where $\Delta:\hat{\Lambda}\to\mathaccent 866{\Lambda^{\otimes 2}}$ is the
coproduct, characterized by $\Delta(p_{i})=1\otimes p_{i}+p_{i}\otimes 1$ for
$i\in\mathbb{N}$. These compatibilities are standard for ordinary
$\mathrm{S}_{n}$ representations: for a discussion of their extension to chain
complexes see for instance [10, §5].

####  8.2. Character computation

To translate Propostion 8.4 into a statement about Frobenius characters, we
need the following.

######  Proposition 8.7 (Stanley [27]).

Let $\mu:\mathbb{N}_{\geq 1}\to\\{1,0,-1\\}$ be the arithmetic Möbius
function, then

$\operatorname{ch}(\mathcal{W})=w:=\sum_{d\geq
1}\frac{\mu(d)}{d}\log(1+p_{d}).$

The element $w$ is the plethysistic inverse of $\sum_{n\geq 1}h_{n}$.

######  Definition 8.8.

We write $\partial_{n}$ for the differential operator on
$\frac{\partial}{\partial p_{n}}:\hat{\Lambda}\to\hat{\Lambda}$, given by
expressing $s\in\hat{\Lambda}$ as a power series in the $p_{i}$ and
differentiating with respect to $p_{n}$. We define

$D_{n}:=\sum_{d|n}\frac{\mu(d)}{d}\partial_{n/d},$

where $\mu:\mathbb{N}\to\\{+1,0,-1\\}$ is the arithmetic Möbius function. To
an integer partition $\lambda=1^{m_{1}}2^{m_{2}}\dots$ we associate a
differential operator, which is a polynomial in the $D_{i}$ defined by

${D\choose\lambda}:=\prod_{i}{D_{i}\choose m_{i}}.$

The main theorem of this section describes
$\operatorname{ch}(\mathrm{K}_{d}(M))$ in terms of $\operatorname{ch}(M)$ and
the differential operators ${D\choose\lambda}$.

######  Theorem 8.9.

Let $M$ be a chain complex of ${\mathbf{FS}}{{}^{\mathrm{op}}}$ modules which
is bounded and finite dimensional in each degree. Let $d\in\mathbb{N},d\geq
1$. Then there is an identity in $\mathaccent 1371{\Lambda\otimes\Lambda}$

$\mathrm{c}\mathrm{h}(\mathrm{K}_{d}(M))=\sum_{\lambda\vdash
d}{D\choose\lambda}\mathrm{c}\mathrm{h}(M)\otimes p_{\lambda}.$

###### Proof.

In terms of symmetric functions Proposition 8.4 implies that

$\sum_{d}\mathrm{c}\mathrm{h}(\mathrm{K}_{d}(M))=\Delta(\operatorname{ch}(M))[w]_{2}.$

The brackets $[-]_{2}$ denote plethysm in the second factor.

To calculate this expression, note that since $\Delta(p_{i})=p_{i}\otimes
1+1\otimes p_{i}$ for any $s\in\hat{\Lambda}$, expanding $s$ as a power series
in the $p_{\lambda}$ we have

$\Delta(s)=\sum_{\lambda\text{ integer partition
}}\frac{\partial_{\lambda}(s)}{\lambda!}\otimes p_{\lambda},$

where if $\lambda=1^{m_{1}}2^{m_{2}}\dots$ then
$\partial_{\lambda}:=\prod_{i}\partial_{i}^{m_{i}}$. Therefore we have

$\Delta(\operatorname{ch}(M))=\sum_{\lambda\text{ integer partition
}}\frac{\partial_{\lambda}(\operatorname{ch}(M))}{\lambda!}\otimes
p_{\lambda}.$

Since $p_{\lambda}[w]=\prod_{i}(w[p_{i}])^{m_{i}}$ and $w[p_{i}]=\sum_{d\geq
1}\frac{\mu(d)}{d}\log(1+p_{id})$ we have

$\sum_{d\geq
1}\mathrm{c}\mathrm{h}(\mathrm{K}_{d}(M))=\sum_{\lambda=1^{m_{1}}2^{m_{2}}\dots}~{}\prod_{i\geq
1}\frac{\partial_{i}^{m_{i}}(\operatorname{ch}(M))}{m_{i}!}\otimes\prod_{i\geq
1}\left(\sum_{d\geq 1}\frac{\mu(d)}{d}\log(1+p_{id})\right)^{m_{i}}.$

To rearrange this sum, we introduce formal variables $t_{i}$ which commute
with $p_{i}$ and consider the formal expression

$\sum_{\lambda=1^{m_{1}}2^{m_{2}}\dots}~{}\prod_{i\geq
1}\frac{t_{i}^{m_{i}}}{m_{i}!}\left(\sum_{d\geq
1}\frac{\mu(d)}{d}\log(1+p_{id})\right)^{m_{i}}=\exp\left(\sum_{i}t_{i}\left(\sum_{d}\frac{\mu(d)}{d}\log(1+p_{id})\right)\right)$
$=\exp\left(\sum_{n\geq
1}\log(1+p_{n})\sum_{d|n}\frac{\mu(d)}{d}t_{n/d}\right)=\prod_{n\geq
1}(1+p_{n})^{T_{n}}$

where $T_{n}:=\sum_{d|n}\frac{\mu(d)}{d}t_{n/d}$ and the final expression is
interpreted as a power series via the binomial theorem
$(1+p_{n})^{T_{n}}=\sum_{k}{T_{n}\choose k}p_{n}^{k}$. Expanding the product,
we see that this expression equals

$\sum_{\lambda=1^{m_{1}}2^{m_{2}}\dots}p_{\lambda}\prod_{i}{T_{i}\choose
m_{i}}.$

Now, we may interpret the formal variables $t_{i}$ (resp. $p_{i}$) as the
operators $\mathaccent 1371{\Lambda\otimes\Lambda}\to\mathaccent
1371{\Lambda\otimes\Lambda}$ defined by $a\otimes
b\mapsto\partial_{i}(a)\otimes b$ (resp. $a\otimes b\mapsto a\otimes p_{i}b$),
because these operators commute with each other. Applying the operator defined
by our formal expression in $p_{i},t_{i}$ to $\operatorname{ch}(M)\otimes 1,$
we obtain an equality

$\sum_{d}\mathrm{c}\mathrm{h}(\mathrm{K}_{d}(M))=\sum_{\lambda}{D\choose\lambda}\operatorname{ch}(M)\otimes
p_{\lambda}.$

Equating degree $d$ pieces in the second factor, we obtain the desired result.
∎

###  9\. Differential equations and characters of bounded type

In this section, motivated by Theorem 8.9, we first describe the solution
space to a system of differential equations involving ${D\choose\lambda}$.
Then we introduce a notion of _type_ of ${\mathbf{FS}}{{}^{\mathrm{op}}}$
module and rephrase the results so far in terms of the type. We also prove
Theorem 1.11.

####  9.1. Solution space to a system differential equations

It is convenient to introduce generators of $\hat{\Lambda}$ adapted to
$D_{i}$.

######  Definition 9.1.

For $n\in\mathbb{N}$ define

$y_{n}:=\sum_{k\geq 1}\frac{p_{nk}}{k}=\sum_{i}-\log(1-x_{i}^{n}).$

The elements $y_{n}$ satisfy several elementary identities: for example
$y_{1}=\log(\sum_{n\geq 0}h_{n})$ and $y_{n}=y_{1}[p_{n}]$. The most important
one follows from Möbius inversion:

######  Proposition 9.2.

For all $n,m\geq 1$ we have $D_{n}(y_{m})=\delta(n,m),$ where $\delta(n,m)$ is
the Kronecker delta.

###### Proof.

We have

$y_{m}=\sum_{e,~{}m|e}(m/e)p_{e}.$

So

$D_{n}(y_{m})=\sum_{d|n}\frac{\mu(n/d)}{n/d}\partial_{d}\sum_{e,~{}m|e}(m/e)p_{e}=\sum_{e,~{}m|e|n}\frac{\mu(n/e)}{(n/e)}(m/e).$

Factoring out $m/n$ and applying Möbius inversion for the poset
$(\mathbb{N}_{\geq 1},|)$ , we see that this sum is $1$ if $n=m$ and zero
otherwise. ∎

A similar Möbius inversion to Proposition 9.2 yields

$p_{n}=\sum_{d\geq 1}\frac{\mu(d)}{d}y_{nd}.$

######  Definition 9.3.

Let $(\hat{\Lambda})^{\rm pow}_{i}$ denote the closure of the span of the
degree $i$ monomials in the power sum symmetric functions. Thus an arbitrary
element of $(\hat{\Lambda})^{\rm pow}_{i}$ takes the form

$\sum_{(m_{1},m_{2},\dots),\sum
m_{j}=i}a_{(m_{1},\dots)}\prod_{j}p_{j}^{m_{j}},$

for some choice of $a_{(m_{1},\dots)}\in\mathbb{Q}$. We also write
$(\hat{\Lambda})^{\rm pow}_{\leq i}:=\bigoplus_{j\leq i}(\hat{\Lambda})^{\rm
pow}_{j}$.

Our main theorem of this section describes the space of solutions to a system
of differential equations.

######  Theorem 9.4.

Let $j_{1}\geq j_{2}\geq\dots\geq j_{r}\in\mathbb{N}$. Then the space of
solutions in $\hat{\Lambda}$ to the system of linear differential equations

(9.5) ${D\choose\lambda_{1}}\dots D\choose\lambda_{r}s=0,\text{ for all
}(\lambda_{1},\dots,\lambda_{r})\text{ integer partitions such that
}|\lambda_{t}|\geq j_{t},$

is the subspace

$\sum_{t=1}^{r}(\hat{\Lambda})^{\rm power}_{t-1}\left\\{\exp(\sum_{i\geq
1}a_{i}y_{i})~{}\bigg{|}~{}A=1^{a_{1}}2^{a_{2}}\dots,~{}|A|<j_{t}\right\\}.$

######  Example 9.6.

Let $r=2$, $j_{1}=3$ and $j_{2}=2$. Then Theorem 9.4 states that the space of
solutions to (9.5) is the closure of the span of

$\\{1,\exp(y_{1}),\exp(y_{2}),\exp(2y_{1})\\}\cup\\{p_{k},p_{k}\exp(y_{1})\\}_{k\in\mathbb{N}}.$

The proof of Theorem 9.4 is straightforward, but occupies some space. We defer
the the proof until the end of this section, and first explain the
consequences of Theorem 9.4.

