollary
7. Then $u_{r}$ is the only symmetric polynomial in $x_{i}$,
$i\in\mathbb{N}_{N}$ that is homogenous of degree $r$ and satisfies the
property that $D_{d}u_{r}$ is the Kronecker $\delta$-symbol $\delta_{d,r}$.

######  Remark 9.

Recall that the partial ordinary Bell polynomial $\hat{B}_{r,t}$ of some
variables $z_{h}$, $1\leq h\leq r-t+1$ can be expressed as
$\sum_{\lambda\vdash r,\
l(\lambda)=t}t!\prod_{h=1}^{r-t+1}z_{h}^{m_{h}}/m_{h}!$, and it is homogenous
of degree $t$ when all the variables have degree 1 (when $z_{h}$ has degree
$h$, the homogeneity degree is $r$), so that inverting the signs of the
variables multiplies $\hat{B}_{r,t}$ by $(-1)^{t}$. As $\tilde{e}_{\lambda}$
is the product $\prod_{h=1}^{r}\tilde{e}_{h}^{m_{h}}$, we deduce that the
coordinate $u_{r}$ from Theorem 8 can be written in short as
$-\sum_{t=1}^{r}\hat{B}_{r,t}(-\tilde{e}_{1},\ldots,-\tilde{e}_{r-t+1})/t$. A
very similar observation will be useful for Proposition 17 below.

###### Proof of Theorem 8.

We begin with the uniqueness of $u_{r}$, recalling that the homogeneity of
degree $r$ implies, via Theorem 1, that $u_{r}$ has a unique presentation as
$\sum_{\lambda\vdash r}c_{\lambda}\tilde{e}_{\lambda}$ (as $r\leq N$, the
condition $l(\lambda^{t})\leq N$ is redundant for partitions $\lambda\vdash
r$). Moreover, since when $d>r$ we have $m_{h}=0$ for every $h\geq d$ in every
partition $\lambda\vdash r$, the condition $D_{d}u_{r}=0$ holds for such $d$
regardless of the choice of the homogenous expression $u_{r}$ of degree $r$.

Now, when $d=r$ the partition $\lambda_{r}$ of $r$ of length 1 contains
$\tilde{e}_{r}$ (in fact, $\tilde{e}_{\lambda_{r}}=\tilde{e}_{r}$), and all
the other partitions are only based on integers $1\leq h<r$. It follows from
Corollary 7 (and even Corollary 6) that if $u_{r}$ is presented as above then
$D_{r}u_{r}$ just gives the coefficient $c_{\lambda_{r}}$. Thus the equality
$D_{r}u_{r}=1$ implies (and is equivalent to) $c_{\lambda_{r}}=1$.

We now prove the uniqueness of the value of $c_{\lambda}$ by decreasing
induction on the maximal number showing up in $\lambda$ (namely on
$l(\lambda^{t})$ as before). We have established the basis of the induction
with $l(\lambda^{t})=r$ above, so assume that we have determined the value of
$c_{\lambda}$ for every $\lambda\vdash r$ with $l(\lambda^{t})>d$ for some
$d<r$ by the equalities $D_{p}u_{r}=\delta_{p,r}$ for every $p>d$, and we wish
to find the values of $c_{\lambda}$ when $l(\lambda^{t})=d$. Note that $D_{p}$
with $p>d$ annihilates $\tilde{e}_{\lambda}$ as above, so that this
coefficient did not show up in the previous calculations, and consider the
equality $D_{d}u_{r}=0$. The expressions $\tilde{e}_{\lambda}$ with
$l(\lambda^{t})<d$ are annihilated by $D_{d}$ once again, and as
$\lambda-d\varepsilon_{h}$ is a partition of $r-d$ for every $d\leq h\leq r$,
we deduce that the image under $D_{d}$ of the determined combination
$\sum_{\lambda\vdash r,\ l(\lambda^{t})>d}c_{\lambda}\tilde{e}_{\lambda}$ is
some combination $\sum_{\rho\vdash r-d}b_{\rho}\tilde{e}_{\rho}$. Now, for
every $\lambda\vdash r$ with $l(\lambda^{t})=d$ Corollary 7 (with the precise
bound $l(\lambda^{t})$ on $h$) yields
$D_{d}\tilde{e}_{\lambda}=m_{d}\tilde{e}_{\lambda-d\varepsilon_{d}}$, and the
map $\lambda\mapsto\lambda-d\varepsilon_{d}$ is an injective map from the
partitions $\lambda\vdash r$ with $l(\lambda^{t})=d$ into the partitions
$\rho\vdash r-d$ (it is a bijection when we impose the condition
$l(\rho^{t})\leq d$ on the latter partitions). Therefore the only choice for
obtaining $D_{d}u_{r}=0$ is by choosing the coefficient $c_{\lambda}$ for such
$\lambda$ to be $-b_{\lambda-d\varepsilon_{d}}/m_{d}$, where
$b_{\lambda-d\varepsilon_{d}}$ is given in terms of the values that we had
already determined by the induction hypothesis. This means that if an
expression homogenous symmetric $u_{r}$ that satisfies
$D_{d}u_{r}=\delta_{d,r}$ for every $d$ exists, then it is unique.

For proving the existence, we simply verify that our explicit formula for
$u_{r}$ has the desired property. We have already seen that $D_{d}u_{r}=0$
when $d>r$ in any case, and since the coefficient associated with
$\lambda_{r}$, having length 1 and only one positive multiplicity $m_{r}=1$,
equals 1, we get $D_{r}u_{r}=1$ as before. So assume that $d<r$, and we have
to verify that $D_{d}u_{r}=0$. Using Corollary 7 (and linearity) we write
$D_{d}u_{r}$ as

$D_{d}\sum_{\lambda\vdash
r}(-1)^{l(\lambda)-1}\frac{\big{(}l(\lambda)-1\big{)}!}{\prod_{h=1}^{r}m_{h}!}\tilde{e}_{\lambda}=\sum_{\lambda\vdash
r}(-1)^{l(\lambda)-1}\frac{\big{(}l(\lambda)-1\big{)}!}{\prod_{h=1}^{r}m_{h}!}\cdot\sum_{h=d}^{N}m_{h}\tilde{e}_{\lambda-d\varepsilon_{h}},$

and for every $\rho\vdash r-d$, in which every $h$ appears with multiplicity
$\mu_{h}$, we have to prove that the total contribution to the multiplier of
$\tilde{e}_{\rho}$ vanishes.

Now, the terms yielding a multiple of $\tilde{e}_{\rho}$ are the ones coming
from $\lambda\vdash r$ and $h\geq d$ such that
$\lambda-d\varepsilon_{h}=\rho$. When $h>d$ this means that one instance of
$h$ was replaced by $h-d$ when we moved from $\lambda$ to $\rho$, meaning that
we have $m_{h}=\mu_{h}+1$, $m_{h-d}=\mu_{h-d}-1$, and $m_{g}=\mu_{g}$ for
every other value of $g$. Thus
$l(\lambda)=\sum_{h=1}^{r}m_{h}=\sum_{h=1}^{r}\mu_{h}=l(\rho)$, and the
additional multiplier is

$\frac{m_{h}}{\prod_{g=1}^{r}m_{g}!}=\frac{m_{h-d}+1}{(m_{h}-1)!(m_{h-d}+1)!\prod_{g\neq
h,h-d}m_{g}!}=\frac{\mu_{h-d}}{\prod_{g=1}^{r-d}\mu_{g}!}$

(we can let $g$ go only up to $r-d$ since $\rho\vdash r-d$). This means that
the contribution of the term arising from such $h$ to the coefficient
multiplying expression $\tilde{e}_{\rho}$ is
$(-1)^{l(\rho)-1}\big{(}l(\rho)-1\big{)}!\big{/}\prod_{g=1}^{r-d}\mu_{g}!$
times $\mu_{h-d}$.

However, we also have the combination with $h=d$ and $\lambda$ with
$\lambda-d\varepsilon_{d}=\rho$, for which going from $\lambda$ to $\rho$
simply means discarding one instance of $d$. Therefore $m_{d}=\mu_{d}+1$ and
$m_{g}=\mu_{g}$ for $g\neq d$, so that $l(\lambda)=l(\rho)+1$ and
$m_{d}\big{/}\prod_{h=1}^{r}m_{h}!$ reduces to
$1\big{/}\prod_{g=1}^{r-d}\mu_{g}!$, and as
$(-1)^{l(\lambda)-1}\big{(}l(\lambda)-1\big{)}!=-(-1)^{l(\rho)-1}\big{(}l(\rho)\big{)}!$,
we find that this is the coefficient
$(-1)^{l(\rho)-1}\big{(}l(\rho)-1\big{)}!\big{/}\prod_{g=1}^{r-d}\mu_{g}!$
times $\tilde{e}_{\rho}$ multiplied by $-l(\rho)$. Gathering all the multiples
of $(-1)^{l(\rho)-1}\big{(}l(\rho)-1\big{)}!\big{/}\prod_{g=1}^{r-d}\mu_{g}!$
times $\tilde{e}_{\rho}$ yields the total value

$\textstyle{\sum_{h=d+1}^{r}\mu_{h-d}-l(\rho)=\sum_{g=1}^{d-r}\mu_{g}-l(\rho)=0},$

as desired. Therefore our $u_{r}$ satisfies the desired equalities, and it was
seen above to be unique once it exists. This proves the theorem. ∎

Note that the proof of the uniqueness in Theorem 8 was not enough to establish
the existence of $u_{r}$, since at the induction step associated with $d$ we
could only make sure to cancel the multiple of $\tilde{e}_{\rho}$ for
$\rho\vdash r-d$ with $l(\rho^{t})\leq d$ by choosing the (unique) appropriate
values for $c_{\lambda}$ for $\lambda\vdash r$ with $l(\lambda^{t})=d$. That
argument alone did not suffice to verify that if $l(\rho^{t})>d$ then the
coefficient of $\tilde{e}_{\rho}$ in that $D_{d}$-image, which cannot be
affected by the $c_{\lambda}$’s in question, indeed vanishes.

The fact that the expression $u_{r}$ from Theorem 8 is $\tilde{e}_{r}$ plus a
polynomial expression in $\\{\tilde{e}_{h}\\}_{h=1}^{r-1}$, with rational
coefficients, implies, via Theorem 1, that $\\{u_{r}\\}_{r=1}^{N}$ also
generate the ring of symmetric functions in $\\{x_{i}\\}_{i\in\mathbb{N}_{N}}$
over $\mathbb{Q}$. We can therefore use them as coordinates for presenting any
symmetric function $\phi$ of $\\{x_{i}\\}_{i\in\mathbb{N}_{N}}$, and with this
presentation the respective derivatives are only based on derivatives of
$\phi$ and denominators, with no multipliers.

######  Corollary 10.

Let $\phi$ be a continuously differentiable symmetric function of $N$ real
variables. If we write $\phi(x_{1},\ldots,x_{N})$ as
$\varphi(u_{1},\ldots,u_{N})$ for some function $\varphi$, with the $u_{r}$’s
being those from Theorem 8, then we have the equality
$\varphi_{u_{d}}=D_{d}\phi$ for every $1\leq d\leq N$, holding at every point
in which the values of the $x_{i}$’s are all distinct.

###### Proof.

For every $1\leq d\leq N$, the operator $D_{d}$ is a differential operator of
degree 1, which can be applied to any differentiable function at any point
$(x_{1},\ldots,x_{N})$ for which $x_{i}\neq x_{j}$ wherever $i\neq j$ (this
condition is required for the denominators in the definition of $D_{d}$ not to
vanish). Therefore when we apply it on
$\phi=\phi(x_{1},\ldots,x_{N})=\varphi(u_{1},\ldots,u_{N})$, we get, by the
chain rule, the expression $\sum_{r=1}^{N}D_{d}u_{r}\cdot\varphi_{u_{r}}$. As
Theorem 8 evaluates $D_{d}u_{r}$ as $\delta_{d,r}$, the latter sum reduces to
$\varphi_{u_{d}}$, as desired. This proves the corollary. ∎

The result of Corollary 10 is equivalently valid for holomorphic symmetric
functions in $N$ complex variables. So will all the results about derivatives
below.

Another consequence that we obtain from Theorem 8 is the following one.

######  Corollary 11.

The Weyl algebra of symmetric functions in $N$ variables is generated by
$\\{u_{r}\\}_{r=1}^{N}$ (all commuting with one another) and
$\\{D_{d}\\}_{d=1}^{N}$ (commuting with one another as well), with the
commutation relation $[D_{d},u_{r}]=\delta_{d,r}$.

###### Proof.

We already saw that Theorem 1 allows us to present the ring of symmetric
polynomials in $N$ variables as the polynomial algebra generated by
$\\{u_{r}\\}_{r=1}^{N}$. Since Theorem 8 shows that $\\{D_{d}\\}_{d=1}^{N}$ is
the dual system of derivations on that algebra, joining them to this algebra
produces the corresponding Weyl algebra (and in particular the commutation of
the $D_{d}$’s with one another). This proves the corollary. ∎

##  2 Properties of the Coordinates $\\{u_{r}\\}_{r=1}^{N}$

In this section we establish some of the properties of coordinates
$\\{u_{r}\\}_{r=1}^{N}$ from Theorem 8, and see alternative ways to
characterize them.

First we state that the homogeneity and the formulae from Corollary 10
essentially characterize the coordinates $\\{u_{r}\\}_{r=1}^{N}$ from Theorem
8.

######  Theorem 12.

Take, for every $1\leq r\leq N$, a homogenous symmetric function $v_{r}$ of
degree $r$ in $\\{x_{i}\\}_{i\in\mathbb{N}_{N}}$. Assume that
$\\{v_{r}\\}_{r=1}^{N}$ are coordinates for symmetric functions, such that if
we write a symmetric function $\phi=\phi(x_{1},\ldots,x_{N})$ as
$\eta(v_{1},\ldots,v_{N})$, then for every $r$ the derivative $\eta_{v_{r}}$
involves derivatives with respect to the $x_{i}$’s and denominators, with no
numerators, and with the terms from $D_{r}\phi$ showing up multiplied by 1.
Then $v_{r}=u_{r}$ for every $1\leq r\leq N$.

While one can prove Theorem 12 now already, we postpone the proof until we
have some formulae from the next section, which will make the proof
significantly simpler. The reader can verify that this theorem is not used
anywhere in the following section, so that there is no circular reasoning in
proving it in this way.

Recall that a diagonal point in a 2-dimensional space is a point where the
values of the two variables are the same. Here we have, in general, more than
two variables, so the only symmetric set that we can define in this manner is
the _total diagonal_ , consisting of the points in $\mathbb{R}^{N}$ at which
all the variables $\\{x_{i}\\}_{i\in\mathbb{N}_{N}}$ take the same value. An
argument generalizing the proof of Theorem 8 yields the following additional
property of our coordinates.

######  Theorem 13.

Take any simple derivative $\prod_{q=1}^{d}\partial_{i_{q}}$ of order $d$
(where some of the indices $i_{q}$, $1\leq q\leq d$ are allowed to coincide).
Then when we apply it to $u_{r}$ with $r\neq d$, the resulting function
vanishes along the total diagonal.

For proving Theorem 13, we first make the following evaluation, of the
derivative from that theorem on an expression $\tilde{e}_{\lambda}$ as in
Corollary 7.

######  Lemma 14.

Let $\lambda$ be a partition of some integer $r$, in which each number $1\leq
h\leq r$ appears with multiplicity $m_{h}$, and consider the operator
$\prod_{q=1}^{d}\partial_{i_{q}}$ from Theorem 13. If
$\iota:\mathbb{N}_{d}\to\mathbb{N}_{N}$ is the map taking $1\leq q\leq d$ to
the index $i_{q}$, then denote by $X_{\iota}$ the set of all partitions of
$\mathbb{N}_{d}$ into a disjoint union of sets
$\\{J_{\nu}\\}_{\nu=1}^{\gamma}$, such that the indices $i_{q}$ for $q$ in the
same set $J_{\nu}$ are all distinct. Then the value of
$\partial^{d}\tilde{e}_{\lambda}\big{/}\prod_{q=1}^{d}\partial_{i_{q}}$ is the
sum over all elements of $X_{\iota}$ of the expression

$\sum_{\begin{subarray}{c}h_{\nu}\geq 1,\ 1\leq\nu\leq\gamma\\\
\sum_{\nu=1}^{\gamma}h_{\nu}\leq
r\end{subarray}}\prod_{\nu=1}^{\gamma}\big{(}m_{h_{\nu}}-\textstyle{\sum_{\sigma=1}^{\nu-1}\delta_{h_{\nu},h_{\sigma}}}\big{)}\displaystyle\tilde{e}_{\lambda-\sum_{\nu=1}^{\gamma}h_{\nu}\varepsilon_{h_{\nu}}}\prod_{\nu=1}^{\gamma}\frac{\tilde{e}_{h_{\nu}-\kappa_{\nu}}(\\{i_{q}|q\in
J_{\nu}\\}^{c})}{N!/(N-\kappa_{\nu})!},$

where $\kappa_{\nu}$ is the size of the set $J_{\nu}$ in the given element of
$X_{\iota}$ for any $1\leq\nu\leq\gamma$, which hence satisfy
$\sum_{\nu=1}^{\gamma}\kappa_{\nu}=d$.

###### Proof.

We argue by induction on $d$. For $d=1$ the derivative $\partial_{i}$ is
simply the operator $D_{I}$ for the singleton $I=\\{i\\}$, where Corollary 6
gives $\partial_{i}e_{h}=e_{h-1}(\\{i\\}^{c})$. Recalling the normalization
from Corollary 6, of $\tilde{e}_{h}$ as $\frac{(N-h)!e_{h}}{N!}$ and thus of
$\tilde{e}_{h-1}(\\{i\\}^{c})$ as $\frac{(N-h)!e_{h-1}(\\{i\\}^{c})}{(N-1)!}$,
we deduce that
$\partial_{i}\tilde{e}_{h}=\frac{\tilde{e}_{h-1}(\\{i\\}^{c})}{N}$. It
follows, via Leibniz’ Rule, that for our $\lambda\vdash r$ we have
$\partial_{i}\tilde{e}_{\lambda}=\sum_{h=1}^{r}m_{h}\tilde{e}_{\lambda-h\varepsilon_{h}}\frac{\tilde{e}_{h-1}(\\{i\\}^{c})}{N}$
in the notation from the proof of Theorem 8. This proves the case with $d=1$.

For higher values of $d$, note that our expression for
$\partial\tilde{e}_{\lambda}/\partial_{i}$ contains $\tilde{e}_{\mu}$’s for
partitions $\mu$, but also $\tilde{e}_{\nu}(\\{i\\}^{c})$ for partitions $\nu$
(in this case of length 1). When we apply another operator $\partial_{j}$, its
action on $\tilde{e}_{\lambda-h\varepsilon_{h}}$ from the $g$th summand will
produce
$\sum_{g=1}^{r-h}(m_{g}-\delta_{g,h})\tilde{e}_{\lambda-h\varepsilon_{h}-g\varepsilon_{g}}\frac{\tilde{e}_{g-1}(\\{j\\}^{c})}{N}$
(the expression with the $\delta$-symbol showing up because the multiplicity
of $h$ in $\lambda-h\varepsilon_{h}$ is the same $m_{g}$ when $g\neq h$, but
equals $m_{h}-1$ if $g=h$ since we took out one instance of $h$), by what we
just saw. However, it can also act on the multiplier
$\frac{\tilde{e}_{h-1}(\\{i\\}^{c})}{N}$, which gives
$\frac{\tilde{e}_{h-2}(\\{i,j\\}^{c})}{N(N-1)}$ if $j\neq i$ by the same
argument, and just 0 in case $j=i$ since that expression does not depend on
$x_{i}$ anymore. Combining these two parts we obtain the result for $d=2$,
since the set of $X_{\iota}$ contains the partition of $\mathbb{N}_{2}$ into
two singletons, as well as the partition into one set if $i\neq j$.

Continuing by induction, assume now that
$\partial^{d}\tilde{e}_{\lambda}\big{/}\prod_{q=1}^{d}\partial_{i_{q}}$ is
expressed via the induction hypothesis, apply another operator
$\partial_{i_{d+1}}$, to obtain the image of $\tilde{e}_{\lambda}$ under a
general such operator of degree $d+1$. Let $\tilde{\iota}$ be the extension of
$\iota$ to $\mathbb{N}_{d+1}$ taking $d+1$ to $i_{d+1}\in\mathbb{N}_{N}$.
Take, inside the summand associated with the element of $X_{\iota}$ containing
the sets $\\{J_{\nu}\\}_{\nu=1}^{\gamma}$ (of respective cardinalities
$\\{\kappa_{\nu}\\}_{\nu=1}^{\gamma}$), the term with the values
$\\{h_{\nu}\\}_{\nu=1}^{\gamma}$, and let $\partial_{i_{d+1}}$ acts on it.

Now, the partition $\lambda-\sum_{\nu=1}^{\gamma}h_{\nu}\varepsilon_{h_{\nu}}$
is of $r-\sum_{\nu=1}^{\gamma}h_{\nu}$, and it contains each possible value of
a new index $1\leq h_{\gamma+1}\leq r-\sum_{\nu=1}^{\gamma}h_{\nu}$ with the
multiplicity
$m_{h_{\gamma+1}}-\sum_{\sigma=1}^{\gamma}\delta_{h_{\gamma+1},h_{\sigma}}\big{)}$.
Therefore $\partial_{i_{d+1}}$ takes
$\tilde{e}_{\lambda-\sum_{\nu=1}^{\gamma}h_{\nu}\varepsilon_{h_{\nu}}}$, by
what we saw above, to the sum over such $h_{\gamma+1}$ of this respective
multiplicity times
$\tilde{e}_{\lambda-\sum_{\nu=1}^{\gamma+1}h_{\nu}\varepsilon_{h_{\nu}}}\frac{\tilde{e}_{h_{\gamma+1}-1}(\\{i_{d+1}\\}^{c})}{N}$
(when this partition is empty, the vanishing of this derivative will be
expressed in terms of the vanishing multiplicities). Multiplying by the
remaining expressions and summing over all the values of
$\\{h_{\nu}\\}_{\nu=1}^{\gamma}$ thus produces the sum corresponding to the
element of $X_{\tilde{\iota}}$ obtained from our element of $X_{\iota}$ by
adding the singleton $J_{\gamma+1}=\\{d+1\\}$.

But the operator $\partial_{i_{d+1}}$ can also act on the multiplier
$\tilde{e}_{h_{\nu}-\kappa_{\nu}}(\\{i_{q}|q\in J_{\nu}\\}^{c})$ for some
$1\leq\nu\leq\gamma$. If $i_{d+1}$ coincides with $i_{q}$ for some $q\in
J_{\nu}$, then this function is independent of $x_{i_{d+1}}$ and the
derivative is 0. When this is not the case, the derivative gives
$\tilde{e}_{h_{\nu}-\kappa_{\nu}-1}(\\{i_{q}|q\in J_{\nu}\\}^{c}\setminus
i_{d+1})/(N-\kappa_{\nu})$ as we saw above, and when we gather all of the
terms obtained in this way, over every choice of
$\\{h_{\nu}\\}_{\nu=1}^{\gamma}$, this yields the sum associated with the
element of $X_{\tilde{\iota}}$ in which $d+1$ is added to the set $J_{\nu}$,
thus keeping the condition that all the indices $i_{q}$ for $q$ in this set
are distinct (and now it is not a singleton), and the other sets remain the
same. When we combine everything together, and sum over all elements of
$X_{\iota}$, we indeed obtain all the sums arising from all the elements of
$X_{\tilde{\iota}}$. This proves the lemma. ∎

###### Proof of Theorem 13.

The result is obvious for $d>r$ by the homogeneity degree of $u_{r}$, so
assume $d<r$. We express $u_{r}$ as in Theorem 8, and then for every
$\lambda\vdash r$ the image of $\tilde{e}_{\lambda}$ showing up in $u_{r}$
under our operator is given by Lemma 14. At a point on the total diagonal, say
where $x_{i}=a$ for every $i\in\mathbb{N}_{N}$, each term $\tilde{e}_{g}(I)$,
of every set $I$, attains the value $\frac{a^{g}}{g!}$, so that the value of
the product of
$\tilde{e}_{\lambda-\sum_{\nu=1}^{\gamma}h_{\nu}\varepsilon_{h_{\nu}}}$ and of
$\tilde{e}_{h_{\nu}-\kappa_{\nu}}(\\{i_{q}|q\in J_{\nu}\\}^{c})$ over
$1\leq\nu\leq\gamma$ from that lemma attains at our point the same value as
that of
$\tilde{e}_{\lambda-\sum_{\nu=1}^{\gamma}\kappa_{\nu}\varepsilon_{h_{\nu}}}$.
Note that since $\lambda\vdash r$ and $\sum_{\nu=1}^{\gamma}\kappa_{\nu}=d$,
this expression corresponds to a partition of $r-d>0$. We thus fix one element
of $X_{\iota}$ from Lemma 14 (and with it the $J_{\nu}$’s and their
cardinalities, the $\kappa_{\nu}$’s), multiply the expression arising from
$\lambda$ by the coefficient
$(-1)^{l(\lambda)-1}\big{(}l(\lambda)-1\big{)}!\big{/}\prod_{h=1}^{r}m_{h}!$
from Theorem 8, sum over $\lambda$, fix a partition $\rho\vdash r-d$, with
multiplicities $\\{\mu_{h}\\}_{h=1}^{r-d}$, and consider the resulting
coefficient of the value of
$\tilde{e}_{\rho}\big{/}\prod_{\nu=1}^{\gamma}\frac{N!}{(N-\kappa_{\nu})!}$ at
our diagonal point.

Now, for fixed $\lambda$ and $\\{h_{\nu}\\}_{\nu=1}^{\gamma}$ such that
$\lambda-\sum_{\nu=1}^{\gamma}\kappa_{\nu}\varepsilon_{h_{\nu}}=\rho$, denote
by $\tilde{\mu}_{h}$ the multiplicity of $h$ in the partition
$\lambda-\sum_{\nu=1}^{\gamma}h_{\nu}\varepsilon_{h_{\nu}}$. Then the product
of the expressions $m_{h_{\nu}}-\sum_{\mu=1}^{\nu-1}\delta_{h_{\nu},h_{\mu}}$
cancels with the denominator $\prod_{h=1}^{r}m_{h}!$ to give a denominator of
$\prod_{h}\tilde{\mu}_{h}!$. We also note that for each $\nu$ such that
$\kappa_{\nu}=h_{\nu}$ we obtain a difference of 1 between $l(\rho)$ and
$l(\lambda)$, and when $\kappa_{\nu}<h_{\nu}$ we obtain a difference of 1
between $\mu_{g_{\nu}}$ and $\tilde{\mu}_{g_{\nu}}$, where
$g_{\nu}=h_{\nu}-\kappa_{\nu}$. Letting $\lambda$ and the $h_{\nu}$’s vary
(but with the equality
$\lambda-\sum_{\nu=1}^{\gamma}\kappa_{\nu}\varepsilon_{h_{\nu}}=\rho$
holding), for every $\nu$ we either run over all the values of $g_{\nu}$ (thus
increasing the multiplicity of $g_{\nu}$, giving a multiplier of
$\mu_{g_{\nu}}$ minus the $\delta$-symbols with the previous ones in the
comparison between $\prod_{h}\tilde{\mu}_{h}!$ and $\prod_{h}\mu_{h}!$ in the
numerator), or add 1 to $l(\rho)$ (and gives a multiplier in the comparison
between $l(\rho)$ and the length $l(\lambda)-\gamma$ of
$\lambda-\sum_{\nu=1}^{\gamma}h_{\nu}\varepsilon_{h_{\nu}}$).

Therefore, by taking out a multiplier of
$(-1)^{l(\rho)-1}\big{(}l(\rho)-1\big{)}!\big{/}\prod_{h=1}^{r}\mu_{h}!$
(which we can do since $r-d>0$ and $\rho$ is non-trivial), we see that if
$\beta$ of the indices $1\leq\nu\leq\gamma$ give numbers $g_{\nu}$ and the
other ones increase the length, then the contribution from such $\lambda$’s
combine to

$(-1)^{\beta}\binom{\gamma}{\beta}\prod_{t=0}^{\gamma-\beta-1}\big{(}l(\rho)+t\big{)}\cdot\sum_{g_{1}\geq
1}\ldots\sum_{g_{\beta}\geq
1}\prod_{\nu=1}^{\beta}\big{(}\mu_{g_{\nu}}-\textstyle{\sum_{\sigma=1}^{\nu-1}\delta_{g_{\nu},g_{\sigma}}}\big{)},$

where the binomial coefficient represents the choice of which indices increase
the length. Now, the sum over $g_{\beta}$ gives $l(\rho)-\beta+1$ for every
choice of the previous ones, then the sum over $g_{\beta-1}$ yields
$l(\rho)-\beta+2$, and so forth, the total sum divided by $\beta!$ from the
binomial coefficient is $\binom{l(\rho)}{\beta}$. Similarly, the product over
$t$ and the sign combine with the denominator $(\gamma-\beta)!$ to the
extended binomial coefficient $\binom{-l(\rho)}{\gamma-\beta}$. By summing
over $\beta$ and recalling that
$\sum_{\beta=0}^{\gamma}\binom{z}{\beta}\binom{w}{\gamma-\beta}$ equals
$\binom{z+w}{\gamma}$ as extended binomial coefficients for every $z$ and $w$,
we obtain $\gamma!\binom{0}{\gamma}$, which vanishes since we always have
$\gamma\geq 1$ when $d\geq 1$. This shows that the combination coming from
$\rho$ comes multiplied by a vanishing coefficient for every $\rho$, so that
$\partial^{d}u_{r}\big{/}\prod_{q=1}^{d}\partial_{i_{q}}$ indeed vanishes at
our point on the total diagonal. This proves the theorem. ∎

######  Corollary 15.

consider the point on the total diagonal where the common value of all the
variables $x_{i}$, $i\in\mathbb{N}_{N}$ is $a$. Then at this point we have
$u_{1}=a$ and $u_{r}=0$ for every $r\geq 2$. The total diagonal is
characterized algebraically in the $N$th symmetric power by the vanishing of
$\\{u_{r}\\}_{r=2}^{N}$, over any field of characteristic 0.