####  9.2. Spaces of characters of type $<J$

To summarize the results so far, we introduce a notion of type for
${\mathbf{FS}}{{}^{\mathrm{op}}}$ modules.

######  Definition 9.7.

Let $M$ be an ${\mathbf{FS}}{{}^{\mathrm{op}}}$ module let $J$ be the integer
partition $j_{1}\geq j_{2}\geq\dots\geq j_{r}$. We say that $M$ has _type
$<J$_ if

$\mathrm{K}_{\ell_{r}}\circ\dots\circ\mathrm{K}_{\ell_{1}}(M)$

is exact for all choices of $\ell_{t}\geq j_{t}$.

######  Remark 9.8.

By Proposition 3.17 and the quasi-isomorphism
$\mathrm{K}_{d}\xrightarrow{\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}}\mathrm{B}_{d}$,
we may equivanlently define $M$ to have type $<J$ if
$\mathrm{K}_{\ell_{1}}\circ\dots\circ\mathrm{K}_{\ell_{r}}(M)$ is exact for
all choices of $\ell_{t}\geq j_{t}$. ∎

In terms of the type, Theorem 1.2 states that if $M$ is a subquotient of an
${\mathbf{FS}}{{}^{\mathrm{op}}}$ module generated in degree $d$, then there
exists an $r\geq 0$ such that $M$ has type $<(d+1)d^{r-1}$. (Here
$(d+1)d^{r-1}$ is the partition $d+1\geq d\geq\dots\geq d$, with $d$ repeated
$r-1$ times). Theorem 8.9 implies that if $M$ has type $<J$, then
$s=\operatorname{ch}(M)$ satisfies the system of differential equations (9.5).
(Note that the collection of operators $\\{{D\choose\lambda}\\}_{\lambda\text{
partition}}$ all commute). Theorem 9.4 describes the space of solutions to
this system of equations, and thus constrains characters of
${\mathbf{FS}}{{}^{\mathrm{op}}}$ modules of type $<J$. We introduce some
terminology in order to restate these results in a convenient way.

######  Definition 9.9.

Let $A=1^{a_{1}}2^{a_{2}}\dots$ be an integer partition and
$r\in\mathbb{N},r>1$. We define $\mathcal{V}_{A,r}\subseteq\hat{\Lambda}$ to
be the subspace

$\mathcal{V}_{A,r}:=\exp(\sum_{i}a_{i}y_{i})(\hat{\Lambda})^{\rm pow}_{<r}.$

By convention $\mathcal{V}_{A,0}=0$. Note that $\mathcal{V}_{A,r}$ is $1$
dimensional for $r=1$ and infinite dimensional for $r>1$.

Let $J=j_{1}\geq\dots\geq j_{r}$ be an integer partition. Theorem 8.9 and
Theorem 9.4 combine to show:

######  Theorem 9.10.

If $M$ is an ${\mathbf{FS}}{{}^{\mathrm{op}}}$ module of type $<J$ for
$J=j_{1}\geq\dots\geq j_{r}$, then

$\operatorname{ch}(M)\in\bigoplus_{A\text{ integer partition
}}\mathcal{V}_{A,t(A,J)}\subseteq\hat{\Lambda},$

where $t(A,J)$ is the number of $i\in\\{1,\dots,r\\}$ such that $j_{i}>|A|$.

###### Proof.

By Theorem 8.9, if $M$ has type $<J$ then

$0=\operatorname{ch}(\mathrm{K}_{\ell_{1}}\circ\mathrm{K}_{\ell_{2}}\dots\circ\mathrm{K}_{\ell_{r}}(M))=\sum_{\lambda_{1}\vdash\ell_{1},\dots,\lambda_{r}\vdash\ell_{r}}\left(\prod_{i=1}^{r}{D\choose\lambda_{i}}\operatorname{ch}(M)\right)\otimes
p_{\lambda_{1}}\otimes\dots\otimes p_{\lambda_{r}},$

for all $\ell_{t}\geq j_{t}$. Since the terms
$p_{\lambda_{1}}\otimes\dots\otimes p_{\lambda_{r}}$ are linearly independent,
$\operatorname{ch}(M)$ satisfies the equations (9.5). Since
$\bigoplus_{A\text{ integer partition }}\mathcal{V}_{A,t(A,J)}$ is simply an
alternate description of the solution space of Theorem 9.4, the result
follows. ∎

In particular, we have the following immediate corollary, in the case $J=d$.

######  Corollary 9.11.

If $M$ is an ${\mathbf{FS}}{{}^{\mathrm{op}}}$ module and $\mathrm{K}_{d}(M)$
is exact, then $\operatorname{ch}(M)$ is contained in the finite dimensional
subspace of $\hat{\Lambda}$ spanned by $\exp(\sum_{i}a_{i}y_{i})$ for all
integer partitions $A=1^{a_{1}}2^{a_{2}}\dots$ such that $|A|<d$.

In fact the statement of Corollary 9.11 is sharp, in the sense that the
characters of projective ${\mathbf{FS}}{{}^{\mathrm{op}}}$ modules generated
in degree $d$ span the subspace.

In contrast, when $J=j_{1}\geq\dots\geq j_{r}$ for $r>1$ the space of
solutions to (9.5), is no longer finite dimensional. For instance the space
$\mathcal{V}_{0,1}$ consists of arbitary linear combinations of the power sum
symmetric functions.

However, for any $d>0$ the image of $\mathcal{V}_{A,r}$ under the ring
homomorphism

$\hat{\Lambda}\to\mathaccent 1371{\mathbb{Q}[p_{1},p_{2},\dots,p_{d}]}$

setting $p_{i}=0$ for $i>d$ is finite dimensional. In the next section, we
combine this fact together with the constraint that the irreducible
representations appearing in a finitely generated
${\mathbf{FS}}{{}^{\mathrm{op}}}$ module have bounded rank, to produce finite
dimensional spaces of ${\mathbf{FS}}{{}^{\mathrm{op}}}$ characters.

Finally, in the special case $J=d^{s-1}(d+1)^{1}$, Theorem 9.10 implies
Theorem 1.11, as follows.

####  9.3. From symmetric functions to class functions

We show how to recover the results of the introduction, stated in terms of
class functions, from the results of the body of the paper, stated in terms of
symmetric functions.

There is a one-to-one correspondence between elements of $\hat{\Lambda}$ and
class functions on $\sqcup_{n}\mathrm{S}_{n}$, under which
$\operatorname{ch}(M)$ corresponds to the character of $M$,
$\sigma\mapsto\operatorname{Tr}(\sigma,M)$. If $s\in\hat{\Lambda}$, then $s$
corresponds to the class function $\sigma_{\lambda}\mapsto\langle
p_{\lambda},s\rangle$, where $\sigma_{|\lambda|}\in\mathrm{S}_{|\lambda|}$ is
an element in the conjugacy class of the integer partition $\lambda.$

The following fact allows us to translate into the language of the
introduction.

######  Proposition 9.12.

Let $A=1^{a_{1}}2^{a_{2}}\dots$ and $\nu=1^{\ell_{1}}2^{\ell_{2}}\dots$
integer partitions. Then symmetric function
$\frac{p_{\nu}}{z_{\nu}}\exp(\sum_{i}a_{i}y_{i})$ corresponds to the class
function

${\mathbb{X}\choose\nu}A^{\mathbb{X}-\nu}=\prod_{n\geq
1}{\mathbb{X}_{n}\choose\ell_{n}}\left(\sum_{d|n}da_{d}\right)^{\mathbb{X}_{n}-\ell_{n}}.$

###### Proof.

Let $\lambda$ be an integer partition. We wish to determine $\langle
p_{\lambda},\frac{p_{\nu}}{z_{\nu}}\exp(\sum_{i}a_{i}y_{i})\rangle,$ in other
words the coefficient of $p_{\lambda}/z_{\lambda}$ in the power series
expansion of $\frac{p_{\nu}}{z_{\nu}}\exp(\sum_{i}a_{i}y_{i}).$ To do this, it
is convenient to introduce the variables $t_{n}:=p_{n}/n$, so that if
$\lambda=1^{m_{1}}2^{m_{2}}\dots$ we have
$\frac{p_{\lambda}}{z_{\lambda}}=\prod_{i}\frac{t_{i}^{m_{i}}}{m_{i}!},$ and

$\exp(\sum_{i\geq 1}a_{i}y_{i})=\prod_{n\geq
1}\exp(t_{n}\sum_{d|n}da_{d})=\left(\prod_{n\geq
1}\sum_{m=0}^{\infty}\frac{(t_{n})^{m}}{m!}(\sum_{d|n}da_{d})^{m}\right).$

Now, if $\nu=1^{\ell_{1}}2^{\ell_{2}}\dots$ we have

$\frac{p_{\nu}}{z_{\nu}}\exp(\sum_{i}a_{i}y_{i})=\prod_{n\geq
1}\left(\sum_{m=0}^{\infty}\frac{t_{n}^{\ell_{n}}}{\ell_{n}!}\frac{(t_{n})^{m}}{m!}(\sum_{d|n}da_{d})^{m}\right).$

So the coefficient of $\prod_{n}\frac{t_{n}^{m_{n}}}{m_{n}!}$ is

$\prod_{n}{m_{n}\choose\ell_{n}}\left(\sum_{d|n}da_{d}\right)^{m_{n}-\ell_{n}},$

as desired. ∎

###### Proof of Theorem 1.11.

Let $M$ be an ${\mathbf{FS}}{{}^{\mathrm{op}}}$ module of class $(d,s)$. Then,
by definition $M$ has type $<d^{s-1}(d+1)^{1}$. Thus by Theorem 9.10

$\operatorname{ch}(M)\in\bigoplus_{A\vdash
d}\mathcal{V}_{A,1}\oplus\bigoplus_{A\text{ integer partition,
}|A|<d}\mathcal{V}_{A,s}.$

Concretely this means that $\operatorname{ch}(M)$ is an infinite sum of terms
proportional to $p_{\nu}\exp(\sum_{i}a_{i}y_{i})$, with restrictions on the
$\nu$ and $a_{i}$ appearing. Using Proposition 9.12 to translate, we obtain
Theorem 1.11. ∎

####  9.4. Proof of Theorem 9.4

###### Proof of Theorem 9.4.

Using the transition matrix between $\\{p_{i}\\}$ and $\\{y_{i}\\}$, we may
write any $s\in\hat{\Lambda}$ as a power series in the variables
$\\{y_{i}\\}_{i\geq 1}$:

$s=\sum_{(d_{1},d_{2},\dots)\in\mathbb{N}^{\oplus\infty}}c_{d_{1},d_{2}\dots}\prod_{i}y_{i}^{d_{i}}.$

We cut down the space of possible solutions to (9.5) in several steps.