###### Proof.

As $u_{r}$ is homogenous of degree $r$, we have the equality
$\sum_{i=1}^{N}x_{i}\partial_{i}u_{r}=ru_{r}$ as functions on
$\mathbb{R}^{N}$. Now, Theorem 13 shows that if $r\geq 2$ then
$\partial_{i}u_{r}=0$ for all $i$ at our point, meaning that the left hand
side vanishes at that point hence so does the right hand side, i.e., so does
$u_{r}$. For $r=1$ both sides reduce to $\sum_{i=1}^{N}x_{i}/N$ (indeed,
$u_{1}=\tilde{e}_{1}=\frac{e_{1}}{N}$, so that $\partial_{i}u_{1}=\frac{1}{N}$
as we saw in, e.g., the proof of Theorem 13), which clearly attains the value
$a$ at our point. The second assertion now follows from the first via the fact
that $\\{u_{r}\\}_{r=1}^{N}$ are independent generators for the ring of
symmetric polynomials over any such field, as we saw from Theorem 1. This
proves the corollary. ∎

######  Remark 16.

Neither the property from Corollary 15 nor that from Theorem 13 characterizes
$\\{u_{r}\\}_{r=1}^{N}$, even up to scalar multiples. For seeing this we
define, for a partition $\eta\vdash r$ with multiplicities
$\\{\mu_{h}\\}_{h=1}^{r}$, the expression
$u_{\eta}:=\prod_{h=1}^{r}u_{h}^{\mu_{h}}$ as usual. It is clear that
$u_{\eta}$ is homogenous of degree $r$, and it follows from Corollary 15 that
if $\mu_{h}\geq 1$ for some $h\geq 2$ (i.e., when $\eta$ is not the partition
of length $r$ of $r$) then $u_{\eta}$ vanishes along the total diagonal. It
follows that if $r\geq 3$ and $\eta$ is any partition with $2\leq l(\eta)\leq
r-1$ then we can change $u_{r}$ by any multiple of $u_{\eta}$ and still get a
set of coordinates with the property from that Corollary 15. As for the
derivatives as in Theorem 13, take any $u_{\eta}$ for some $\eta\vdash r$ with
$l(\eta)\geq 2$ that does not contain the value 1 (the smallest such partition
is that of 4 as $2+2$), and consider an operator of the form from Theorem 13
that has some degree $d<r$. As applying it to $u_{\eta}$ will give linear
combinations of $\\{u_{r}\\}_{r=2}^{N}$ and their derivatives, all of which
vanish along the total diagonal (by Theorem 13 and Corollary 15), we deduce
that such $u_{\eta}$ also has the property from Theorem 13. A characterization
of our coordinates $\\{u_{r}\\}_{r=1}^{N}$ via derivatives (along the total
diagonal and elsewhere) is presented in Theorem 18 below.

Next we consider the dependence of our coordinates on $N$, and the variation
with $N$. Recall that the classical definition of the ring of symmetric
functions (say over $\mathbb{Q}$), in infinitely many variables, is given in
[M] and others as the direct sum over $d$ of the inverse limit of the spaces
$\mathbb{Q}[x_{1},\ldots,x_{N}]_{d}^{S_{N}}$ of symmetric functions in $N$
variables that are homogenous of degree $d$, where the maps in the inverse
limit construction are the ones going from
$\mathbb{Q}[x_{1},\ldots,x_{N+1}]^{S_{N+1}}$ to
$\mathbb{Q}[x_{1},\ldots,x_{N}]^{S_{N}}$ (as graded rings) and taking $x_{i}$
with $i\leq N$ to itself and $x_{n+1}$ to 0. We denote this ring, which is
also graded by the degrees, by $\Lambda_{\mathbb{Q}}$ (since it is the
extension of scalars from $\mathbb{Z}$ to $\mathbb{Q}$ from the ring denoted
by $\Lambda$ in [M]).

Now, for a fixed degree $d$, the maps used in the inverse limit construction
for the $d$-homogenous part of $\Lambda_{\mathbb{Q}}$ are isomorphisms for
every $N\geq d$. Therefore as maps of additive groups they can be inverted,
thus producing the $d$-homogenous part of $\Lambda_{\mathbb{Q}}$ (and, in
fact, all of $\Lambda_{\mathbb{Q}}$ at once) as a direct limit as well. The
corresponding map from $\mathbb{Q}[x_{1},\ldots,x_{N}]_{d}^{S_{N}}$ to
$\mathbb{Q}[x_{1},\ldots,x_{N+1}]_{d}^{S_{N+1}}$ takes the basic symmetric
monomial $m_{\eta}(x_{1},\ldots,x_{N})$ associated with some partition $\eta$
of $d$ to $\frac{1}{N+1-l(\eta)}\sum_{\sigma\in
S_{N+1}/S_{N}}\sigma(m_{\eta})$, as an element of
$\mathbb{Q}[x_{1},\ldots,x_{N+1}]_{d}^{S_{N+1}}$. Indeed, the term associated
with every $\sigma\in S_{N+1}/S_{N}$ is well-defined (because
$m_{\eta}(x_{1},\ldots,x_{N})$ is already $S_{N}$-invariant), the sum is
clearly $S_{N+1}$-invariant and is based only on monomials from $m_{\eta}$,
and the multiplying coefficient is one over the number of summands $\sigma$
which produce any fixed monomial from $m_{\eta}$ (it is also the ratio between
the sizes of the stabilizers of a monomial in $m_{\eta}$ in $S_{N}$ and in
$S_{N+1}$). Using these maps we can view each
$\mathbb{Q}[x_{1},\ldots,x_{N}]_{d}^{S_{N}}$ as contained in
$\mathbb{Q}[x_{1},\ldots,x_{N+1}]_{d}^{S_{N+1}}$, or, more importantly, in
$\Lambda_{\mathbb{Q}}$.

We set $\hat{u}_{r}:=\hat{u}_{r}^{(N)}:=\frac{N!}{(N-r)!}u_{r}$ (in $N$
variables), as well as $\hat{D}_{r}:=\frac{(N-r)!}{N!}D_{r}$, which equals the
average $\sum_{I\subseteq\mathbb{N}_{N},\ |I|=r}D_{I}\big{/}\binom{N}{r}$ of
the operators $D_{I}$ from Lemma 3 over all the relevant sets
$I\subseteq\mathbb{N}_{N}$, and prove the following result.

######  Proposition 17.

The coordinates $\\{\hat{u}_{r}^{(N)}\\}_{r=1}^{N}$ generate the ring of
symmetric functions in $N$ variables over $\mathbb{Q}$, and if we express a
continuously differentiable symmetric function $\phi(x_{1},\ldots,x_{N})$ as
$\hat{\varphi}(\hat{u}_{1},\ldots,\hat{u}_{N})$, then the derivative
$\hat{\varphi}_{\hat{u}_{d}}$ equals $\hat{D}_{d}\phi$ for every $1\leq d\leq
N$. Moreover, for any $r\in\mathbb{N}$, the images of the symmetric function
$\hat{u}_{r}^{(N)}$ in $\Lambda_{\mathbb{Q}}$ converges with $N$ to the
symmetric function $(-1)^{r-1}\frac{p_{r}}{r}$, where $p_{r}$ is the usual
power sum symmetric function.

###### Proof.

The fact that each $\hat{u}_{r}$ is a scalar multiple of the corresponding
$u_{r}$, and that $\hat{D}_{r}$ is $D_{r}$ divided by the same scalar, makes
the first two statements immediate consequences of Theorem 8 (via Theorem 1)
and Corollary 10.

Now, the dependence of $u_{r}$ on $N$ is via the definition of the normalized
expressions $\tilde{e}_{h}$ and $\tilde{e}_{\lambda}$. Since $\tilde{e}_{h}$
was obtained from $e_{h}$ by dividing by the monic polynomial
$\frac{N!}{(N-h)!}=\prod_{j=0}^{h-1}(N-j)$ of degree $h$ in $N$ means, it
follows that when $\lambda\vdash r$, the expression $\tilde{e}_{\lambda}$ is
$e_{\lambda}$ divided a the monic polynomial of degree $r$ in $N$, namely
$P_{\lambda}(N):=\prod_{h=1}^{N}\prod_{j=0}^{h-1}(N-j)^{m_{h}}=\prod_{j=0}^{h-1}(N-j)^{\sum_{h=j+1}^{N}m_{h}}$.
Therefore in $\hat{u}_{r}$ each expression $e_{\lambda}$ with $\lambda\vdash
r$ is multiplied by the coefficient
$(-1)^{l(\lambda)-1}\big{(}l(\lambda)-1\big{)}!\big{/}\prod_{h=1}^{r}m_{h}!$
from Theorem 8, as well as by the quotient $\frac{N!/(N-r)!}{P_{\lambda}(N)}$
of two monic polynomials of degree $r$ in $N$. Since the limit of every such
quotient as $N\to\infty$ exists and equals to 1, the convergence of
$\hat{u}_{r}^{(N)}$ in $\Lambda_{\mathbb{Q}}$ follows.

It is also clear that the limit of the $\hat{u}_{r}^{(N)}$’s is given by the
same formula from Theorem 8, but with each $\tilde{e}_{\lambda}$ replaced by
the ordinary $e_{\lambda}$. Moreover, Remark 9 allows us to write this
symmetric function as
$-\sum_{t=1}^{r}\hat{B}_{r,t}(-e_{1},\ldots,-e_{r-t+1})/t$. But the Newton
identities express $p_{r}$ as
$(-1)^{r}r\sum_{t=1}^{r}\hat{B}_{r,t}(-e_{1},\ldots,-e_{r-t+1})/t$, so that
this limit function is indeed the asserted one. This proves the proposition. ∎

We remark that the elementary symmetric function $e_{r}$ itself, associated
with the partition of length 1 of $r$, is multiplied in $\hat{u}_{r}^{(N)}$ by
1 for every $N$ (as both the multiplier from Theorem 8 and the quotient of the
two monic polynomials reduce to 1 for this partition). It is important to note
that while the $\hat{u}_{r}^{(N)}$ tend with $N$ to
$(-1)^{r-1}\frac{p_{r}}{r}$, and the equality with that limit holds for any
$N$ if $r=1$, such an equality occurs for no $N$ when $r\geq 2$. For example,
$\hat{u}_{2}^{(N)}$ equals $e_{2}-\frac{(N-1)}{2N}e_{1}^{2}$, which for $N=2$
and the variables $x$ and $y$ becomes
$\frac{xy}{2}-\frac{x^{2}+y^{2}}{4}=(x-y)^{2}/4$ (giving a multiple of the
coordinate form Lemma 2.3 of [EKZ]), and when $N=3$ (and the third variable
$z$) it reduces to $(xy+xz+yz-x^{2}-y^{2}-z^{2})/3$, neither of which are
multiples of $p_{2}$.

Recall from Remark 16 that the properties from Theorem 13 and Corollary 15 are
not sufficient for characterizing the $u_{r}$’s up to scalar multiples. For
doing so, we consider derivatives like from those results, but now of order
$r$. Since applying a derivative of order $r$ to a symmetric function of
degree $r$ yields a constant, we may consider the behavior of this constant as
a function of $N$, when the symmetric function is given in terms of
expressions that depend on $N$, like the $\tilde{e}_{\lambda}$’s from our
formulae. The proof of Proposition 17 shows that the
$\Lambda_{\mathbb{Q}}$-images of the $u_{r}$’s themselves, written as
$u_{r}^{(N)}$ for emphasizing the dependence on $N$, decay like
$O\big{(}\frac{1}{N^{r}}\big{)}$ as $N\to\infty$ there. This is why the
normalization $\hat{u}_{r}^{(N)}$ can be more appropriate in some situations
(see also Corollary 28 and Theorem 30 below).

Using this property we obtain the following characterization of our
coordinates $\\{u_{r}\\}_{r=1}^{N}$, up to scalars.

######  Theorem 18.

For every $r\geq 1$, the expression $u_{r}$ is the only linear combination of
$\\{\tilde{e}_{\lambda}\\}_{\lambda\vdash r}$ with coefficients that do not
depend on $N$ such that its image under every differential operator
$\prod_{q=1}^{r}\partial_{i_{q}}$ of order $r$ in which not all the indices
$i_{q}$, $1\leq q\leq r$ decays, in the limit $N\to\infty$, as
$O\big{(}\frac{1}{N^{r+1}}\big{)}$, normalized such that
$\partial_{i}^{r}u_{r}=(-1)^{r-1}\frac{(r-1)!}{N^{r}}$ for some, hence any,
$i\in\mathbb{N}_{N}$.

Explicit calculations suggest that for some derivatives the decay is even
faster than the one established in Theorem 18—see Conjecture 19 below.

###### Proof.

As in the proof of Theorem 8, we begin with proving uniqueness, and then show
that the $u_{r}$’s satisfy the required property.