We first note that if $s$ is a solution to (9.5) and
$(d_{i})\in\mathbb{N}^{\oplus\infty}$ satisfies $\sum_{i\geq j_{1}}d_{i}\geq
r$ then $c_{d_{1},d_{2},\dots}=0$. To see this apply the operator
$D^{\prime}:=\prod_{i}\frac{D_{i}^{d_{i}}}{d_{i}!}$ to $s$. On the one hand,
the constant term of $D^{\prime}(s)$ is $c_{d_{1},d_{2}\dots}$. On the other
hand by our hypothesis on $(d_{i})$ we may choose a monomial
$y_{i_{1}}y_{i_{2}}\dots y_{i_{r}}$ dividing $\prod_{i}y_{i}^{d_{i}}$ with
$i_{t}\geq j_{1}$. Then $D_{i_{1}}\dots D_{i_{r}}$ divides $D^{\prime}$ and
taking $\lambda_{t}=i_{t}$ we see that (9.5) implies that $D_{i_{1}}\dots
D_{i_{r}}(s)=0$ hence $D^{\prime}(s)=0$.

Thus assuming that $s$ satisfies (9.5) we may write

(9.13) $s=\sum_{w}wf_{w}(y_{1},\dots,y_{j_{1}-1})$

where the sum is over monomials $w$ of degree $<d$ in the variables
$y_{j_{1}},y_{j_{1}+1}\dots$. (Throughout this proof, if
$\prod_{i}y_{i}^{d_{i}}$ is a monomial we define its degree to be
$\deg(m):=\sum_{i}d_{i}$). Now consider the operators ${D_{i}\choose
j_{1}}^{r}$ for $i=1,\dots,j_{1}-1$. Equation (9.5) implies that
${D_{i}\choose j_{1}}^{r}(s)$ equals zero. Because $D_{i}$ only affects the
$y_{i}$ variable, we have that

${D_{i}\choose j_{1}}^{r}f_{w}=0,\text{ for all $i=1,\dots,j_{1}-1$.}$

These $j_{1}-1$ equations have the following consequence.

######  Lemma 9.14.

The function $f_{w}$ is a linear combination of functions of the form

$\exp(\sum_{i=1}^{j_{1}-1}a_{i}y_{i})\prod_{i=1}^{j_{1}-1}y_{i}^{d_{i}},$

where $(a_{i}),(d_{i})\in\mathbb{N}^{j_{1}-1}$, and $a_{i}<j_{1}$ and
$d_{i}<r$ for all $i$.

For each $A=1^{a_{1}}2^{a_{2}}\dots$ let
$\mathcal{V}_{A,r}\subseteq\hat{\Lambda}$ be the subspace of functions of the
form

(9.15) $\exp(\sum_{i}a_{i}y_{i})\left(\sum_{m}c_{m}m\right),$

where the sum is over monomials $m$ of degree $<r$ and $c_{m}\in\mathbb{Q}$.
Then (9.13) and Lemma 9.14 imply the crude statement that if $s$ is a solution
to (9.5) then

$s\in\sum_{A=(a_{i})\in\\{0,\dots,j_{1}-1\\}^{r}}\mathcal{V}_{A,r}.$

Note that the linear independence of finitely many exponentials of distinct
orders implies that the right hand sum is in fact a direct sum. And for any
$A=1^{a_{1}}2^{a_{2}}\dots$, the action of $\\{D_{i}\\}_{i\in\mathbb{N}}$
preserves $\mathcal{V}_{A,r}$. This means that the solution space to the
system (9.5) is a sum over $(a_{i})\in\\{0,\dots,j-1\\}^{r}$ to the space of
solutions in $\mathcal{V}_{A,r}$. We compute these spaces of solutions in the
following lemma.

######  Lemma 9.16.

Let $(a_{i})\in\mathbb{N}^{\oplus\infty}$. Let $\sum_{m}c_{m}m$,
$c_{m}\in\mathbb{Q}$ be a sum over monomials in the $y_{i}$ of bounded degree.
By convention we set $j_{t}=0$ if $t>r$.

The symmetric function $s:=\exp(\sum a_{i}y_{i})(\sum_{m}c_{m}m)$ is a
solution to (9.5) if and only if for every monomial $m$ appearing in the sum
with $c_{m}\neq 0$ we have $\sum_{i}ia_{i}<j_{\deg(m)+1}$.

Assuming the Lemma, we have that the space of solutions is spanned by
functions of the form $\exp(\sum_{i}a_{i}y_{i})(\sum_{m}c_{m}m)$ where the sum
is over monomials $m$ in the $y_{i}$ of degree $t-1$ and
$(a_{i})\in\mathbb{N}^{\oplus\infty}$ satisfies $\sum_{i}ia_{i}<j_{t}$. Since
the change of coordinates between $p_{i}$ and $y_{i}$ is linear, this
completes the proof. ∎

###### Proof of Lemma 9.14.

Set $j:=j_{1}$ and $f:=f_{w}$. Note that the space of solutions to the one-
variable differential equation ${{\partial_{u}}\choose j}^{r}=0$ in the ring
$\mathbb{Q}[[u]]$ is spanned by the functions $u^{d}\exp(au)$ for
$a,d\in\mathbb{N}$, $d<r$ and $a<j$, since ${{\partial_{u}}\choose j}^{r}$ is
proportional to
$\partial_{u}^{r}(\partial_{u}-1)^{r}\dots(\partial_{u}-j+1)^{r}$.

To obtain the multivariable statement from the single variable one, first
expand $f(y_{1},\dots,y_{j})$ as $\sum_{m}mf_{m}(y_{1})$ where $m$ is a
monomial in $y_{2},\dots,y_{j}$. Then each $f_{m}(y_{1})$ satisfies the single
variable equation, and hence is a linear combination of
$y_{1}^{d_{1}}\exp(a_{1}u)$ for $a_{1}<j$ and $d_{1}<r$. So we can write

$f=\sum_{m}m\sum_{a_{1},d_{1}}c_{m}(a_{1},d_{1})y_{1}^{d_{1}}\exp(a_{1}y_{1})=\sum_{a_{1},d_{1}}y^{d_{1}}\exp(a_{1}y_{1})f_{a_{1},d_{1}}(y_{2},\dots,y_{j-1}),$

where $c_{m}(a_{1},d_{1})\in\mathbb{Q}$ is some choice of coefficients; the
sum is over $a_{1}$ (resp. $d_{1}$) ranging from $0$ to $j-1$ (resp. $r-1$);
and $f_{a_{1},d_{1}}(y_{2},\dots y_{j})=\sum_{m}c_{m}(a_{1},d_{1})m$. Applying
this same process to the variable $y_{2}$ in the function $f_{a_{1},d_{1}}$
and continuing on like this, expanding out the resulting sums, we obtain the
result. ∎

###### Proof of Lemma 9.16.

First suppose that there is a monomial $m$ in the sum such that
$\sum_{i}ia_{i}\geq j_{\deg(m)+1}$. We may assume that $m$ has maximal degree,
since any monomial of degree $\geq\deg(m)$ will also satisfy the inequality.
Write $m=\prod_{i}y_{i}^{d_{i}}$ and $d=\deg(m)$.

We will choose a sequence of partitions
$\lambda_{1},\dots,\lambda_{\max(r,d)}$ such that $|\lambda_{t}|\geq j_{t}$
and $\prod_{t=1}^{\max(r,d)}{D\choose\lambda_{t}}$ does not annihilate $s$.
This suffices for the forward direction, because it shows that
$\prod_{t=1}^{r}{D\choose\lambda_{t}}$ does not annihilate $s$. Let
$\nu:=1^{a_{1}}2^{a_{2}}\dots$, and write $\prod_{i}y_{i}^{d_{i}}$ as
$y_{i_{1}}\dots y_{i_{d}}$. For $1\leq t\leq d$ we choose
$\lambda_{t}=i_{t}^{\kappa_{t}}$ where $\kappa_{t}\in\mathbb{N}$ is
sufficiently large that $\kappa_{t}>a_{i_{t}}$ and $\kappa_{t}i_{t}\geq
j_{t}$. If $d<t\leq r$, we set $\lambda_{t}:=\nu$. In both cases, by
construction we have $|\lambda_{t}|\geq j_{t}$, since
$|\lambda_{t}|=\kappa_{t}i_{t}\geq j_{t}$ and
$|\lambda_{t}|=\sum_{i}ia_{i}\geq j_{d+1}\geq j_{t}$ respectively.

That the operator corresponding to this choice of
$(\lambda_{t})_{t=1}^{\max(r,d)}$ does not annihilate $s$ follows from two
claims:

  1. (1)

${D\choose\lambda_{1}}\dots D\choose\lambda_{d}(s)$ is a nonzero scalar times
$\exp(\sum_{i}a_{i}y_{i})$.