Now, for evaluating the image of a combination of the $\tilde{e}_{\lambda}$’s
under such a differential operator, we can again apply Lemma 14. Moreover, the
same argument from the proof of Theorem 13 shows that for a fixed $\lambda$,
we must take the $\kappa_{\nu}$’s such that
$\lambda-\sum_{\nu=1}^{\gamma}\kappa_{\nu}\varepsilon_{h_{\nu}}$ gives a
partition of $r-d$, where here $d=r$ so that we only have the empty partition.
Thus $\lambda-\sum_{\nu=1}^{\gamma}h_{\nu}\varepsilon_{h_{\nu}}$ is also the
empty partition, so that $\gamma=l(\lambda)$ and the $h_{\nu}$’s are those
showing up in $\lambda$, and we also get $\kappa_{\nu}=h_{\nu}$ for every
$1\leq\nu\leq\gamma=l(\lambda)$, so that the functions
$\tilde{e}_{h_{\nu}-\kappa_{\nu}}$ are trivial. Therefore, every element of
$X_{\iota}$ in which there are $l(\lambda)$ sets, with the set $J_{\nu}$ of
indices having distinct $\iota$-images having size $h_{\nu}$ as shows up in
$\lambda$ for every $1\leq\nu\leq l(\lambda)$ (in the decreasing ordering of
the entries of $\lambda$, say), contributes to the expression from Lemma 14 a
single instance of
$\prod_{\nu=1}^{l(\lambda)}\frac{m_{h_{\nu}}!}{N!/(N-h_{\nu})!}$, more simply
expressed as
$\prod_{h=1}^{r}\frac{m_{h}!}{N!/(N-h)!}=\big{(}\prod_{h=1}^{r}m_{h}!\big{)}\big{/}P_{\lambda}(N)$
in the notation from the proof of Proposition 17.

We therefore denote by $X_{\iota}^{\lambda}$ the subset of $X_{\iota}$
consisting of partitions of $\mathbb{N}_{d}$ into sets
$\\{J_{\nu}\\}_{\nu=1}^{l(\lambda)}$ satisfying these conditions, and it is
clear that if the indices $\\{i_{q}\\}_{q=1}^{r}$ have the coincidence
relation such that there are $t$ distinct indices, say $j_{p}$, $1\leq p\leq
l$, and the index $j_{p}$ appears $s_{p}$ times among the $i_{q}$’s, then we
express this coincidence pattern via that the partition $\sigma$ of $r$ as
$\sum_{p=1}^{l}s_{p}$. It is clear that the size of $X_{\iota}^{\lambda}$
depends on $\iota$ only through the partition $\sigma$ (indeed, since our
derivative of order $r$ takes any symmetric function of degree $r$ to a
constant, we can evaluate this constant at any point, and applying such our
derivative to any function $\phi$ gives, at such a point, the expression
denoted by $\partial_{H}^{\sigma}\phi$ in the next section, where
$H\subseteq\mathbb{N}_{N}$ is any subset containing all the $i_{q}$’s). Now,
we have $|X_{\iota}^{\lambda}|>0$ if and only if the indices with pattern
$\sigma$ can be organized in sets of disjoint indices whose sizes are
determined by $\lambda$, which is easily seen to be equivalent to $\sigma^{t}$
dominating $\lambda$ in the natural dominance order on partitions of $r$, a
statement that we shall denote by $\lambda\leq\sigma^{t}$.

Consider thus a linear combination $\sum_{\lambda\vdash
r}c_{\lambda}\tilde{e}_{\lambda}$, with coefficients
$\\{c_{\lambda}\\}_{\lambda\vdash r}$ that are independent of $N$, not all of
which are 0. Take a differential operator $\prod_{q=1}^{r}\partial_{i_{q}}$,
corresponding to $\iota:\mathbb{N}_{r}\to\mathbb{N}_{N}$ and thus to a
partition $\sigma\vdash r$, and then its action sends our combination to
$\sum_{\lambda\vdash
r}c_{\lambda}|X_{\iota}^{\lambda}|\big{(}\prod_{h=1}^{r}m_{h}!\big{)}\big{/}P_{\lambda}(N)$,
where we saw that we can restrict the sum to be taken only over $\lambda$ that
are dominated by $\sigma^{t}$. Moreover, as all the numerators are independent
of $N$, and all the denominators are monic polynomials of degree $r$ in $N$,
this expression is
$\sum_{\lambda\leq\sigma^{t}}c_{\lambda}|X_{\iota}^{\lambda}|\big{(}\prod_{h=1}^{r}m_{h}!\big{)}\big{/}N^{r}+O\big{(}\frac{1}{N^{r+1}}\big{)}$.
We are therefore interested in those combinations $\sum_{\lambda\vdash
r}c_{\lambda}\tilde{e}_{\lambda}$ for which the last numerator vanishes for
every $\sigma$ which is not the maximal partition in terms of dominance, of
length 1 (i.e., where $\sigma^{t}$ is not the minimal partition, of length
$r$).

Consider first the case where $\sigma$ is such that $\sigma^{t}$ is one of the
(possibly several) minimal partitions $\lambda\vdash r$ for which
$c_{\lambda}\neq 0$. Then our numerator is the sum over one non-zero element,
meaning that the order of decay of our derivative with $N$ is precisely as
$O\big{(}\frac{1}{N^{r}}\big{)}$. Under our assumption this is allowed only if
$\sigma$ is the maximal partition, corresponding to the derivative of order
$r$ with respect to a single variable, and therefore the multiplier of the
element $\tilde{e}_{1}^{r}$ corresponding to the minimal partition cannot
vanish. For this $\sigma$ the corresponding derivative is
$\partial_{i}^{r}u_{r}$, sending any combination to $\frac{r!}{N^{r}}$ times
the coefficient of $\tilde{e}_{1}^{r}$, so that in our normalization this
coefficient has to be $(-1)^{r-1}/r$, and we have to see that our condition
determines the remaining coefficients. We do this by induction using the
dominance order, take a partition $\lambda\vdash r$ that is not the one that
we just normalized, and assume that we have already determined all the
coefficients $c_{\eta}$ for $\eta\leq\lambda$ except for $c_{\lambda}$ itself.
As the vanishing of the numerator arising from the derivative associated with
$\sigma=\lambda^{t}$ expresses a positive multiple of $c_{\lambda}$ in terms
of the coefficients $c_{\eta}$ that we already know, the uniqueness statement
follows (in fact, so does the existence, but without the explicit form of the
combination satisfying it).

It thus remains to prove that our combination $u_{r}$ does have this property.
After multiplying by $\frac{N!}{(N-r)!}$, this is equivalent, for rational
functions of $N$, to all the derivatives of order $r$ of the expression
$\hat{u}_{r}^{(N)}$ from Proposition 17 vanishing as $N\to\infty$, and the
derivative with respect to a single variable tending to $(-1)^{r-1}(r-1)!$.
But that proposition shows that in this limit, $\hat{u}_{r}^{(N)}$ tends to
$(-1)^{r}\frac{p_{r}}{r}$, so that its derivatives tend to those of the latter
symmetric function. Recalling that $p_{r}$ is the power sum function
$\sum_{i=1}^{N}x_{i}^{r}$ (i.e., it is a trace function in the terminology of
[B]), it is indeed annihilated by any mixed derivative, and the pure
derivative indeed gives the required (non-vanishing) value. This completes the
proof of the theorem. ∎

One can determine the expression for all the derivatives of order $r$ of
$u_{r}$ for small $r$ explicitly. Based on doing so for all $r\leq 6$ and
$\iota:\mathbb{N}_{r}\to\mathbb{N}_{N}$ (or more precisely $\sigma\vdash r$),
we pose the following conjecture

######  Conjecture 19.

Let $\prod_{q=1}^{r}\partial_{i_{q}}$ be a differential operator of order $r$
that corresponds to a partition $\sigma\vdash r$ as above. Then the total
expression of $\partial^{d}u_{r}\big{/}\prod_{q=1}^{d}\partial_{i_{q}}$ decays
like $O\big{(}\frac{1}{N^{r+l(\sigma)-1}}\big{)}$ as $N\to\infty$.

Note that while for Theorem 18 it suffices to consider the limit of
$\hat{u}_{r}^{(N)}$ as $N\to\infty$, for doing the finer analysis required for
investigating Conjecture 19 in a similar way one needs to remove the parts
associated with partitions that are dominating $\sigma$. As, for example,
subtracting $(-1)^{r}\frac{p_{r}}{r}\big{/}N^{r}$ and subtracting
$(-1)^{r}\frac{p_{r}}{r}\big{/}\frac{N!}{(N-r)!}$ from $u_{r}$ and multiplying
by $N^{r+1}$ (or by $\frac{N!}{(N-r-1)!}$) would give different limits as
$N\to\infty$, this seems to be a more delicate question, that is therefore
left for future research.

##  3 Derivatives at Diagonal Points

The expression $D_{d}\phi$ from Corollary 10 is only defined at points where
the variables $x_{i}$, $i\in\mathbb{N}_{N}$ have distinct values. We now turn
to evaluate these derivatives in case some (or all) of the values of the
$x_{i}$’s coincide. We shall assume throughout that $\phi$ is a symmetric
function of $N$ real variables $x_{i}$, $i\in\mathbb{N}_{N}$ (expressed as
$\varphi(u_{1},\ldots,u_{N})$), and that $\phi$ (and $\varphi$) have the
derivatives of every order that will show up. All the results will hold
equally well for holomorphic symmetric functions of $N$ complex variables, but
since smooth functions of complex variables can be non-holomorphic, the real
case is more natural for presenting our results.

We begin by observing the symmetry in the derivatives of $\phi$ where points
coincide.

######  Lemma 20.

Assume that $J\subseteq\mathbb{N}_{N}$ is a set of size $p$, and that we
differentiate at a point where the values of all the variables $x_{i}$ with
$i\in J$ is the same. Assign, to each $i\in J$, a multiplicity $h_{i}$, and
set $g:=\sum_{i\in J}h_{i}$. Then the value of the derivative
$\partial^{g}\phi\big{/}\prod_{i\in J}\partial x_{i}^{h_{i}}$ is invariant
under the action of the symmetric group of $J$ on the sets of multiplicities.

###### Proof.

Consider first the case where $J$ is the set $\\{i,j\\}$, of size 2. For every
$\delta\neq 0$, the value of $\phi$ obtained by adding $\delta$ to $x_{i}$ and
leaving the other variables invariant coincides with the value attained when
$x_{j}$ is taken to $x_{j}+\delta$ and the rest of the variables are left
invariant (by symmetry). Therefore the limits defining the derivatives
$\phi_{i}$ and $\phi_{j}$ at our point are the same, as desired. Similarly,
the limit defining $\phi_{i}$ at the point where $x_{i}$ is replaced by
$x_{i}+\delta$ coincides (again by symmetry) with the one producing $\phi_{j}$
with $x_{j}+\delta$ in place of $x_{j}$. As this is the case for all $\delta$,
we obtain $\phi_{ii}=\phi_{jj}$. More generally, the value of
$\partial_{i}^{h-1}\phi$ with $x_{i}+\delta$ instead of $x_{i}$ will be the
same as that of $\partial_{j}^{h-1}\phi$ when $x_{j}$ is replaced by
$x_{j}+\delta$, so that by the taking the limit we established the result when
$|J|=2$ and one of the multiplicities vanishes.

Let now $J$ and the multiplicities by general. Since the permutation group is
generated by transpositions, take two indices $i$ and $j$ from $J$, and it
suffices to prove that $\partial^{g}\phi\big{/}\prod_{i\in J}\partial
x_{i}^{h_{i}}$ is invariant under the transposition interchanging $i$ and $j$.
Assume, without loss of generality, that $h_{i}\geq h_{j}$, let
$h:=h_{i}-h_{j}$, and consider the function
$\psi:=\partial^{g-h}\phi\big{/}\partial x_{i}^{h_{j}}\partial
x_{j}^{h_{j}}\prod_{k\in J\setminus\\{i,j\\}}\partial x_{k}^{b_{k}}$. Since
$\phi$ and the differential operator taking it to $\psi$ are symmetric under
interchanging $i$ and $j$, so is $\psi$, and thus the previous paragraph gives
us $\partial_{i}^{h}\psi=\partial_{j}^{h}\psi$. But as the latter two
functions are the asserted one $\partial^{g}\phi\big{/}\prod_{i\in J}\partial
x_{i}^{h_{i}}$ and its image under our transposition respectively, the desired
equality follows. This proves the lemma. ∎

Note that for second derivatives, Lemma 20 gives $\phi_{ii}=\phi_{jj}$ when
$x_{i}=x_{j}$ (as well as $\phi_{ik}=\phi_{jk}$ for a third, distinct index
$k$), but these derivatives are not related to $\phi_{ij}$. As one example for
this, let $\phi$ be a _trace function_ in the terminology of [B], i.e., a
function of the form $\sum_{i=1}^{N}f(x_{i})$ for a function $f$ of a single
variable. Then $\phi_{ij}=0$ but $\phi_{ii}$ and $\phi_{jj}$ equal
$f^{\prime\prime}(x_{i})$ (and $x_{i}=x_{j}$), which need not vanish. As
another example, take $\phi$ be the symmetric function $e_{h}$ for some $h\geq
2$. Then it is clear (e.g., using Lemma 2) that
$\phi_{ij}=e_{h-2}(x_{I^{c}})\neq 0$ for $I=\\{i,j\\}$, while
$\phi_{ii}=\phi_{jj}=0$.

It follows from Lemma 20 that if $\sigma$ is the partition of the total order
$g$ of the derivative mentioned there into the parts $\\{h_{i}\\}_{i\in J}$
(with $l(\sigma)\leq|J|$), written classically as $g=\sum_{q=1}^{l}h_{q}$
(ignoring the vanishing $h_{i}$’s, say), then by assigning the multiplicities
$h_{q}$, $1\leq q\leq l$ to arbitrary distinct indices from $J$ the resulting
derivative $\partial^{g}\phi\big{/}\prod_{i\in J}\partial x_{i}^{h_{i}}$ of
$\phi$ depends only on $J$ and $\sigma$, and not on the assignment. We can
thus denote this derivative by $\partial_{J}^{\sigma}\phi$, and it is well-
defined at points like in that lemma (we can define it using an arbitrary
assignment, or using an average, at every point in $\mathbb{R}^{N}$, but we
shall consider such expressions only under the assumption that the values of
the variables $x_{i}$, $i\in J$ are all the same, where we now know it to be
well-defined and independent of any additional data). In particular
$\partial_{J}^{(1)}\phi$ is just $\phi_{i}$ for $i\in J$, where the
superscript stands for the unique partition of 1, and this is clearly the only
simple derivative of order 1 that we can define based on $J$. Similarly, if
$J$ contains at at least two elements and $i$ and $j$ are in $J$ then the
derivatives $\phi_{ij}$ and $\phi_{ii}$, whose difference was mentioned in the
Introduction, can be written as $\partial_{J}^{(1,1)}\phi$ and
$\partial_{J}^{(2)}\phi$ respectively. The combination of third derivatives
that shows up there, when $|J|\geq 3$, is therefore
$\partial_{J}^{(1,1,1)}\phi-\frac{3}{2}\partial_{J}^{(2,1)}\phi+\frac{1}{2}\partial_{J}^{(3)}\phi$
in this notation.