  2. (2)

$\exp(\sum_{i}a_{i}y_{i})$ is an eigenvector for ${D\choose\nu}$ with nonzero
eigenvalue.

To see claim (1), we factor

${D\choose\lambda_{1}}\dots
D\choose\lambda_{d}={D_{i_{1}}\choose\kappa_{1}}\dots
D_{i_{d}}\choose\kappa_{d}$

as $D^{\prime}\prod_{i}(D_{i}-a_{i})^{d_{i}}$, where $D^{\prime}$ is a nonzero
scalar times a product of terms of the form $(D_{i}-b_{i})$ for $b_{i}\neq
a_{i}$. Let $n=\prod_{i}y_{i}^{e_{i}}$ be a monomial distinct from $m$. Then
by maximality of the degree of $m$ there exists an $i_{0}\geq 1$ such that
$e_{i_{0}}<d_{i_{0}}$, so the factor $(D_{i_{0}}-a_{i_{0}})^{d_{i_{0}}}$
annihilates $n\exp(\sum_{i}a_{i}y_{i})$. Therefore we have that
${D\choose\lambda_{1}}\dots D\choose\lambda_{d}(s)={D\choose\lambda_{1}}\dots
D\choose\lambda_{d}(c_{m}m\exp(\sum_{i}a_{i}y_{i})).$ Further, we have that
$\left(\prod_{i}(D_{i}-a_{i})^{d_{i}}m\exp(\sum_{i}a_{i}y_{i})\right)$ is a
nonzero scalar times $\exp(\sum_{i}a_{i}y_{i})$. And because
$(D-b_{k})\exp(\sum_{i}a_{i}y_{i})=(b_{k}-a_{k})\exp(\sum_{i}a_{i}y_{i})$ we
see that $\exp(\sum_{i}a_{i}y_{i})$ is an eigenvector of $D^{\prime}$ with
nonzero eigenvalue, establishing the first claim.

To see the second claim, notice that ${D\choose\nu}$ is a nonzero scalar times
a product of operators of the form $(D_{i}-b_{i})$ for $b_{i}<a_{i}$.

For the converse, let $\prod_{i}y_{i}^{d_{i}}$ be a monomial of degree $d$ and
suppose that $\sum_{i}ia_{i}<j_{d+1}$. (Note that this implies $d<r$). Let
$(\lambda_{1},\dots,\lambda_{d+1})$ be a sequence of integer partitions with
$|\lambda_{t}|\geq j_{t}$. We show that

(9.17)
$\left(\prod_{t=1}^{d+1}{D\choose\lambda_{t}}\right)\exp(\sum_{i}a_{i}y_{i})(\prod_{i}y_{i}^{d_{i}})=0.$

The converse follows from this statement via linearity and the fact that if
the product of the first $d$ operators annihilates $s$ then so does the full
product.

To see that equation (9.17) holds, note that for each
$\lambda_{t}=1^{m_{1,t}}2^{m_{2,t}}\dots$ there must exist an $i$ such that
$m_{i,t}>a_{i}$, since otherwise we would have

$|\lambda_{t}|=\sum_{i}im_{i,t}\leq\sum_{i}ia_{i}<j_{d+1}\leq j_{t},$

contradicting the hypothesis on $\lambda_{t}$. For each $t$ choose such an
$i$, and denote it by $i_{t}$. Then $\prod_{t=1}^{d+1}(D_{i_{t}}-a_{t})$ is a
factor of $\left(\prod_{t=1}^{d+1}{D\choose\lambda_{t}}\right)$. Because
$(D_{k}-a_{k})(m\exp(\sum_{i}a_{i}y_{i})=\exp(\sum_{i}a_{i}y_{i})D_{k}(m)$, we
see that each factor in the product $\prod_{t=1}^{d+1}(D_{i_{t}}-a_{t})$
reduces the degree of the coefficent of $\exp(\sum_{i}a_{i}y_{i})$ by one.
Since there are $d+1$ factors and $\prod_{i}y_{i}^{d_{i}}$ has degree $d$ we
have that

$\left(\prod_{t=1}^{d+1}(D_{i_{t}}-a_{t})\right)\exp(\sum_{a_{i}}y_{i})(\prod_{i}y_{i}^{d_{i}})=0,$

finishing the proof of the converse. ∎

###  10\. Characters of bounded rank

In §8 and §9 we examined the constraints on the characters of
${\mathbf{FS}}{{}^{\mathrm{op}}}$ modules of type $<J$ for $J$ an integer
partition. The purpose of this section is to study an additional constraint on
the characters of ${\mathbf{FS}}{{}^{\mathrm{op}}}$ modules.

######  Definition 10.1.

Let $M$ be an $\mathrm{S}_{\bullet}$ module over $\mathbb{Q}$. Let
$k\in\mathbb{N}$. We say that $\operatorname{rank}(M)\leq k$ if, in the
decomposition of $M_{n}$ into irreducibles, every irreducible representation
appearing corresponds to a Young diagram with $\leq k$ rows. We say that an
${\mathbf{FS}}{{}^{\mathrm{op}}}$ module has rank $\leq k$ when its underlying
$\mathrm{S}_{\bullet}$ representation does.

######  Definition 10.2.

We write $\mathcal{F}_{\leq k}\subseteq\hat{\Lambda}$ for the closure of the
span of Schur functions $s_{\lambda}$ such that the Young diagram of $\lambda$
has $\leq k$ rows.

Notice that an $\mathrm{S}_{\bullet}$ representation $M$ has rank $\leq k$ if
and only if $\operatorname{ch}(M)\in\mathcal{F}_{\leq k}$. The reason that the
rank is useful is the following folklore proposition, which appears in [17,
Theorem 4.1].

######  Proposition 10.3.

Let $M$ be an ${\mathbf{FS}}{{}^{\mathrm{op}}}$ module over $\mathbb{Q}$. If
$M$ is a subquotient of a module generated in degree $\leq d$, then
$\operatorname{rank}(M)\leq d$.

Therefore, by Theorem 1.2 and Theorem 9.10, if $M$ is a subquotient of an
${\mathbf{FS}}{{}^{\mathrm{op}}}$ module generated in degree $d$ there exists
an $r\in\mathbb{N}$ such that $\operatorname{ch}(M)\in\hat{\Lambda}$ lies in
the subspace

$\mathcal{F}_{\leq d}\cap\left(\bigoplus_{A\vdash
d}\mathcal{V}_{A,1}\oplus\bigoplus_{A\text{ integer partition,
}|A|<d}\mathcal{V}_{A,r}\right).$

In this section, we prove that this intersection is finite dimensional, and
characterize it using the Hall inner product.

######  Definition 10.4.

Write $\epsilon_{k}:\hat{\Lambda}\to\mathaccent
1371{\mathbb{Q}[p_{1},\dots,p_{k}]}$ for the specialization homomorphism
$p_{i}\mapsto p_{i}$ for $i\leq k$ and $p_{i}\mapsto 0$ for $i>k$.

We prove the following theorem over the course of this section.

######  Theorem 10.5.

Let $A_{i},i\in I$ be a finite collection of distinct integer partitions, and
$r_{i}\in\mathbb{N},i\in I$ be a finite collection of natural numbers (not
necessarily distinct). Let $k\in\mathbb{N}$ and $k\geq\max_{i}(|A_{i}|)$. Then
we have

$\mathcal{F}_{\leq k}\cap\bigoplus_{i\in
I}\mathcal{V}_{A_{i},r_{i}}=\bigoplus_{i\in I}\mathcal{F}_{\leq
k}\cap\mathcal{V}_{A_{i},r_{i}}.$

Further, given any element $s\in\hat{\Lambda}$, there is a unique element
$\pi_{k}(s)\in\mathcal{F}_{\leq k}$ such that

$\epsilon_{k}(\pi_{k}(s))=\epsilon_{k}(s).$

And if $s\in\mathcal{V}_{A_{i},r_{i}}$, then
$\pi_{k}(s)\in\mathcal{V}_{A_{i},r_{i}}$.

Because the space $\epsilon_{k}(\mathcal{V}_{A,r})$ is finite dimensional,
Theorem 10.5 implies that $\mathcal{F}_{\leq k}\cap\mathcal{V}_{A,r}$ is
finite dimensional, since the elements of this intersection are uniquely
determined by their image in $\epsilon_{k}(\mathcal{V}_{A,r})$. In particular
the elements $\pi_{k}(p_{\nu}\exp(\sum_{i}a_{i}y_{k}))$ where $p_{\nu}$ ranges
over degree $<r$ monomials in $p_{1},\dots,p_{k}$ form a basis of
$\mathcal{F}_{\leq k}\cap\mathcal{V}_{A,r}$, because
$\epsilon_{k}(p_{\nu}\exp(\sum_{i}a_{i}y_{k}))$ is a basis for
$\epsilon_{k}(\mathcal{V}_{A,r})$.

Under the correspondence between class functions and symmetric functions, the
specialization $\epsilon_{k}$ corresponds to restricting the class function to
elements $\sigma\in\mathrm{S}_{n}$ whose cycles all have length $\leq k$. So
Theorem 10.5 provides unique lift from restricted class functions to class
functions.

Assuming Theorem 10.5 we can prove Theorem 1.13.

###### Proof of Theorem 1.13.

Let $(d,s)\in\mathbb{N}^{2}$. If $M$ has type $(d,s)$ then by Proposition 10.3
we have that the rank of $M$ is $\leq d$ and $M$ has type $<J$ for
$J=d^{s-1}(d+1)$. Define

$\mathcal{U}_{d,s}:=\bigoplus_{A\vdash d}\mathcal{F}_{\leq
d}\cap\mathcal{V}_{A,1}\oplus\bigoplus_{A\text{ integer partition,
}|A|<d}\mathcal{F}_{\leq d}\cap\mathcal{V}_{A,s}.$

By Theorem 10.5 and Theorem 9.10 we have that
$\operatorname{ch}(M)\in\mathcal{U}_{d,s}$. Since
$\pi_{k}(p_{\nu}\exp(\sum_{i}a_{i}y_{k}))$ where $p_{\nu}$ ranges over degree
$<r$ monomials in $p_{1},\dots,p_{k}$ form a basis of $\mathcal{F}_{\leq
k}\cap\mathcal{V}_{A,r}$, we see that $\mathcal{U}_{d,s}$ has dimension
$p(d)+{d+s-1\choose s-1}\sum_{i<d}p(i)$ where $p(i)$ is the number of integer
partitions of $i$. ∎

To prove Theorem 10.5 and describe the projection operator
$\pi_{k}:\hat{\Lambda}\to\mathcal{F}_{\leq k}$, we characterize the spaces
$\mathcal{F}_{\leq k}$ and $\mathcal{V}_{A,r}$ using the Hall inner product.

######  Definition 10.6.

If $W$ is a subspace of $\Lambda$, we write $W^{\perp}\subseteq\hat{\Lambda}$
for the _perpendicular space_ with respect to the Hall inner product,
characterized by the property $t\in W^{\perp}$ if and only if $\langle
w,t\rangle=0$ for all $w\in W$.

We record the following general fact from linear algebra.

######  Proposition 10.7.

Let $W\subseteq\Lambda$. The Hall inner product induces an isomorphism
$W^{\perp}\to(\Lambda/W)^{*}$, by $t\mapsto\langle-,t\rangle$. In particular,
there is a unique element of $W^{\perp}$ corresponding to every linear
function $L:\Lambda/W\to\mathbb{Q}$ namely

$\sum_{\lambda\text{ integer partition}}L([p_{\lambda}])~{}p_{\lambda}.$

where $[p_{\lambda}]$ denotes the equivalence class of $p_{\lambda}$ mod $W$.

The next two propositions characterize $\mathcal{F}_{\leq k}$ and
$\mathcal{V}_{A,r}$ as the perpendicular spaces to two different ideals in
$\Lambda$. The first is certainly known, but we do not have a reference so we
include the proof.

######  Proposition 10.8.

Let $(e_{k+1},e_{k+2}\dots)\Lambda$ be the ideal of $\Lambda$ generated by
$e_{i}$ for $i>k$. Then

$\mathcal{F}_{\leq k}=((e_{k+1},e_{k+2}\dots)\Lambda)^{\perp},$

with respect to the Hall inner product.

###### Proof.

For $\lambda$ an integer partition, we write $\operatorname{rank}(\lambda)$
for the number of rows in the Young diagram of $\lambda$. Write
$t=\sum_{\lambda}b_{\lambda}s_{\lambda}$ for $b_{\lambda}\in\mathbb{Q}$. Let
$w\in\Lambda$ and $i>k$. By [26, 7.15.4]

$\langle we_{i},t\rangle=\sum_{\lambda}b_{\lambda}\langle
we_{i},s_{\lambda}\rangle=\sum_{\lambda}b_{\lambda}\langle
w,s_{\lambda/1^{i}}\rangle.$

If $t\in\mathcal{F}_{\leq k}$ then $b_{\lambda}=0$ for all $\lambda$ of rank
$>k$. By the (transpose of) the Pieri rule [26, 7.15.9] we have that
$s_{\lambda/1^{i}}$ is zero if $\lambda$ does not have rank $\geq i$. Hence
$\langle we_{i},t\rangle=0$, and so $\mathcal{F}_{\leq
k}\subseteq(e_{k+1},\dots)\Lambda^{\rm\perp}$.

For the converse note that the degree $n$ component of the ideal
$(e_{k+1},\dots)\Lambda$ is spanned by the set of $e_{\lambda}$ where
$\lambda\vdash n$ is an integer partition with $>k$ columns in its Young
diagram. Thus its perpendicular space in $\hat{\Lambda}_{n}$ has the same
dimension as the number of integer partitions of $n$ with $\leq k$ columns.
Taking a transpose, we see that this is the same as the dimension of the
degree $n$ component of $\mathcal{F}_{\leq k}$. Thus the containment
$\mathcal{F}_{\leq k}\subseteq(e_{k+1},\dots)\Lambda^{\rm\perp}$ is an
equality. ∎

######  Proposition 10.9.

Let $A=1^{a_{1}}2^{a_{2}}\dots$ and let $r\in\mathbb{N}$, $r>1$. Define
$u_{n}:=p_{n}-\sum_{i|n}ia_{i}$ for all $n\geq 1$. Then Let
$(u_{1},u_{2},\dots)^{r}\Lambda$ be the $r^{\rm th}$ power of the ideal
generated by the $u_{n}$. Then

$\mathcal{V}_{A,r}=\left((u_{1},u_{2},\dots)^{r}\Lambda\right)^{\perp},$

with respect to the Hall inner product.

###### Proof.

Any element $t\in\hat{\Lambda}$ can be expanded uniquely as

$t=\sum_{\lambda\text{ integer
partition}}c_{\lambda}p_{\lambda}\exp(\sum_{i}a_{i}y_{i})$

for $c_{\lambda}\in\mathbb{Q}$, because the lowest degree term of
$p_{\lambda}\exp(\sum_{i}a_{i}y_{i})$ is $p_{\lambda}$. Then
$t\in\hat{\Lambda}$ if and only if $c_{\lambda}=0$ for all $\lambda$ whose
Young diagrams have $\geq r$ rows.

For $\lambda=1^{m_{1}}2^{m_{2}}\dots$ put
$u_{\lambda}:=\prod_{i}u_{i}^{m_{i}}.$

######  Lemma 10.10.

We have that

$\left\langle\frac{u_{\lambda}}{z_{\lambda}},~{}p_{\nu}\exp(\sum_{i}a_{i}y_{i}))\right\rangle=\delta(\lambda,\nu),$

where $\delta(\lambda,\nu)$ is the Kronecker delta.

###### Proof.

First notice that for any $\lambda,\nu$ we have

$\langle\partial_{i}(p_{\lambda}/z_{\lambda}),p_{\nu}\rangle=\langle
p_{\lambda}/z_{\lambda},(p_{i}/i)p_{\nu}\rangle,$

using the fact that $\langle
p_{\lambda}/z_{\lambda},p_{\nu}\rangle=\delta(\lambda,\nu)$. By linearity, it
follows that the operator $\partial_{i}$ and multiplication by $p_{i}/i$ are
adjoint. Thus for any pair of symmetric functions
$f\in\Lambda,g\in\hat{\Lambda}$ we have that

$\langle f,g\exp(\sum_{d\geq 1}a_{d}y_{d})\rangle=\langle f,g\exp(\sum_{d\geq
1}a_{d}d\sum_{d|k}p_{k}/k)\rangle=\langle\exp(\sum_{d}a_{d}d\sum_{d|k}\partial_{k})f,g\rangle.$

Taylor expanding with respect to $p_{k}$, we see that for any
$k\in\mathbb{N}$, the operator $\exp(\partial_{k})$ acts on $\Lambda$ by the
ring homomorphism $p_{k}\mapsto p_{k}+1$ and $p_{n}\mapsto p_{n}$ for $d\neq
k$. Thus we have that $\exp(\sum_{d}da_{d}\sum_{k}\partial_{k})$ acts on
$\Lambda$ by the substitution $p_{n}\mapsto p_{n}+\sum_{d|n}da_{d}$. In
particular we have that
$\exp(\sum_{d}da_{d}\sum_{k}\partial_{k})u_{\lambda}=p_{\lambda}$. So

$\langle
u_{\lambda}/z_{\lambda},p_{\nu}\exp(\sum_{d}a_{d}y_{d})\rangle=\langle
p_{\lambda}/z_{\lambda},p_{\nu}\rangle=\delta(\lambda,\nu),$

as desired. ∎

Applying Lemma 10.10 to the expansion of $t$, we see that
$t\in\mathcal{V}_{A,r}$ if and only if $\langle
u_{\lambda}/z_{\lambda},t\rangle=0$ for all $\lambda=1^{m_{1}}2^{m_{2}}\dots$
such that $\sum_{i}m_{i}\geq r$. Rescaling by $z_{\lambda}$ we see that
$t\in\mathcal{V}_{A,r}$ if and only if every degree $\geq r$ monomial in the
$u_{i}$ pairs to zero with it. Taking linear combinations of monomials
finishes the proof. ∎

Since sums of subspaces correspond to intersections of perpendicular spaces,
Propositions 10.8 and 10.9 imply.

######  Proposition 10.11.

Let $A=1^{a_{1}}2^{a_{2}}\dots$, let $u_{n}=p_{n}-\sum_{i|n}ia_{i}$, and let
$r\in\mathbb{N}$. Then

$\mathcal{V}_{A,r}\cap\mathcal{F}_{\leq
k}={((u_{1},u_{2},\dots)^{r}+(e_{k+1},e_{k+2},\dots))}^{\perp}.$

Next we need

######  Proposition 10.12.

The ring homomorphism

$\iota_{k}:\mathbb{Q}[p_{1},\dots,p_{k}]\to\Lambda/(e_{k+1},\dots),$

$\iota_{k}(p_{i}):=p_{i}$ is an isomorphism. In particular, the images of
$\\{p_{\lambda}/z_{\lambda}\\}_{\lambda=1^{m_{1}}\dots k^{m_{k}}}$ in
$\frac{\Lambda}{(e_{k+1},\dots)}$ form a basis. Here $\lambda$ ranges over
partitions whose associated Young diagrams have $\leq k$ columns.

###### Proof.

We have that
$\frac{\Lambda}{(e_{k+1},\dots)}=\mathbb{Q}[e_{1},\dots,e_{k}]\cong\mathbb{Q}[x_{1},\dots,x_{k}]^{\mathrm{S}_{k}}$
is isomorphic to ring of symmetric functions in $k$ variables. It is well
known that the power sums $p_{1},\dots,p_{k}$ form an independent set of
generators of this ring. ∎

Now Propositions 10.7, 10.8 and 10.12 allow us to define $\pi_{k}(s)$.