The combinations of higher-order derivatives were obtained from the action of
an operator like $D_{I}$ at points where some variables coincide, and due to
the vanishing denominators, they can be obtained by taking the limit of the
expressions for points with distinct variables as some variables tend to the
value of others. We will see that when the values of $p$ variables coincide,
we shall obtain, in general, derivatives of all orders $1\leq g\leq p$, and
the form of the $g$th derivative is always the same. This means that for every
$g\in\mathbb{N}$ (recall that in this paper the notation $\mathbb{N}$ only
includes positive integers, without 0, like the set $\mathbb{N}_{N}$ from
above) there are fixed coefficients $\\{c_{\sigma}\\}_{\sigma\vdash g}$ such
that wherever $J$ is a set of $p$ indices, with $p\geq g$, the derivatives of
order $g$ of $\phi$ that are based on indices from $J$ is always the same
combination $\partial_{J}^{g}\phi=\sum_{\sigma\vdash
g}c_{\sigma}\partial_{J}^{\sigma}\phi$ of the derivatives
$\partial_{J}^{\sigma}\phi$ with $\sigma\vdash g$. The derivative
$\partial_{J}^{1}\phi$ is just $\partial_{J}^{(1,1)}\phi=\phi_{i}$ from above,
$\partial_{J}^{2}\phi$ will be the difference
$\partial_{J}^{(1,1)}\phi-\partial_{J}^{(2)}\phi$ from above, and the
expression from the end of the previous paragraph will be
$\partial_{J}^{3}\phi$.

For determining the explicit value of $\partial_{J}^{g}\phi$, we shall need
some notation. Let a finite set $K$ of indices be given, as well as a non-
negative integer $\nu$. We then define the set $A_{K,\nu}$ to be the set of
tuples $(a,\vec{b})$, with $a\in\mathbb{N}$ and $\vec{b}=\\{b_{k}\\}_{k\in
K}\in\mathbb{N}^{K}$, satisfying the equality $a+\sum_{k\in
K}b_{k}=|K|+\nu+1$. Note that $K$ can be empty, and then $A_{\nu}$ only
consists of the single 1-tuple with entry $a=\nu+1$. Also note that for
$\nu=0$ the set $A_{K,\nu}$ contains again only one element, in which $a$ and
all the $b_{k}$’s equal 1.

We now aim to prove the following result.

######  Theorem 21.

Assume that $I$ is a set containing $J$, with $|I|=d$ and $|J|=p\geq 1$, and
consider a point where the variables $x_{i}$, $i\in J$ all take the value $y$,
and that the values of the variables $x_{i}$, $i\in I\setminus J$ are distinct
from $y$ and from one another. At such a point set

$\partial_{J}^{g}\phi:=\sum_{\sigma\vdash
g}(-1)^{g-l(\sigma)}\frac{g\cdot\big{(}l(\sigma)-1\big{)}!}{\prod_{h=1}^{g}h!^{m_{h}}m_{h}!}\partial_{J}^{\sigma}\phi$

for every $1\leq g\leq p$ (independently of $I$), where for $\sigma\vdash g$
and $1\leq h\leq p$ the symbol $m_{h}$ stands for the multiplicity with which
$h$ appears in $\sigma$, and we have the equality

$D_{I}\phi=\sum_{k\in I\setminus J}\frac{\phi_{k}}{(y-x_{k})^{p}\prod_{l\in
I\setminus(J\cup\\{k\\})}(x_{l}-x_{k})}+\sum_{(a,\vec{b})\in A_{I\setminus
J,p-1}}\frac{(-1)^{p-a}\partial_{J}^{a}\phi}{\prod_{k\in I\setminus
J}(x_{k}-y)^{b_{k}}}.$

######  Remark 22.

The derivative $\partial_{J}^{\sigma}\phi$ is the value at our point of the
image of $\phi$ under an operator $\prod_{q=1}^{g}\partial_{i_{q}}$ for some
choice of $i_{q}$’s such that the associated map
$\iota:\mathbb{N}_{g}\to\mathbb{N}_{n}$ as in Lemma 14 has image in $J$ and
corresponds to $\sigma$. Choosing $g$ distinct elements of $J$, say $j_{h}$,
$1\leq h\leq g$, and normalizing the maps $\iota$ yielding the different
$\sigma\vdash r$ according to these $j_{h}$’s, this means that in the
terminology from Remark 9, we can express the operator $\partial_{J}^{g}$ in
terms of the usual (exponential) partial Bell polynomials $B_{g,t}$ in the
derivatives $\partial_{j_{h}}$, $1\leq h\leq g$ as
$(-1)^{g}g\sum_{t=1}^{g}B_{g,t}(-\partial_{j_{1}},\ldots,-\partial_{j_{g-t+1}})$.

We will prove Theorem 21 by induction on $p$. For this we shall need the
following preliminary calculations.

######  Lemma 23.

Let $\psi$ be a function of the variable $x_{j}$ that is differentiable at
least $\nu$ times, and let $\\{x_{k}\\}_{k\in K}$ be distinct real numbers,
where $K$ is a finite set of indices not containing $j$. Recall the set
$A_{K,\nu}$ from above, and then the equality

$\frac{d^{\nu}}{dx_{j}^{\nu}}\frac{\psi^{\prime}(x_{j})}{\prod_{k\in
K}(x_{k}-x_{j})}=\sum_{(a,\vec{b})\in
A_{K,\nu}}\frac{\nu!\psi^{(a)}(x_{j})}{(a-1)!\prod_{k\in
K}(x_{k}-x_{j})^{b_{k}}}$

holds wherever $x_{j}$ is different from the values of the $x_{k}$’s.

Lemma 23 is clear when $K$ is empty, as it reduces to the immediate equality
$\frac{d^{\nu}}{dx_{j}^{\nu}}\psi^{\prime}(x_{j})=\psi^{(\nu+1)}(x_{j})$.
However, we shall require it for non-empty $K$ later.

###### Proof.

We argue by induction on $\nu$, with the case $\nu=0$ being trivial, since the
set $A_{K,0}$ only contains the tuple with $a=1$ and $b_{k}=1$ for every $k\in
K$ (note the initial derivative of $\psi$ in the numerator). Now, assume that
the result holds for some $\nu$, and since differentiating the numerator in
one summand for the $\nu$th derivative increases $a$ by 1 and differentiating
$\frac{1}{(x_{k}-x_{j})^{b_{k}}}$ increases $b_{k}$ by 1, we indeed obtain a
sum over the expressions associated with $A_{K,\nu+1}$, and we only have to
check the multiplying coefficients. So take an element $(a,\vec{b})\in
A_{K,\nu+1}$, and we saw that the derivatives producing multiples of this term
are obtained from terms associated with tuples in $A_{K,\nu+1}$ in which
exactly one of the entries is one less than the corresponding entry of
$(a,\vec{b})$, and the other entries are like those of $(a,\vec{b})$. Now,
from the term with $b_{k}-1$ the derivative of
$\frac{1}{(x_{k}-x_{j})^{b_{k}-1}}$ gives our term multiplied by $b_{k}-1$
(thus covering the case with $b_{k}=1$ not appearing in this construction),
and all of these summands showed up in the $\nu$th derivative with the
coefficient $\frac{\nu!}{(a-1)!}$. The last contribution is, when $a>1$, from
the term with $a-1$, where there is no coefficient from the derivative but the
multiplier is $\frac{\nu!}{(a-2)!}=\frac{\nu!}{(a-1)!}(a-1)$ (which indeed
vanishes when $a=1$ and we can ignore this restriction as well). The total
numerical coefficient is thus $\frac{\nu!}{(a-1)!}$ times $a-1+\sum_{k\in
K}(b_{k}-1)$, and as the latter sum equals $\nu+|K|+2-1-|K|=\nu+1$ for
$(a,\vec{b})\in A_{K,\nu+1}$, we indeed obtain the desired coefficient
$\frac{(\nu+1)!}{(a-1)!}$. Thus proves the lemma. ∎

Recall that the definition of $\partial_{J}^{p}\phi$ from Theorem 21 remains
the same when $J$ is increased to a larger set. In particular, the expressions
$\partial_{J}^{h}\phi$ are defined for every $1\leq h\leq p$. The next
relation that we shall need is among these expressions.

######  Lemma 24.

Assume that all the variables $x_{i}$ with $i\in J$ equal $y$, and choose some
$j$ which is not in $J$. Take some $1\leq g\leq p$, and then as $x_{j}\to y$
as well, the limit of the expression
$\partial_{j}^{g}\phi+\sum_{\mu=0}^{g-1}\frac{(g-1)!}{\mu!}(-1)^{g-\mu}\partial_{j}^{\mu}\partial_{J}^{g-\mu}\phi$
vanishes. Equivalently, the expression
$\partial_{j}^{g}\phi+\sum_{\mu=1}^{g-1}\frac{(g-1)!}{\mu!}(-1)^{g-\mu}\partial_{j}^{\mu}\partial_{J}^{g-\mu}\phi$
takes the limit $-(-1)^{g}(g-1)!\partial_{J}^{g}\phi$ as $x_{j}\to y$ for
every such $g$, and when we consider this expression as a multiple of
$\partial_{J\cup\\{j\\}}^{g}\phi$, the latter statement holds also for
$g=p+1$.

###### Proof.

For each $\mu\geq 1$ we write $\partial_{J}^{g-\mu}\phi$, via the definition
in Theorem 21, as the sum $\sum_{\tau\vdash
g-\mu}(-1)^{g-\mu-l(\tau)}(g-\mu)\cdot\big{(}l(\sigma)-1\big{)}!\partial_{J}^{\tau}\phi\big{/}\prod_{h=1}^{g-\mu}h!^{\tilde{m}_{h}}\tilde{m}_{h}!$,
where for $\tau$ we use $\tilde{m}_{h}$ for the multiplicities. When we
differentiate it $\mu$ times with respect to $j$, at the limit $x_{j}\to y$
the derivative $\partial_{j}^{\mu}\partial_{J}^{\tau}$ becomes, by definition,
$\partial_{J\cup\\{j\\}}^{\sigma}$ where $\sigma$ is the partition of $g$ such
that $\sigma-\mu\varepsilon_{\mu}=\tau$ in the notation used in, e.g.,
Corollary 7. We thus take a partition $\sigma\vdash g$, and see with which
multiplicity the expression $\partial_{J\cup\\{j\\}}^{\sigma}\phi$ shows up in
the limit of
$\partial_{j}^{g}\phi+\sum_{\mu=1}^{g-1}\frac{(g-1)!}{\mu!}(-1)^{g-\mu}\partial_{j}^{\mu}\partial_{J}^{g-\mu}\phi$
as $x_{j}\to y$. The first term only produces the derivative corresponding to
the partition of $g$ as $g$ (with length 1), and as for every $1\leq\mu\geq
g-1$ both $j$ and indices different from $j$ show up in
$\partial_{j}^{\mu}\partial_{J}^{\tau}\phi$ for any $\tau\vdash g-\mu$, the
sum over $\mu$ only contributes to partitions $\sigma\vdash g$ with
$l(\sigma)\geq 2$.

Now, given such $\sigma$ we have $l(\sigma^{t})\leq g-1$, and we get a
contribution exactly from those $1\leq\mu\leq g-1$ and $\tau\vdash g-\mu$ such
that $\tau=\sigma-\mu\varepsilon_{\mu}$, namely from those indices $\mu$ that
show up in $\sigma$, and then $\tau$ is determined. When $m_{h}$ (resp.
$\tilde{m}_{h}$) is the multiplicity with which $h$ shows up in $\sigma$
(resp. $\tau$), we get that $\tilde{m}_{h}$ can be expressed succinctly as
$m_{\mu}-\delta_{h,\mu}$, and $l(\tau)=l(\sigma)-1$. Moreover, we can write
$1\big{/}\prod_{h=1}^{g-\mu}h!^{\tilde{m}_{h}}\tilde{m}_{h}!$, expressing the
denominator of the numerical coefficient multiplying
$\partial_{J}^{\tau}\phi$, as
$\mu!m_{\mu}\big{/}\prod_{h=1}^{g-1}h!^{m_{h}}m_{h}!$ using the multiplicities
for $\sigma$ (and we can take the product over $h$ to go up to $g$, since
$m_{g}=0$ when $\sigma\vdash g$ and $l(\sigma)\geq 2$). Altogether, since the
part $(-1)^{g-\mu}$ of the sign with which $\partial_{J}^{\tau}\phi$ appears
in $\partial_{J}^{g-\mu}\phi$ cancels with the sign multiplier of the $\mu$th
summand, and the $\mu!$ in the latter expression cancels with the denominator
of $\frac{(g-1)!}{\mu!}$, the total coefficient with which
$\partial_{J\cup\\{j\\}}^{\sigma}\phi$ shows up in our expression is $(g-1)!$
times

$-\frac{(-1)^{l(\sigma)}\big{(}l(\sigma)-2\big{)}!}{\prod_{h=1}^{g-\mu}h!^{m_{h}}m_{h}!}\bigg{[}\sum_{\mu=1}^{g-1}(g-\mu)m_{\mu}\bigg{]}=-(-1)^{g}\frac{(-1)^{g-l(\sigma)}g\big{(}l(\sigma)-1\big{)}!}{\prod_{h=1}^{g-\mu}h!^{m_{h}}m_{h}!}$

(this is so, since the vanishing of the multiplier $m_{\mu}$ when $\mu$ does
not show up in $\sigma$ allows us to consider the sum over all $1\leq\mu\leq
g-1$, and by adding $\mu=g$ in the same trivial manner we recall that
$\sum_{\mu=1}^{g}m_{\mu}=l(\sigma)$ and $\sum_{\mu=1}^{g}m_{\mu}\mu=g$ when
$\sigma\vdash g$). Note that in the remaining partition of length 1 the latter
expression (including the multiplier $(g-1)!$) reduces to the coefficient 1
from the previous paragraph. As this is $-(-1)^{g}(g-1)!$ times the
coefficient multiplying $\partial_{J\cup\\{j\\}}^{\sigma}\phi$ in
$\partial_{J\cup\\{j\\}}^{g}\phi$ for every $\sigma\vdash g$, now without the
restriction on $l(\sigma)$ (also when $g=p+1$), we obtain the second
assertion, from which the first one follows since when $g\leq p$ we can
represent the summand $(g-1)!(-1)^{g}\partial_{J}^{g}\phi$ cancels with
$-(g-1)!(-1)^{g}\partial_{J\cup\\{j\\}}^{g}\phi$ by Lemma 20. This proves the
lemma. ∎

The combination of these results that we shall need is the following one.

######  Proposition 25.

At a point like in Lemma 24, with $\phi$ differentiable at least $p$ times,
take an index set $K$ that is disjoint from $J\cup\\{j\\}$, and let
$\\{x_{k}\\}_{k\in K}$ be as in Lemma 23. Then the function

$\frac{\phi_{j}}{\prod_{k\in
K}(x_{k}-x_{j})}+\sum_{c=0}^{p-1}(x_{j}-y)^{c}\sum_{(a,\vec{b})\in
A_{K,c}}\frac{(-1)^{a}\partial_{J}^{a}\phi}{\prod_{k\in K}(x_{k}-y)^{b_{k}}}$

of $x_{j}$, and all of its derivatives up to order $p-1$, vanish at the limit
$x_{j}\to y$. The value of its $p$th derivative is $-p!\sum_{(a,\vec{b})\in
A_{K,p}}(-1)^{a}\partial_{J}^{a}\phi\big{/}\prod_{k\in K}(x_{k}-y)^{b_{k}}$.

###### Proof.

Take some $\nu\geq 0$, and consider the $\nu$th derivative of this function of
$x_{j}$. The derivative of the first term is given in Lemma 23, where $\psi$
is the restriction of $\phi$ to the variable $x_{j}$ with the other ones fixed
with our conditions, so that each $\psi^{(a)}(x_{j})$ is
$\partial_{j}^{a}\phi$. In the second term we only differentiate the powers of
$x_{j}-y$ and the numerators $\partial_{J}^{a}\phi$, since the denominators
are independent of $x_{j}$. The Generalized Leibniz’ Rule shows that the
$\nu$th derivative of $(x_{j}-y)^{c}\partial_{J}^{a}\phi$ is
$\sum_{\mu}\binom{\nu}{\mu}\frac{c!}{(c-\nu+\mu)!}(x_{j}-y)^{c-\nu+\mu}\partial_{j}^{\mu}\partial_{J}^{a}\phi$,
with $\max\\{0,\nu-c\\}\leq\mu\leq\nu$. Interchanging the sum over $c$ and
$\mu$, and replacing $c$ by $c+\nu-\mu$, we find that the $\nu$th derivative
of the second term is

$\sum_{\mu=\nu+1-p}^{\nu}\binom{\nu}{\mu}\sum_{c=0}^{p-1-\nu+\mu}(x_{j}-y)^{c}\frac{(c+\nu-\mu)!}{c!}\sum_{(a,\vec{b})\in
A_{K,c+\nu-\mu}}\frac{(-1)^{a}\partial_{j}^{\mu}\partial_{J}^{a}\phi}{\prod_{k\in
K}(x_{k}-y)^{b_{k}}},$

where the binomial coefficient restricts $\mu$ to
$\max\\{0,\nu+1-p\\}\leq\mu\leq\nu$. As the terms with $c>0$ clearly vanish as
$x_{j}\to y$, it remains to consider the limit of

$\sum_{(a,\vec{b})\in
A_{K,\nu}}\frac{\nu!\partial_{j}^{a}\phi}{(a-1)!\prod_{k\in
K}(x_{k}-x_{j})^{b_{k}}}+\sum_{\mu=\nu+1-p}^{\nu}\frac{\nu!}{\mu!}\sum_{(a,\vec{b})\in
A_{K,\nu-\mu}}\frac{(-1)^{a}\partial_{j}^{\mu}\partial_{J}^{a}\phi}{\prod_{k\in
K}(x_{k}-y)^{b_{k}}}$

(again with the sum over $\mu$ effectively being over
$\max\\{0,\nu+1-p\\}\leq\mu\leq\nu$), where we can substitute $x_{j}=y$ in the
denominators of the summands in the first term.

Take some values of $\\{b_{k}\\}_{k\in K}$ that show up in an element of
$A_{K,\nu}$. The value of $a$ in the corresponding summand in the first term
is $|K|+\nu+1-\sum_{k\in K}b_{k}$, an expression that we denote by $g$, and
satisfies $g\geq 1$ since $(g,\vec{b})\in A_{K,\nu}$ by assumption. The values
of $\mu$ for which this element shows up in $A_{K,\nu-\mu}$ are those for
which the required value of $a$, which is now $g-\mu$, is positive, namely
$\mu\leq g-1$. Assuming that $\nu\leq p-1$, so that the effective lower bound
on $\mu$ is indeed 0, these terms in our expression are
$\nu!\big{/}(g-1)!\prod_{k\in K}(x_{k}-y)^{b_{k}}$ times the combination
$\partial_{j}^{g}\phi+\sum_{\mu=0}^{g-1}\frac{(g-1)!}{\mu!}(-1)^{g-\mu}\partial_{j}^{\mu}\partial_{J}^{g-\mu}\phi$,
and $g\leq\nu+1\leq p$. As this combination vanishes as $x_{j}\to y$ by the
first assertion in Lemma 24, this establishes the first assertion. If we now
take $\nu=p$, then the combination that we get is the same, but with the sum
over $\mu$ starting from 1, and the bound $g\leq p+1$. We apply the second
assertion of Lemma 24 for the limit as $x_{j}\to y$, and merge with the
external multiplier (with $\nu=p$) to get
$(-1)^{g+1}p!\partial_{J}^{g}\phi\big{/}\prod_{k\in K}(x_{k}-y)^{b_{k}}$. As
$g$ was seen to be the value of $a$ complementing the $b_{k}$’s to an element
of $A_{K,\nu}=A_{K,p}$, the second assertion follows as well. This proves the
proposition. ∎

###### Proof of Theorem 21.

We prove the desired formula by induction on $p$. The case $p=1$ is just the
definition of $D_{I}\phi$ where all the points are distinct, since
$A_{I\setminus J,0}$ has a single simple element and
$\partial_{J}^{1}\phi=\phi_{i}$ where $J$ is the singleton $\\{i\\}$. We shall
assume that $d>p$, and, by induction, that the formula for $D_{I}\phi$ is
valid when $|J|=p$. We take some index $j\in I\setminus J$, view the values of
the variable $x_{k}$ for $k\in I\setminus(J\cup\\{j\\})$ as fixed, and allow
the variable $x_{j}$ to tend to the common value $y$ of $x_{i}$, $i\in J$. In
the terms involving $\phi_{k}$ for $k\in I\setminus(J\cup\\{j\\})$, this limit
is obtained by simply replacing the multiplier $x_{j}-x_{k}$ by another power
of $y-x_{k}$, thus yielding the desired sum over such $k$ in $D_{I}\phi$ for
$p+1$.

For analyzing the remaining terms, set $K:=I\setminus(J\cup\\{j\\})$, so that
$I\setminus J$ is $K\cup\\{j\\}$, and note that the value of $b_{j}$ in an
element of $A_{I\setminus J,p-1}$ lies between 1 and $\nu$, and that when we
fix that value, the remaining entries give an element of $A_{K,p-b_{j}}$.
Having the denominator $(y-x_{j})^{p}$ in the term involving $\phi_{j}$, we
replace $b_{j}$ by $c=p-b_{j}$, and the equality that we have to show that the
limit of

$\frac{1}{(x_{j}-y)^{p}}\Bigg{[}\frac{(-1)^{p}\phi_{j}}{\prod_{k\in
K}(x_{k}-x_{j})}+\sum_{c=0}^{p-1}(x_{j}-y)^{c}\sum_{(a,\vec{b})\in
A_{K,c}}\frac{(-1)^{p-a}\partial_{J}^{a}\phi}{\prod_{k\in
K}(x_{k}-y)^{b_{k}}}\Bigg{]}$

as $x_{j}\to y$ exists and equals $\sum_{(a,\vec{b})\in
A_{K,p}}(-1)^{p+1-a}\partial_{J\cup\\{j\\}}^{a}\phi\big{/}\prod_{k\in
K}(x_{k}-y)^{b_{k}}$. We claim that we can do it by applying L’Hôpital’s Rule
$p$ successive times.

Indeed, the denominator is $(x_{j}-y)^{p}$, which vanishes along with its
first $p-1$ derivatives as $x_{j}\to y$, and the numerator is $(-1)^{p}$ times
the function from Corollary 25, which also has the same vanishing properties.
Therefore the limit is the quotient of the $p$th derivative of the numerator
divided by that of the denominator, which is $p!$. As $\frac{(-1)^{p}}{p!}$
times the expression for the $p$th derivative from Proposition 25 is the
asserted value, the result required for the induction step follows. This
proves the theorem. ∎

It is clear from the proof of Theorem 21 that the only property of the
explicit expression for $\partial_{J}^{p}\phi$ is the relation from Lemma 24.
As this lemma expresses $\partial_{J}^{g}\phi$, and with it
$\partial_{J\cup\\{j\\}}^{g}\phi$, in terms of the derivatives of lower order
(and their derivatives with respect to $x_{j}$), this relation can be used for
constructing these expressions inductively.

Theorem 21 considers one operator $D_{I}$, with $|I|=d$, at a point where the
the values of the variables associated with one set $J$, of size $p$,
coincide, while all the other values are distinct. This is, indeed, the most
generic assumptions under which an expression like $\partial_{J}^{p}\phi$, for
this value of $p$, shows up. We can view this situation as a partition of $I$
into $d-p+1$ sets, one of which is $J=J_{1}$ and the other ones are singletons
$J_{\alpha}$, $2\leq\alpha\leq d-p+1$, and denote by $y_{\alpha}$ the value of
$x_{i}$ for $i\in J_{\alpha}$ (this is $y$ when $\alpha=1$ and $x_{k}$ if
$\alpha>1$ and $J_{\alpha}$ is the singleton $\\{k\\}$). All the terms in the
expression for $D_{I}\phi$ in Theorem 21 involve (up to a multiplying
numerical coefficient which is a sign) a derivative
$\partial_{J_{\alpha}}^{g}\phi$ with $1\leq g\leq|J_{\alpha}|$ (this means
$g=1$ for $\alpha>1$), over a denominator which is a product over
$\beta\neq\alpha$ of $(y_{\beta}-y_{\alpha})^{c_{\beta}}$, where
$c_{\beta}\geq|J_{\beta}|$ and $g+\sum_{\beta\neq\alpha}c_{\beta}=d$.

At a general point in $\mathbb{R}^{N}$ we may well have several sets of points
coinciding. For generalizing the interpretation of Theorem 21 from the
previous paragraph, we consider a collection $\vec{J}$ of non-empty disjoint
finite sets $\\{J_{\alpha}\\}_{\alpha=1}^{s}$ and an index $1\leq\alpha\leq
s$, and define the set $B_{\vec{J},\alpha}$ to be the set of tuples of
positive integers the form $(a,\vec{c})$ in which
$\vec{c}=\\{c_{\beta}\\}_{\beta\neq\alpha}$ is such that
$c_{\beta}\geq|J_{\beta}|$ for every $\beta\neq\alpha$, and $a\in\mathbb{N}$
satisfies $a+\sum_{\beta\neq\alpha}c_{\beta}=\sum_{\alpha=1}^{s}|J_{\alpha}|$.
Note that when $J_{\beta}$ is a singleton for every $\beta\neq\alpha$ and
$I=\bigcup_{\alpha=1}^{s}J_{\alpha}$, the set $B_{\vec{J},\alpha}$ is the same
as $A_{I\setminus J_{\alpha},|J_{\alpha}|-1}$ (as it should be, by comparing
Proposition 26 below to Theorem 21), so that it is indeed a more general
definition. Using these sets, we obtain the following formula for $D_{I}\phi$
at any point.

######  Proposition 26.

Assume that the index set $I$ is the disjoint union of non-empty sets
$\\{J_{\alpha}\\}_{\alpha=1}^{s}$, and consider a point in $\mathbb{R}^{N}$ at
which $x_{i}$ takes the same value $y_{\alpha}$ for every $i\in J_{\alpha}$,
but the values $\\{y_{\alpha}\\}_{\alpha=1}^{s}$ are distinct. At such a point
we have

$D_{I}\phi=\sum_{\alpha=1}^{s}\sum_{(a,\vec{c})\in
B_{\vec{J},\alpha}}\Bigg{[}\prod_{\beta\neq\alpha}\binom{c_{\beta}-1}{|J_{\beta}|-1}\Bigg{]}\frac{(-1)^{|J_{\alpha}|-a}\partial_{J_{\alpha}}^{a}\phi}{\prod_{\beta\neq\alpha}(y_{\beta}-y_{\alpha})^{c_{\beta}}}.$

###### Proof.

As in the proof of Theorem 21, we can evaluate our $D_{I}\phi$ by considering
the ordinary $D_{I}\phi$ from Lemma 3 at points that are close to the desired
one and in which all the variables take different values, and then take the
limit at our point in a convenient way. For this we decompose $D_{I}\phi$ from
Lemma 3 as the sum over $1\leq\alpha\leq s$ of $\sum_{i\in
J_{\alpha}}\partial_{i}\phi\big{/}\prod_{i\neq j\in I}(x_{j}-x_{i})$, and in
the part associated with $\alpha$ we first take the limit where all the
variables $x_{i}$ with $i\in J_{\alpha}$ tend to $y_{\alpha}$, and then
consider the limit in the other variables. Theorem 21 shows that the
$\alpha$th part becomes $\sum_{(a,\vec{b})\in A_{I\setminus
J_{\alpha},|J_{\alpha}|-1}}(-1)^{|J_{\alpha}|-a}\partial_{J}^{a}\phi\big{/}\prod_{k\in
I\setminus J_{\alpha}}(x_{k}-y_{\alpha})^{b_{k}}$ at the first limit, and at
the second limit, each denominator $\prod_{k\in I\setminus
J_{\alpha}}(x_{k}-y_{\alpha})^{b_{k}}$ becomes
$\prod_{\beta\neq\alpha}(y_{\beta}-y_{\alpha})^{c_{\beta}}$, with
$c_{\beta}:=\sum_{k\in J_{\alpha}}b_{k}$.