######  Definition 10.13.

Let $s\in\hat{\Lambda}$. We define $\pi_{k}(s)\in\mathcal{F}_{\leq k}$ to be
the element corresponding (as in Proposition 10.7) to the linear function,
$\frac{\Lambda}{(e_{k+1},\dots)}\to\mathbb{Q}$ defined by
$p_{\lambda}/z_{\lambda}\mapsto\langle p_{\lambda}/z_{\lambda},s\rangle$ for
all integer partitions $\lambda=1^{m_{1}}\dots k^{m_{k}}$, whose Young
diagrams have $\leq k$ columns.

Because $\langle p_{\lambda}/z_{\lambda},t\rangle$ is precisely the
coefficient of $p_{\lambda}$ in the power-sum expansion of $t$, it is clear
that $s$ and $\pi_{k}(s)$ have the same image under $\epsilon_{k}$. And by
Propositions 10.7, 10.8 and 10.12, $\pi_{k}(s)$ is the only element of
$\mathcal{F}_{\leq k}$ with this property.

To actually compute $\pi_{k}(s)$ as a sum of power functions, as in
Proposition 10.7, we need to know how to write $p_{\nu}$ as a sum of monomials
in $p_{1},\dots,p_{k}$ under the specialization
$\Lambda\to\frac{\Lambda}{(e_{k+1},\dots)}\cong\mathbb{Q}[x_{1},\dots,x_{k}]^{\mathrm{S}_{k}}$.

######  Proposition 10.14.

Let $A=1^{a_{1}}2^{a_{2}}\dots$ be an integer partition and $r,k\in\mathbb{N}$
and let $u_{n}$ denote $p_{n}-\sum_{d|n}da_{d}$. Suppose that $k\geq n$. Let
$\overline{(u_{1},u_{2},\dots)^{r}}$ be the image of $(u_{1},u_{2}\dots)^{r}$
in $\Lambda/(e_{k+1},\dots)$. Then
$\iota_{k}^{-1}\left(\overline{(u_{1},u_{2},\dots)^{r}}\right)=(u_{1},\dots,u_{k})^{r}$.

###### Proof.

It suffices to show that
$\iota_{k}^{-1}\left(\overline{(u_{1},u_{2}\dots)}\right)=(u_{1},\dots,u_{k})$,
where $\overline{(u_{1},u_{2},\dots)}$ is the image of ${(u_{1},u_{2},\dots)}$
in $\Lambda/(e_{k+1},\dots)$.

Now $\iota_{k}^{-1}\left(\overline{(u_{1},u_{2}\dots)}\right)$ clearly
contains $(u_{1},\dots,u_{k})$ which is a maximal ideal. So it suffices to
prove that $\overline{(u_{1},u_{2}\dots)}$ is a proper ideal of
$\Lambda/(e_{k+1},\dots)$. To do this, we construct a nontrivial homomorphism
$\Lambda/(e_{k+1},\dots)\to\mathbb{Q}$ under which which $u_{i}$ vanishes.

Consider the $\mathbb{Z}$-set $S:=\bigsqcup_{d\geq
1}(\mathbb{Z}/d\times[a_{d}])$. Linearizing it we obtain an $|A|$ dimensional
vector space $\mathbb{Q}S$ and an operator $T:\mathbb{Q}S\to\mathbb{Q}S$
defined to be the action of $+1\in\mathbb{Z}$. Then there is a specialization
homomorphism $\kappa:\Lambda\to\mathbb{Q}$ defined by

$p_{n}\mapsto\operatorname{Tr}(T^{n},\mathbb{Q}S).$

Since $\mathbb{Q}S$ has dimension $|A|\leq k$ we have that
$\wedge^{i}\mathbb{Q}S=0$ for all $i>k$. Thus
$\kappa(e_{i})=\operatorname{Tr}(T,\wedge^{i}\mathbb{Q}S)=0$ for all $i>k$. So
$\kappa$ factors through $\Lambda/(e_{k+1},\dots)$. Finally, $\kappa(p_{n})$
is the number of fixed points of the action of $+n\in\mathbb{Z}$ on $S$. If
$d|n$ then $+n$ fixes all of the points of $\mathbb{Z}/d$, otherwise it fixes
none of them. So we see that the number of fixed points of $+n$ is
$\sum_{d|n}da_{d}$. Therefore
$\kappa(u_{n})=\kappa(p_{n}-\sum_{d|n}da_{d})=0$.

Thus $\kappa$ is the desired homomorphism with kernel
$(u_{1},u_{2}\dots)+(e_{k+1},\dots)$ and so we are done. ∎

######  Remark 10.15.

The homomorphism $\kappa$ constructed in the proof of Proposition 10.14 can
also be directly by $p_{n}\mapsto p_{n}(\zeta_{1},\zeta_{2},\dots,0,0,\dots)$
where the $\zeta_{i}$ are the roots of unity obtained by diagonalizing $T$. ∎

Proposition 10.14 has the following immediate consequence.

######  Proposition 10.16.

Let $A=1^{a_{1}}2^{a_{2}}\dots$ and let $u_{n}=p_{n}-\sum_{i|n}ia_{i}$. There
is an isomorphism

$\frac{\mathbb{Q}[u_{1},\dots,u_{k}]}{(u_{1},\dots,u_{k})^{r}}\to\frac{\Lambda}{(u_{1},u_{2},\dots)^{r}+(e_{k+1},e_{k+2},\dots)}$

induced by the map $u_{i}\mapsto u_{i}$ for $i=1,\dots,k$.

###### Proof.

The morphism is surjective because its image equals the image of
$\mathbb{Q}[p_{1},\dots,p_{k}]$, and it is injective by Proposition 10.14. ∎

######  Proposition 10.17.

Let $A$ be an integer partition such that $|A|\leq k$. If
$s\in\mathcal{V}_{A,r}$ then $\pi_{k}(s)\in\mathcal{V}_{A,r}$.

###### Proof.

Let $A=1^{a_{1}}2^{a_{2}}\dots$ and let $u_{i}=p_{i}-\sum_{i}ia_{i}$. By
Proposition 10.9, it suffices to show that $\langle b,\pi_{k}(s)\rangle=0$ for
every $b\in(u_{1},u_{2}\dots)^{r}$. Consider
$\iota_{k}^{-1}(b)\in\mathbb{Q}[p_{1},\dots,p_{k}]$. By Proposition 10.14 it
lies in the ideal $(u_{1},\dots,u_{k})^{r}$. By Propositions 10.9 and 10.7, it
suffices to show that $\langle m,\pi_{k}(s)\rangle=0$, where $m$ is any degree
$>r$ monomial in $u_{1},\dots,u_{k}$. Because $m$ is a linear combination of
monomials in $p_{1},\dots,p_{k}$, by the definition of $\pi_{k}(m)$ we have
that $\langle m,\pi_{k}(s)\rangle=\langle m,s\rangle=0$ . ∎

###### Proof of Theorem 10.5.

We have already constructed $\pi_{k}$ and shown that it is unique. Proposition
10.17 states that if $\pi_{k}$ preserves the subspace $\mathcal{V}_{A,r}$ when
$k\geq|A|$. It remains to show that $\mathcal{F}_{\leq k}\cap\bigoplus_{i\in
I}\mathcal{V}_{A_{i},r_{i}}=\bigoplus_{i\in I}\mathcal{F}_{\leq
k}\cap\mathcal{V}_{A_{i},r_{i}}$. Suppose that we have
$s_{i}\in\mathcal{V}_{A_{i},r_{i}}$ such that $\sum_{i\in
I}s_{i}\in\mathcal{F}_{\leq k}$. The operator $\pi_{k}$ is projection onto the
subspace $\mathcal{F}_{\leq k}$, so we have $\sum_{i\in
I}s_{i}=\pi_{k}(\sum_{i\in I}s_{i})=\sum_{i}\pi_{k}(s_{i})$. By linear
independence of $\mathcal{V}_{A_{i},r_{i}}$ and the fact that
$\pi_{k}(s_{i})\in\mathcal{V}_{A_{i},r_{i}}$ it follows that
$\pi_{k}(s_{i})=s_{i}$ for all $i$. Thus all of the $s_{i}$ are in
$\mathcal{F}_{\leq k}$. This completes the proof. ∎

Finally we include an alternate description of $\mathcal{F}_{\leq k}$, that
can be used to compute $\pi_{k}$ quickly some cases.

######  Proposition 10.18.

Let $(\hat{\Lambda})^{\rm homog}_{\leq k}$ denote the closure of the span of
monomials of degree $\leq k$ in the homogenous symmetric functions. Then
$\mathcal{F}_{\leq k}=(\hat{\Lambda})^{\rm homog}_{\leq k}$.

###### Proof.

We need to show that the closure of the span $s_{\lambda}$ where $\lambda$ are
integer partitions with $\leq d$ rows equals $(\hat{\Lambda})^{\rm
homog}_{\leq d}$. By the Pieri rule [26, 7.15.9], we have that every monomial
$h_{i}$ of length $\leq d$ is a sum of such $s_{\lambda}$, showing one
direction of the containment. To prove equality, observe that in degree $n$
the two subspaces have the same dimension: the number of integer partitions of
$n$ with $\leq d$ rows. ∎

We conclude this section with an example.

######  Example 10.19.

Consider the space $\mathcal{V}_{1,2}$, which is the closure of the span of
$\exp(y_{1})$ and $\\{p_{n}\exp(y_{1})\\}_{n\geq 1}$. Then because

$\exp(y_{1})=\sum_{n\geq 0}h_{n},$

we have that $\pi_{2}(\exp(y_{1}))=\exp(y_{1})$ and
$\pi_{2}(p_{1}\exp(y_{1}))=p_{1}\exp(y_{1})$. We also have that
$\pi_{2}(p_{n}\exp(y_{1}))=0$ for $n\geq 3$, because
$\epsilon_{2}(p_{n}\exp(y_{1}))=0$. So the interesting case is
$\pi_{2}(p_{2}\exp(y_{1}))$. Observe that we have

$\sum_{n\geq 2}p_{n}\exp(y_{1})=\sum_{n\geq 2}nh_{n}$

by Newtons identity. Hence this element lies in $\mathcal{F}_{\leq 2}$ and its
image under $\epsilon_{2}$ agrees with $p_{2}\exp(y_{1})$. So it must be
$\pi_{2}(p_{2}\exp(y_{1}))$.