Now, from the conditions on elements of $A_{I\setminus
J_{\alpha},|J_{\alpha}|-1}$ we clearly have the equality
$a+\sum_{\beta\neq\alpha}c_{\beta}=a+\sum_{k\in I\setminus J}=d=|I|$, and as
$b_{k}\geq 1$ for each $k\in I\setminus J_{\alpha}$ we get
$c_{\beta}\geq|J_{\beta}|$ for every $\beta\neq\alpha$. The limit of the
$\alpha$th part of $D_{I}\phi$ is thus indeed the sum over $(a,\vec{c})\in
B_{\vec{J},\alpha}$ of the corresponding function (with the asserted sign),
times a coefficient counting how many elements $(a,\vec{b})\in A_{I\setminus
J_{\alpha},|J_{\alpha}|-1}$ that become our element $(a,\vec{c})\in
B_{\vec{J},\alpha}$ under the map taking such an element of $A_{I\setminus
J_{\alpha},|J_{\alpha}|-1}$ to that of $B_{\vec{J},\alpha}$ having the same
$a$ and for which $c_{\beta}$ is $\sum_{k\in J_{\alpha}}b_{k}$. But as the
$b_{k}$’s are positive and free up to the sum conditions, we can consider the
question for each $\beta$ separately, and by subtracting 1 from each we need
the number of options to add $|J_{\beta}|$ ordered non-vanishing integers and
get $c_{\beta}-|J_{\beta}|$, which is $\binom{c_{\beta}-1}{|J_{\beta}|-1}$ by
a standard exercise in Discrete Mathematics (put $|J_{\beta}|-1$ sticks
between $c_{\beta}-|J_{\beta}|$ balls to represent a solution, and count the
options to do this). This proves the proposition. ∎

Using Proposition 26, we can obtain the form of the action of our operators
$D_{d}$ from Proposition 5 and Theorem 8, and with them of the derivatives
from Corollary 10, at any point in $\mathbb{R}^{N}$. Take an index
$1\leq\alpha\leq M$, and assume that we are given integers $c_{\beta}\geq 0$
for all $\beta\neq\alpha$. We then denote by $\Xi_{\alpha,\vec{c}}$ the set of
all integers $\\{\kappa_{\beta}\\}_{\beta\neq\alpha}$ such that
$\kappa_{\beta}=0$ when $c_{\beta}=0$ and $1\leq\kappa_{\beta}\leq c_{\beta}$
if $c_{\beta}>0$.

######  Proposition 27.

Let now the full set $\mathbb{N}_{N}$ of indices be the disjoint union
$\bigcup_{\alpha=1}^{M}H_{\alpha}$ of non-empty sets, and take a point where
$x_{i}=y_{\alpha}$ for any variable $x_{i}$ associated with $i\in H_{\alpha}$,
and that $y_{\alpha}\neq y_{\beta}$ when $\alpha\neq\beta$. Take some $1\leq
d\leq N$, and then we have the equality $D_{d}\phi$ is $d!$ times the sum over
$1\leq\alpha\leq M$ of

$D_{d}\phi=d!\sum_{\alpha=1}^{M}\sum_{\sum_{\beta\neq\alpha}c_{\beta}<d}C_{\alpha,\vec{c}}\frac{\partial_{H_{\alpha}}^{d-\sum_{\beta\neq\alpha}c_{\beta}}\phi}{\prod_{\beta\neq\alpha}(y_{\beta}-y_{\alpha})^{c_{\beta}}},$

where the combinatorial multipliers are given by

$C_{\alpha,\vec{c}}:=\sum_{\vec{\kappa}\in\Xi_{\alpha,\vec{c}}}(-1)^{\sum_{\beta\neq\alpha}(c_{\beta}-\kappa_{\beta})}\binom{|H_{\alpha}|}{d-\sum_{\beta\neq\alpha}\kappa_{\beta}}\prod_{\begin{subarray}{c}\beta\neq\alpha\\\
c_{\beta}>0\end{subarray}}\binom{c_{\beta}-1}{\kappa_{\beta}-1}\binom{|H_{\beta}|}{\kappa_{\beta}}.$

###### Proof.

Recall that $D_{d}\phi$ is $d!$ times the sum of $D_{I}\phi$ for subsets
$I\subseteq\mathbb{N}_{N}$ of size $d$. Take such a set $I$, set
$J_{\alpha}:=I\cap H_{\alpha}$ for every $\alpha$ (some of which might be
empty), and Proposition 26 describes $D_{I}\phi$ in terms of the sets
$\\{J_{\alpha}\\}_{\alpha=1}^{M}$ (ignoring the empty ones). Now, the values
$y_{\alpha}$ and $y_{\beta}$ are the same from those coming from $H_{\alpha}$
and $H_{\beta}$, and the derivative $\partial_{J_{\alpha}}^{a}\phi$ equals
$\partial_{H_{\alpha}}^{a}\phi$, so that the only remaining dependence on the
sets $\\{J_{\alpha}\\}_{\alpha=1}^{M}$ are through its cardinalities.
Therefore $D_{d}\phi$ is indeed a combination of the expressions
$\partial_{H_{\alpha}}^{a}\phi\big{/}\prod_{\beta\neq\alpha}(y_{\beta}-y_{\alpha})^{c_{\beta}}$,
where the equality $a+\sum_{\beta\neq\alpha}c_{\beta}=d$ holds, and where
$\partial_{H_{\alpha}}^{a}\phi$ shows up in a term from $D_{I}\phi$ if and
only if $J_{\alpha}\neq\emptyset$, while in this case $y_{\beta}-y_{\alpha}$
appears with a non-trivial power if and only if $J_{\beta}\neq\emptyset$ as
well.

This means that for a given term, the sets $I$ for which $D_{I}\phi$
contributes to the coefficient of said term are those in which the sets
$J_{\beta}$ that are non-empty are precisely those with $\beta=\alpha$ or with
$y_{\beta}-y_{\alpha}$ showing up in the denominator, and by the definition of
$B_{\vec{J},\alpha}$ in that proposition, we must have $|J_{\beta}|\leq
c_{\beta}$ for every $\beta\neq\alpha$ as well as
$a+\sum_{\beta\neq\alpha}=d=|I|=\sum_{\beta=1}^{M}|J_{\beta}|$. The inequality
$|J_{\alpha}|\geq a$, required in Proposition 26, indeed follows, and the
exponent $|J_{\alpha}|-a$ in the sign from that proposition equals
$\sum_{\beta\neq\alpha}(c_{\beta}-|J_{\beta}|)$. Therefore the cardinalities
$\kappa_{\beta}$ of $J_{\beta}$ for all $\beta\neq\alpha$ (including the
indices of empty sets) lie in the corresponding set $\Xi_{\alpha,\vec{c}}$,
and for an element of this set the total contribution to the coefficient
$C_{\alpha,\vec{c}}$ is the asserted sign times the product of the binomial
coefficients $\binom{c_{\beta}-1}{\kappa_{\beta}-1}$ from Proposition 26,
times the number of options to choose $J_{\beta}$ from $H_{\beta}$ for every
$\beta$ (including $\alpha$) in order to form $I$. As the choices for
different $\beta$’s are independent, and there are
$\binom{|H_{\beta}|}{\kappa_{\beta}}$ such options for every $\beta\neq\alpha$
(which means one option when $\kappa_{\beta}=0$, and we can discard those in
the product), and the remaining binomial coefficient represents the number of
choices of $J_{\alpha}$, of size $d-\sum_{\beta\neq\alpha}\kappa_{\beta}$,
from $H_{\alpha}$, this is indeed the required contribution. Thus
$C_{\alpha,\vec{c}}$ is the sum of those expressions over elements of
$\Xi_{\alpha,\vec{c}}$, and we have the external multiplier $d!$ in the first
equation. This proves the proposition. ∎

As for the combinatorial coefficients from Proposition 27, note that the set
$\Xi_{a,\vec{c}}^{\alpha}$ is a singleton if $c_{\beta}\leq 1$ for every
$\beta$, but not otherwise. In the former case the unique element is with
$\kappa_{\beta}=c_{\beta}$ for all $\beta$ and hence
$d-\sum_{\beta\neq\alpha}\kappa_{\beta}=a$, so that the coefficient
$C_{a,\vec{c}}$ equals just
$\binom{|H_{\alpha}|}{a}\prod_{\\{\beta|c_{\beta}=1\\}}|H_{\beta}|$ (this is,
in particular, the case when $a=d$ and all the $c_{\beta}$’s vanish, where
$\partial_{H_{\alpha}}^{a}\phi$ simply comes multiplied by
$\binom{|H_{\alpha}|}{d}$). However, even in the next simplest case, where
$c_{\gamma}=2$ and the other $c_{\beta}$’s are 0 and 1, the set
$\Xi_{a,\vec{c}}^{\alpha}$ contains two elements, and $C_{a,\vec{c}}$ takes
the value
$\big{[}\binom{|H_{\alpha}|}{a}\binom{|H_{\gamma}|}{2}-\binom{|H_{\alpha}|}{a+1}|H_{\gamma}|\big{]}\prod_{\\{\beta|c_{\beta}=1\\}}|H_{\beta}|$,
and the expression in parentheses expands as
$\binom{|H_{\alpha}|}{a}|H_{\gamma}|\cdot\frac{(a+1)|H_{\gamma}|-2|H_{\alpha}|+(a-1)}{2a+2}$,
which does not seem to simplify in any reasonable manner. When the
$c_{\beta}$’s have larger values these coefficients are, of course, more
complicated. Therefore in the generality that Proposition 27 is given we can
write $D_{1}\phi$ as
$\sum_{\alpha=1}^{M}|H_{\alpha}|\partial_{H_{\alpha}}^{1}\phi$, and we have

$D_{2}\phi=2\sum_{\alpha=1}^{M}\sum_{\beta\neq\alpha}|H_{\alpha}|\cdot|H_{\beta}|\cdot\frac{\partial_{H_{\alpha}}^{1}\phi}{y_{\beta}-y_{\alpha}}+\sum_{\alpha=1}^{M}|H_{\alpha}|(|H_{\alpha}|-1)\partial_{H_{\alpha}}^{2}\phi,$

but $D_{3}$ already contains coefficients like
$3|H_{\alpha}|\cdot|H_{\beta}|(|H_{\alpha}|-|H_{\beta}|)$ in front of
$\partial_{H_{\alpha}}^{1}\phi/(y_{\beta}-y_{\alpha})^{2}$. However, since the
derivatives are local, at a given point we can find coordinates that respect
only the partial symmetry existing around this point, which will produce a
very simple form of all the derivatives of the function $\phi$ near that
point—see Theorem 30 below.

For finding these coordinates, we first consider the special case of interest
in Proposition 27 where the point lies along the total diagonal. Then we get
the following consequence, which we phrase in terms of the normalized
differential operators $\hat{D}_{r}$ from Remark 17.

######  Corollary 28.

At the point in the total diagonal of $\mathbb{R}^{N}$ where all the variables
$x_{i}$, $i\in\mathbb{N}_{N}$ attain the same value $a$, the operator
$\hat{D}_{d}$ from Remark 17 takes $\phi$ to
$\partial_{\mathbb{N}_{N}}^{d}\phi$. When $\phi$ is the trace function taking
$(x_{1},\ldots,x_{N})$ to $\sum_{i\in\mathbb{N}_{N}}f(x_{i})$ for some
function $f$, the value of $\hat{D}_{d}\phi$ is
$\frac{(-1)^{d-1}}{(d-1)!}f^{(d)}(a)$.

###### Proof.

We apply Proposition 27 for the partition of $\mathbb{N}_{N}$ into one single
set $H_{\alpha}=\mathbb{N}_{N}$. Then there are no $\beta\neq\alpha$, meaning
that for the unique value of $\alpha$, we only get a multiple of
$\partial_{\mathbb{N}_{N}}^{d}\phi$ in the formula for $D_{d}\phi$. Since the
single set $H_{\alpha}$ is of size $N$, the multiplying coefficient
$C_{d,\emptyset}$ reduces to $\binom{N}{d}$, and we have the multiplier $d!$.
As the normalization from Remark 17 gives
$\hat{D}_{d}=\frac{(N-d)!D_{d}}{N!}$, the two multipliers cancel and the first
assertion follows. When $\phi$ is a trace function, all the derivatives
$\partial_{\mathbb{N}_{N}}^{\sigma}\phi$ for $\sigma\vdash d$ with
$l(\sigma)\geq 2$ vanish, so that the formula for
$\partial_{\mathbb{N}_{N}}^{d}\phi$ in the definition in Theorem 21 reduces to
a single term, with the coefficient $(-1)^{d-1}/(d-1)!$. As the derivative
itself is the pure one of order $d$ in some variable $x_{i}$, its value at our
total diagonal point is $f^{(d)}(a)$. This proves the corollary. ∎

The simple formulae from Corollary 28 do not extend to compositions—see Remark
31 below.

######  Remark 29.

There might be different normalizations for the $u_{r}$’s and the $D_{d}$’s
that might produce neater results in some situations. For example, if we
replace $\hat{u}_{r}$ by $(-1)^{r-1}r\hat{u}_{r}$, then the limit from
Proposition 17 becomes simply $p_{r}$, and the coefficients in Theorem 8 will
be integers, and co-prime ones (this is related to the Newton identities
mentioned in the proof of Proposition 17, via Remark 9). Then the $g$th dual
operator which will now be $\frac{(-1)^{g-1}}{g}\hat{D}_{g}$, will be defined
along the total diagonal as in Theorem 28 by the formula from Theorem 21 but
without the $g$ in the numerator and with the sign $(-1)^{l(\sigma)-1}$, and
the image of as trace function arising from $f$ under the $d$th operator will
simply be the Taylor coefficient $f^{(d)}(a)/d!$. Alternatively, one could
have taken $\frac{(-1)^{r-1}}{(r-1)!}\hat{u}_{r}$, so that the operator
$(-1)^{d-1}(d-1)!\hat{D}_{d}$ will take a trace function along the total
diagonal just to the $d$th derivative of the basic univariate function.
However, the normalization that we initially chose for $D_{d}$ is the one with
which the basic calculation in Corollary 6 involves no coefficients, and
$u_{r}$ are the dual coordinates in this normalization.

At a generic point in $\mathbb{R}^{N}$ the variables take different values.
Therefore at a small neighborhood of that point one can distinguish the
variables according to their values, and at such a point a much simpler way to
obtaining a full set of $N$ derivatives at the point would simply be the
ordinary partial ones with respect to the initial variables (i.e., the
derivative obtained by fixing all the variables but one), regardless of the
symmetry property. This was seen (e.g., in Lemma 20) not to give enough
derivatives when some values coincide, but merging the results that we got
with the latter simple observation gives the following, simpler choice of
coordinates around any given point in $\mathbb{R}^{N}$.