It is also possible to compute $\pi_{2}(p_{2}\exp(y_{1}))$ directly from
Definition 10.13. We describe a systematic method for computing $\pi_{k}$ in
§11.4.

###  11\. Generating functions valued in the dual character space

We begin with a summary of what we know so far. First Theorem 10.5 and Theorem
9.10 immediately imply the following.

######  Theorem 11.1.

Let $J=j_{1}\geq\dots\geq j_{r}$ be an integer partition, and let
$k\in\mathbb{N}$ satisfy $k\geq j_{1}$. Then if $M$ is an
${\mathbf{FS}}{{}^{\mathrm{op}}}$ module of type $<J$ and rank $\leq k$, we
have

(11.2) $\operatorname{ch}(M)\in\bigoplus_{A\text{ integer partition
}}\mathcal{F}_{\leq k}\cap\mathcal{V}_{A,t(A,J)},$

where $t(A,J)=\\#\\{i\in\\{1,\dots r\\}~{}|~{}j_{i}>|A|\\}$.

In this section, we study each summand of (11.2) individually. Thus, for the
remainder of this section, we fix an integer partition
$A=1^{a_{1}}2^{a_{2}}\dots$, and $r,k\in\mathbb{N}$ such that $|A|\leq k$. We
set $u_{n}:=p_{n}-\sum_{d|n}da_{d}$ and $u_{\lambda}:=\prod_{i}u_{i}^{m_{i}}$
for $\lambda=1^{m_{1}}\dots$ an integer partition. We will consider
$\mathcal{V}_{A,r}\cap\mathcal{F}_{\leq k}$ and the projection operator
$\pi_{k}:\mathcal{V}_{A,r}\to\mathcal{V}_{A,r}\cap\mathcal{F}_{\leq k}$.

Now, Propositions 10.11 and 10.16 show that there are isomorphisms.

(11.3) $\mathcal{V}_{A,r}\cap\mathcal{F}_{\leq
k}\cong\left(\frac{\Lambda}{(u_{1},u_{2},\dots)^{r}+(e_{k+1},e_{k+2},\dots)}\right)^{*}\cong\left(\frac{\mathbb{Q}[u_{1},\dots,u_{k}]}{(u_{1},\dots,u_{k})^{r}}\right)^{*},$

which take an element $s\in\mathcal{F}_{\leq k}\cap\mathcal{V}_{a,r}$ to the
function $\langle-,s\rangle:\Lambda\to\mathbb{Q},$ and to the restriction of
this function to degree $<r$ monomials in $u_{1},\dots,u_{k}$.

####  11.1. An interpretation

We can interpret the isomorphisms of (11.3) in the following way, which we
find clarifying. Let $\operatorname{ch}(V)\in\hat{\Lambda}$ be the character
of some sequence of $\mathrm{S}_{n}$ representations $V$. Then elements
$m\in\Lambda$ correspond to different _measurements_ that we can perform on
$V$, by taking the inner product $\langle m,\operatorname{ch}(V)\rangle$. For
example:

  * •

The element $m=p_{\lambda}$ measures the trace of an element of
$\mathrm{S}_{|\lambda|}$ with cycle type $\lambda$,

$\langle
p_{\lambda},\operatorname{ch}(V)\rangle=\operatorname{Tr}(\sigma_{\lambda},V_{|\lambda|}).$

  * •

The element $m=h_{n}$ measures the dimension of the space of $\mathrm{S}_{n}$
invariants:

$\langle h_{n},\operatorname{ch}(V)\rangle=\dim(V_{n})^{\mathrm{S}_{n}}.$

  * •

The element $m=u_{\lambda}$ measures the coefficient $c_{\lambda}$ in the
unique expansion

$\operatorname{ch}(V)=\sum_{\lambda\text{ integer
partition}}c_{\lambda}\frac{p_{\lambda}}{z_{\lambda}}\exp(\sum_{i\geq
1}a_{i}y_{i}),$

by Lemma 10.10.

In these terms, the first isomorphism states that if
$\operatorname{ch}(V)\in\mathcal{V}_{A,r}\cap\mathcal{F}_{\leq k}$ then the
value of the measurement $m\in\Lambda$ only depends on its image in the
quotient $\frac{\Lambda}{(u_{1},u_{2},\dots)^{r}+(e_{k+1},e_{k+2},\dots)}$.
(Measurements that were independent for arbitrary characters are correlated
for elements of $\mathcal{V}_{A,r}\cap\mathcal{F}_{\leq k}$). And the second
isomorphism shows that to determine the value of an arbitrary measurement
$m\in\Lambda$ we can proceed in two steps: (1) compute the measurements which
are monomials in $u_{1},\dots,u_{k}$, (2) express the image of $m$ in the
quotient ring as a linear combination of monomials in $u_{1},\dots,u_{k}$.

Step (2) is nontrivial, since we have to actually compute the isomorphism
which Proposition 10.16 provides abstractly. One can use Gröbner bases to
algorithmically reduce $m$ into a normal form, but this does not yield
explicit formulas. The purpose of this section is to introduce a strategy for
accomplishing step (2) in the cases $m=u_{\lambda}$ and $m=h_{n}$, using
generating functions. We use this strategy to construct an explicit basis for
$\mathcal{V}_{A,r}\cap\mathcal{F}_{\leq k}$, to describe how to compute
$\pi_{k}$, and to prove Theorem 1.17.

####  11.2. A generating function identity

We let $R:=\frac{\Lambda}{(u_{1},u_{2},\dots)^{r}+(e_{k+1},e_{k+2},\dots)}$ be
the quotient ring, and write $f:\Lambda\to R$ for the quotient homomorphism.

Let $\lambda$ be an integer partition. We wish to express $f(u_{\lambda})$ as
a linear combination of $f(u_{\nu})$ where $\nu=1^{\ell_{1}}\dots
k^{\ell_{k}}$ has $\leq k$ columns (in other words $u_{\nu}$ is a monomial in
$u_{1},\dots,u_{k}$). Our strategy is to first compute this expression for
$f(u_{n})$ and then use the homomorphism property to compute it for
$f(u_{\lambda})$. Accordingly, we compute an expression for the $R$-valued
generating function

$\sum_{n\geq 0}\frac{f(u_{n})}{n}t^{n}\in R[[t]].$

We will also consider generating functions valued in $\Lambda$ (i.e. elements
of $\Lambda[[t]]$). There is a specialization homomorphism $f:\Lambda[[t]]\to
R[[t]]$, given by applying $f:\Lambda\to R$ coefficient wise. Now, consider

$\exp(-\sum_{n}(u_{n}t^{n})/n)\in\Lambda[[t]].$

We have

$\displaystyle\exp(-\sum_{n\geq 1}(u_{n}t^{n})/n)$
$\displaystyle=\exp(-\sum_{n\geq 1}p_{n}t^{n}/n)\exp(\sum_{n\geq
1}\sum_{i|n}ia_{i}t^{n}/n)=\exp(-\sum_{n\geq 1}p_{n}t^{n}/n)\exp(\sum_{e,i\geq
1}a_{i}\frac{(t^{i})^{e}}{e})$ (11.4) $\displaystyle=(1+\sum_{n\geq
1}(-1)^{n}e_{n}t^{n})\exp(\sum_{i\geq
1}-a_{i}\log(1-t^{i}))=\frac{1+\sum_{n\geq 1}(-1)^{n}e_{n}t^{n}}{\prod_{i\geq
1}(1-t^{i})^{a_{i}}}$

Where we have used the well-known identity,

(11.5) $(1+\sum_{n\geq 1}(-1)^{n}e_{n}t^{n})=\exp(-\sum_{n\geq
1}p_{n}t^{n}/n),$

which can be proved by writing $p_{n}=\sum_{j}x_{j}^{n}$ and using the power
series expansion of $\log$.

Applying the specialization homomorphism $f:\Lambda[[t]]\to R[[t]]$ we obtain
the identity:

$\exp\left(-\sum_{n}\frac{f(u_{n})}{n}t^{n}\right)=\frac{1-f(e_{1})t+\dots+(-1)^{k}f(e_{k})t^{k}}{\prod_{i\geq
1}(1-t^{i})^{a_{i}}}$

Now, we wish to write express the numerator in terms of the variables
$u_{1},\dots,u_{k}\in R$. Before doing this, we introduce some notation.

######  Definition 11.6.

For $m\in\mathbb{Z}$, let $c_{m}$ be the coefficient of $t^{m}$ in
$\prod_{i\geq 1}(1-t^{i})^{a_{i}}.$ Concretely $c_{m}$ is an alternating sum
of products of binomial coefficients. (We may also interpret it as the trace
of an operator constructed as in the proof of Proposition 10.14).

######  Definition 11.7.

For $m\in\mathbb{N}$ we define $E_{m}\in\Lambda$ to be
$E_{m}:=-(-1)^{m}e_{m}+c_{m}$. Notice that $E_{0}=0$ and
$E_{m}=-(-1)^{m}e_{m}$ for $m>|A|$. We extend this definition to integer
partitions, by defining $E_{\lambda}:=\prod_{i}E_{i}^{m_{i}}$ if
$\lambda=1^{m_{1}}2^{m_{2}}\dots$.

By definition, we have that

$1+\sum_{n\geq 1}(-1)^{n}e_{n}t^{n}=-\sum_{m\geq 0}E_{m}t^{m}+\prod_{i\geq
1}(1-t^{i})^{a_{i}}.$

Therefore:

$\exp(-\sum_{n\geq 1}u_{n}t^{n}/n)=1-\frac{\sum_{m\geq
1}E_{m}t^{m}}{\prod_{i\geq 1}(1-t^{i})^{a_{i}}}.$

######  Proposition 11.8.

We have that

$E_{m}=-\sum_{n=1}^{m}(-1)^{n}c_{m-n}\sum_{\lambda\vdash
n}\operatorname{sgn}(\lambda)\frac{u_{\lambda}}{z_{\lambda}}.$

In particular $E_{m}\in(u_{1},u_{2},\dots)\Lambda$.