######  Theorem 30.

Take some point in $\mathbb{R}^{N}$, and define the partition of
$\mathbb{N}_{N}$ into sets $\\{H_{\alpha}\\}_{\alpha=1}^{M}$ such that
$x_{i}=y_{\alpha}$ wherever $i\in H_{\alpha}$ and $y_{\alpha}\neq y_{\beta}$
when $\alpha\neq\beta$ as in Proposition 27. For each such $\alpha$ we define
$|H_{\alpha}|$ coordinates $\hat{u}_{r}^{|H_{\alpha}|}(x_{H_{\alpha}})$, with
the variables being taken only from $H_{\alpha}$, and $\hat{u}_{r}$ being the
functions from Proposition 17, with the normalizing coefficient there, as well
as those in the expressions $\tilde{e}_{\lambda}$ showing up in Theorem 10 for
these variables, naturally being based on $|H_{\alpha}|$ rather than the full
number $N$ of variables. Then, expressing any function $\phi$ which is
differentiable enough times around our point using the set of coordinates
$\big{\\{}\hat{u}_{r}^{|H_{\alpha}|}(x_{H_{\alpha}})\big{\\}}_{1\leq\alpha\leq
M,\ 1\leq r\leq|H_{\alpha}|}$, the respective derivatives of our representing
function of these coordinates are
$\\{\partial^{d}_{H_{\alpha}}\phi\\}_{1\leq\alpha\leq M,\ 1\leq
d\leq|H_{\alpha}|}$. If $\phi$ is the trace function arising from the
univariate function $f$ then for each $\alpha$ these derivatives are
$\\{(-1)^{d-1}f^{(d)}(y_{\alpha})/(d-1)!\\}_{d=1}^{|H_{\alpha}|}$.

Of course, replacing $\hat{u}_{r}^{|H_{\alpha}|}(x_{H_{\alpha}})$ by some
other normalization, like the ones from Remark 29, one can obtain, for a trace
function, the Taylor coefficients $f^{d}(y_{\alpha})/d!$, or just the
derivatives $f^{(d)}(y_{\alpha})$ themselves, for every $1\leq
d\leq|H_{\alpha}|$ (and every $1\leq\alpha\leq M$). The coordinates used in
Lemma 2.4 of [EKZ] are, in some other scalar normalization, the ones arising
from Theorem 30 when the sizes from that theorem being either 1 or 2.

###### Proof.

At any point in a small enough neighborhood of our point, we can distinguish
the variables associated with $i\in H_{\alpha}$ for any $1\leq\alpha\leq M$ as
those whose values are close enough to $y_{\alpha}$. We thus take such
$\alpha$, fix all the variables $x_{i}$ with $i\not\in H_{\alpha}$ as the
corresponding value $y_{\beta}$ (with $\beta\neq\alpha$), and consider the
resulting function of $\\{x_{i}\\}_{i\in H_{\alpha}}$. It is a symmetric
function of these variables, considered around the total diagonal point where
they all equal $y_{\alpha}$. We therefore express this function using the
coordinates
$\big{\\{}\hat{u}_{r}^{|H_{\alpha}|}(x_{H_{\alpha}})\big{\\}}_{r=1}^{|H_{\alpha}|}$,
and the respective derivatives are
$\\{\partial^{d}_{H_{\alpha}}\phi\\}_{d=1}^{|H_{\alpha}|}$ (or the asserted
derivatives of $f$ at $y_{\alpha}$ alone in case $\phi$ is the trace function
associated with $f$) by Theorem 28.

Now, this description of the derivatives of our function of $|H_{\alpha}|$
variables give derivatives of our function representing $\phi$ as long as the
other coordinates that we take for $\phi$ do not involve the variables
$\\{x_{i}\\}_{i\in H_{\alpha}}$. As our set of coordinates has the property
that a coordinate with index $\alpha$ is only based on the variables
$\\{x_{i}\\}_{i\in H_{\alpha}}$, gathering these coordinates together indeed
gives a set of coordinates for $\phi$ in which the total set of derivatives is
the union of the ones arising from each $\alpha$ separately. As this yields
the asserted formulae (also in the trace function case), this proves the
theorem. ∎

We now complete a missing proof from above.

###### Proof of Theorem 12.

We already saw that $\\{u_{r}\\}_{r=1}^{N}$ generate, over $\mathbb{Q}$, all
the symmetric functions in $\\{x_{i}\\}_{i\in\mathbb{N}_{N}}$. It follows, by
homogeneity, that we can write every $v_{r}$ as a polynomial $p_{r}$ in
$\\{u_{d}\\}_{d=1}^{r}$. Moreover, $p_{r}$ is of the form $c_{r}u_{r}$ plus a
polynomial in $\\{u_{d}\\}_{d=1}^{r-1}$ (homogeneity again), and for the
$v_{r}$’s to be coordinates, i.e., algebraically independent, we must have
$c_{r}\neq 0$ for every $r$.

We now write $D_{d}\phi$ as
$\partial_{u_{d}}\varphi(u_{1},\ldots,u_{N})=\partial_{u_{d}}\eta(v_{1},\ldots,v_{N})$
via Corollary 10 and the fact that $\varphi(u_{1},\ldots,u_{N})$ and
$\eta(v_{1},\ldots,v_{N})$ both represent the same expression
$\phi(x_{1},\ldots,x_{N})$, and then the chain rule expresses the latter
derivative as $\sum_{r=d}^{N}\frac{\partial p_{r}}{u_{d}}\cdot\eta_{v_{r}}$,
with the summation starting from $r=d$ since $v_{r}=p_{r}(u_{1},...,u_{r})$
does not depend on $u_{d}$ with $d>r$. Moreover, $\frac{\partial
p_{d}}{u_{d}}$ is the scalar $c_{d}\neq 0$, meaning that the column vector
with entries $\\{D_{d}\phi\\}_{d=1}^{N}$ is obtained from
$\\{\eta_{v_{r}}\\}_{r=1}^{N}$ via multiplication by an invertible upper-
triangular matrix $M$. Moreover, the entry $M_{dr}$ of $M$, with $r\geq d$, is
the derivative $\frac{\partial p_{r}}{u_{d}}$, which is homogenous of degree
$r-d$ in $\\{x_{i}\\}_{i\in\mathbb{N}_{N}}$.

Thus, to obtain the derivatives $\eta_{v_{r}}$, $1\ \leq r\leq N$ themselves,
we multiply the vector of the $D_{d}\phi$’s by the inverse matrix $M^{-1}$.
The degree $r-d$ homogeneity of $M_{dr}$ implies that the entry $M^{-1}_{rd}$
of $M^{-1}$, with $d\geq r$, is homogenous of degree $d-r$ (as so are all the
combinations of products of entries of $M$ that go into this inverse entry).
Writing the symmetric polynomial appearing in $M^{-1}_{rd}$ as
$q_{d,r}(x_{1},\ldots,x_{n})$, we deduce that
$\eta_{v_{r}}=\sum_{d=r}^{N}q_{d,r}\cdot D_{d}\phi$, where $q_{d,d}$ is the
constant $\frac{1}{c_{d}}$.

Now, the function $q_{d,r}$ is symmetric of degree $d-r$, and when it
multiplies $D_{d}$, the statement that $\eta_{v_{r}}$ has no non-constant
numerator means that at every point in $\mathbb{R}^{N}$, and in every term in
the presentation of $D_{d}$ at that point, the denominator from that
presentation of $D_{d}$ has to cancel with the form that $q_{d,r}$ takes at
that point. For every $0\leq s\leq N-1$ consider the set $Y_{s}$ where all the
variables $x_{i}$ with $i>s$ have to take the same value, and the variables
$x_{i}$ with $1\leq i\leq s$ are free. It is clear that $Y_{s}\subseteq
Y_{s+1}$, that $Y_{0}$ is the total diagonal, and $Y_{N-1}$ is the full space.
Assume that $q_{d,r}\neq 0$ for some $r$ and $d>r$, take $s$ to be minimal
such that some such $q_{d,r}$ does not vanish identically on $Y_{p}$ (the
existence of such a $s$ is clear), and let $d$ be maximal such that $q_{d,r}$
does not vanish identically on this $Y_{s}$ for some $r<d$. For such an $r$
(yielding $s$ and $d$) we shall consider the expression for $\eta_{v_{r}}$ at
a point in $Y_{s}$ such that all the variables $x_{i}$ with $1\leq i\leq s+1$
take distinct values, meaning that in the partition from Proposition 27 we can
take $M=s+1$, with $H_{i}$ being the singleton $\\{i\\}$ for $1\leq i\leq s$,
and $H_{s+1}$ containing all the other indices from $\mathbb{N}_{N}$, and is
thus of size $N-s$. We also have $y_{i}=x_{i}$ for all $1\leq i\leq s$, and we
denote by $y$ the common value $y_{s+1}$ of all $x_{i}$ with $i\in H_{s}$,
i.e., $i>s$.

We will be interested in the summands associated with $\alpha=s+1$ in
Proposition 27. Since $|H_{\beta}|=1$ for every $\beta\neq\alpha$, we only
obtain contributions to $C_{\alpha,\vec{c}}$ from the element of
$\Xi_{\alpha,\vec{c}}$ in which $\kappa_{\beta}=1$ wherever $c_{\beta}>0$ (and
$\kappa_{\beta}=0$ when $c_{\beta}=0$ by definition). Assuming that
$|\\{1\leq\beta\leq s|c_{\beta}>0\\}|$ is some number $1\leq t\leq s$, this
coefficient is a sign times $\binom{N-s}{d-t}$. For maximizing the order of
the differential operator, we consider the summands in which every positive
$c_{\beta}$ equals 1 (yielding a positive $C_{\alpha,\vec{c}}$ multiplying
$\partial_{H_{s}}^{d-t}\phi$ divided by a product of distinct linear terms),
and we also concentrate on the summands in which $t$ is minimal (but
$C_{\alpha,\vec{c}}\neq 0$).

Now, if $d\leq N-s$ then we can take $t=0$, and obtain a non-trivial numerator
times a multiple of $\partial_{H_{s}}^{d}\phi$, contradicting our assumption
on such numerators at every point. This covers the case where $s=0$. Otherwise
we have $d=N-s+t$ for some $1\leq t\leq s$, we take this value of $t$ (with
$d-t=N-s$), and we find that the numerator $q_{d,r}$ has to cancel to a
constant when divided by any product of $t$ of the expressions $x_{i}-y$ for
$1\leq i\leq s$. But in the polynomial ring in $x_{i}$, $1\leq i\leq s$ (for
fixed $y$) this means that $q_{d,r}$ is divisible by all this linear
expressions, hence by their product, and thus if $t<s$ then yet again we
obtain $\partial_{H_{s}}^{d-t}\phi=\partial_{H_{s}}^{N-s}\phi$ times a non-
trivial numerator. It remains to consider the case where $t=s$, i.e., $d=N$,
and then when we divide $q_{d,r}$ by the product $\prod_{i=1}^{s}(x_{i}-y)$
then we must get a constant. This determines the degree $d-r$ of $q_{d,r}$ to
be $s$, and as $d=N$ this can only happen for $t=N-s$.

This function $q_{N,N-s}$ is symmetric in $N$ variables and homogenous of
degree $s$, but reduces to $\prod_{i=1}^{s}(x_{i}-y)$ when $x_{i}=y$ for all
$i>s$. But this function contains the monomial $\prod_{i=1}^{s}x_{i}$ with the
coefficient 1, and the monomials $\prod_{i=1}^{s-1}x_{i}\cdot x_{j}$ with
$j>s$ with coefficients that sum, over $j>s$, to $-1$. As this function cannot
be symmetric in all $N$ variables, we reach a contradiction, meaning that
$q_{d,r}$ must vanish for every $d>r$ under the hypothesis of no non-constant
numerators. This means that our matrix $M^{-1}$, and with it $M$, are
diagonal, and therefore the polynomial $p_{r}$ does not depend on $u_{d}$ for
any $d<r$. But this implies that $v_{r}=c_{r}u_{r}$, and as the reciprocal
$\frac{1}{c_{r}}$ on the diagonal is the coefficient with which $D_{r}\phi$
shows up in $\eta_{v_{r}}$, our assumption on that coefficient yields
$c_{r}=1$ for every $1\leq r\leq N$. This implies that $v_{r}=u_{r}$ for every
such $r$ as desired. This completes the proof of the theorem. ∎

######  Remark 31.

One may ask about composing the operators $D_{d}$ along the total diagonal,
for obtaining, for example a shorter proof for Theorem 13. Indeed, we can
define compositions $D_{\lambda}=\prod_{h=1}^{d}D_{h}^{m_{h}}$ for some
partition $\lambda\vdash d$ (with $m_{h}$ being the multiplicity of $h$ in
$\lambda$ as always), and they are all linearly independent by Corollary 11.
Then applying such a $D_{\lambda}$ to $u_{r}$ when $d<r$ would annihilate it
(just use some $D_{h}$ with $m_{h}>0$ and the fact that $D_{h}u_{r}=0$ since
$h<r$), and in case one could replace, along the total diagonal, every $D_{h}$
by the formula from Theorem 28, then the $D_{\lambda}$’s for $\lambda\vdash d$
would have given linearly independent operators of order $d$ there, using
which the proof of Theorem 13 would become much easier. To see that this
argument is wrong, we consider the case $d=r$, and Proposition 17. Such an
argument would have implied that the only such derivative $D_{\lambda}$ that
does not annihilate $u_{r}$ would have been the one with $l(\lambda)=1$, and
thus the derivatives from that proposition, which are constants and therefore
can be evaluated, in particular, along the total diagonal, would be based on
the coefficients in the presentation of these operators in terms of the
$D_{\lambda}$’s, which do not depend on $N$. The much finer dependence on $N$
of the values from the proof of that proposition exemplifies that this is not
the case, and an argument using the compositions of the derivatives from
Theorem 28 for compositions of the $D_{d}$’s is not valid. The reason is that
for compositions of differential operators, we need their formulae on full
neighborhoods, and the formulae from Theorem 28 are valid only at total
diagonal points (indeed, comparing the values of the higher-order derivatives
from that theorem with those of $D_{d}$ on general symmetric functions
produces different values, due to the latter operating via Leibniz’ Rule on
products, which the former involve more expressions).

We conclude with a question about more general settings. Our coordinates are
those for symmetric powers of the affine line, which are the invariant spaces
of the action of $S^{N}$ on the usual $N$th power of the affine line. One may
ask for other group actions on affine spaces, and good coordinates for
differentiation on their spaces of invariants. Note that in many natural
examples, like the symmetric powers of an affine space of dimension $M>1$
(where $S_{N}$ acts on a space of dimension $MN$), or the alternating group
$A_{N}$ acting on such a space (now with $M\geq 1$), the quotient spaces tend
to be singular, including the GIT quotients (since the algebra of invariants
is no longer a polynomial ring, like Theorem 1 establishes for the case
considered in this paper). We leave such questions for further research.

## References

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  * [M] MacDonald, I. G., Symmetric Functions and Hall Polynomials (second edition), Oxford Mathematical Monographs, Oxford Science Publications, x+475pp (1995).

Einstein Institute of Mathematics, the Hebrew University of Jerusalem, Edmund
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