###### Proof.

By (11.4) we have:

$\exp(-\sum_{n\geq 1}u_{n}t^{n}/n)\prod_{i\geq
1}(1-t^{i})^{a_{i}}=1-\sum_{n\geq 1}(-1)^{n}e_{n}t^{n}.$

We have the well-known identity

$\exp(-\sum_{n\geq 1}v_{n}t^{n}/n)=\sum_{n}(-1)^{n}\sum_{\lambda\vdash
n}\operatorname{sgn}(\lambda)\frac{v_{\lambda}}{z_{\lambda}},$

for any family of independent variables $v_{n}$. (In particular, for
$v_{n}=p_{n}$ this identity follows from (11.5) and the character of the sign
representation).

Thus the coefficient of $t^{m}$ in $\exp(-\sum
u_{n}t^{n}/n)\prod_{i}(1-t^{i})^{a_{i}}$ is

$c_{m}+\sum_{n\geq 1}c_{m-n}(-1)^{n}\sum_{\lambda\vdash
n}\operatorname{sgn}(\lambda)\frac{u_{\lambda}}{z_{\lambda}}.$

Subtracting $c_{m}$ and multiplying by $-1$, we obtain the result. ∎

Since $|A|\leq k$ we have that $E_{m}=\pm e_{m}$ for all $m>k$. Hence
$f(E_{m})=0$ for all $m\geq k$, and specializing we obtain:

$\exp\left(-\sum_{n}f(u_{n})t^{n}/n\right)=1-\frac{\sum_{m=1}^{k}f(E_{m})t^{m}}{\prod_{i\geq
1}(1-t^{i})^{a_{i}}}.$

Therefore we have that

$\sum_{n}\frac{f(u_{n})}{n}t^{n}=-\log\left(1-\frac{\sum_{m=1}^{k}E_{m}t^{m}}{\prod_{i\geq
1}(1-t^{i})^{a_{i}}}\right).$

Because the numerator $\sum_{m=1}^{k}f(E_{m})t^{m}$ is nilpotent of order $r$
we obtain a finite Taylor expansion

$\displaystyle\sum_{n}\frac{f(u_{n})}{n}t^{n}$
$\displaystyle=\frac{\sum_{m=1}^{k}f(E_{m})t^{m}}{\prod_{i\geq
1}(1-t^{i})^{a_{i}}}+\frac{(\sum_{m=1}^{k}f(E_{m})t^{m})^{2}}{2\prod_{i\geq
1}(1-t^{i})^{2a_{i}}}+\dots+\frac{(\sum_{m=1}^{k}f(E_{m})t^{m})^{r-1}}{(r-1)\prod_{i\geq
1}(1-t^{i})^{(r-1)a_{i}}}$ (11.9) $\displaystyle=\sum_{\lambda\in{\rm
Part}(r,k)}f(E_{\lambda})t^{|\lambda|}\frac{(\operatorname{rank}(\lambda)-1)!}{\lambda!\left(\prod_{i}(1-t^{i})^{a_{i}}\right)^{\operatorname{rank}(\lambda)}},$

where the sum in the second line is over all integer partitions whose Young
diagrams have $\leq k$ columns and $<r$ rows. In the passage to the second
line we have used the multinomial identity

$\left(\sum_{i=1}^{k}x_{i}\right)^{n}=\sum_{\lambda=1^{m_{1}}\dots
k^{m_{k}},~{}\sum_{i}m_{i}=n}{n\choose m_{1},\dots,m_{k}}x_{\lambda}.$

Equation (11.9) is our desired expression for
$\sum_{n}\frac{f(u_{n})}{n}t^{n}$. The remaining subsections are independent
of each other. In the next two subsections we apply (11.9) to construct a
basis for $\mathcal{V}_{A,r}\cap\mathcal{F}_{\leq k}$, then describe how to
use (11.9) to compute the operator $\pi_{k}$. After that, we prove Theorem
1.17 using a related identity. Finally, we discuss the relationship between
characters of ${\mathbf{FS}}{{}^{\mathrm{op}}}$ and $\mathbf{FI}_{r}$ modules.

####  11.3. A convenient basis for $\mathcal{V}_{A,r}\cap\mathcal{F}_{\leq k}$

######  Definition 11.10.

Let ${\rm Part}(r,k)$ be the set of integer partitions whose Young diagrams
have $\leq k$ columns and $<r$ rows.

From the point of view of character functions, the elements
$\pi_{k}(p_{\nu}\exp(\sum_{i}a_{i}y_{k}))$ for $\nu\in{\rm Part}(r,k)$ are a
natural basis for $\mathcal{V}_{A,r}$. However, we do not know a simple
expression for these elements. In this subsection we construct a different
basis, better adapted to Equation (11.9).

######  Proposition 11.11.

There is an isomorphism

$R\cong\frac{\mathbb{Q}[E_{1},\dots,E_{k}]}{(E_{1},\dots,E_{k})^{r}}$

defined by $E_{i}\mapsto f(E_{i})$ for $i=1,\dots,k$. Consequently, the set
$\\{f(E_{\nu})\\}_{\nu\in\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{t}(r,k)}$ is a
basis for $R$.

###### Proof.

We already know, by Proposition 10.16, that $u_{i}\mapsto f(u_{i})$ for
$i=1,\dots,k$ gives an isomorphism
$\frac{\mathbb{Q}[u_{1},\dots,u_{k}]}{(u_{1},\dots,u_{k})^{r}}\to R$. Since
$E_{m}\in(u_{1},\dots)$ we have that $f(s)=0$ for all
$s\in(E_{1},\dots,E_{k})^{r}$ thus the homomorphism is well defined.

To see that it is surjective, note that by Proposition 11.8

$E_{m}\equiv u_{m}/m+\sum_{1\leq
n<m}c_{m-n}u_{n}/n\mod(u_{1},u_{2}\dots)^{2}.$

Using this equation, by upper-triangularity we may write any $f(u_{m})$ as a
linear combination of $f(E_{m})$ modulo $(u_{1},\dots,u_{k})^{2}$. Since any
collection of elements
$x_{1},\dots,x_{k}\in\mathbb{Q}[u_{1},\dots,u_{k}]/(u_{1},\dots,u_{k})^{r}$
satisfying $x_{i}\equiv u_{i}\mod(u_{1},\dots,u_{k})^{2}$ generate the ring,
it follows that these linear combinations of $f(E_{m})$ generate. Thus the
homomorphism is surjective, and comparing dimensions it is also injective. ∎

Since by Proposition 11.11 the elements $\\{f(E_{\nu})\\}_{\nu\in{\rm
Part}(r,k)}$ form a basis of $R$, we may make the following definition.

######  Definition 11.12.

We define
$\\{L_{\nu}\\}_{\nu\in\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{t}(r,k)}$ to be
the basis of $\mathcal{V}_{A,r}\cap\mathcal{F}_{\leq k}$ which is dual to
$\\{f(E_{\nu})\\}_{\lambda\in{\rm Part}(r,k)}$.

Thus $L_{\nu}$ is the unique element of
$\mathcal{V}_{A,r}\cap\mathcal{F}_{\leq k}$ satisfying $\langle
E_{\lambda},L_{\nu}\rangle=\delta(\lambda,\nu)$ for all integer partitions
$\lambda,\nu\in{\rm Part}(r,k)$. We will now an explicit epxression for
$L_{\nu}$.

######  Definition 11.13.

Let $B=1^{b_{1}}2^{b_{2}}\dots$ be an integer partition. For $n\in\mathbb{Z}$
we define $g_{B}(n)$ be the coefficient of $t^{n}$ in
$\prod_{i}(1-t^{i})^{-b_{i}}$. In particular $g_{B}(n)=0$ for $n<0$. For
$n\geq 0$ we have

$g_{B}(n)=\sum_{\lambda\vdash
n,~{}\lambda=1^{m_{2}}2^{m_{2}}\dots}~{}\prod_{i\geq
1}\left(\kern-3.00003pt\left(\genfrac{}{}{0.0pt}{}{b_{i}}{m_{i}}\right)\kern-3.00003pt\right).$

The function $g_{B}(n)$ equals a quasi-polynomial of degree
$\operatorname{rank}(B)$, for all $n\geq 0$.

######  Definition 11.14.

If $\nu=1^{\ell_{1}}2^{\ell_{2}}\dots$ and
$\nu^{\prime}=1^{\ell_{1}^{\prime}}2^{\ell_{2}^{\prime}}\dots$ we write
$\nu^{\prime}+\nu$ for the partition
$1^{\ell_{1}+\ell_{1}^{\prime}}2^{\ell_{2}+\ell_{2}^{\prime}}\dots$, making
the set of partitions into a commutative monoid (corresponding to
multiplication of monomials). For $n\in\mathbb{N}$ write $n\cdot\nu$ for the
$n$-fold addition of $\nu$ in this monoid.

######  Definition 11.15.

Define $H_{\nu}(n):=n~{}g_{\operatorname{rank}(\nu)\cdot
A}(n-|\nu|)\frac{(\operatorname{rank}(\nu)-1)!}{\nu!}$

By Equation (11.9), we have that
$f(u_{n})=\sum_{\nu\in\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{t}(r,k),|\nu|>0}H_{\nu}(n)f(E_{\nu}).$
So for $\lambda=n_{1}\geq n_{2}\geq\dots$ we have that

$f(u_{\lambda})=\prod_{i}\left(\sum_{\nu\in\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{t}(r,k)}H_{\nu}(n_{i})f(E_{\nu})\right)=\sum_{\nu\in\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{t}(r,k)}f(E_{\nu})\sum_{\nu=\sum_{i}\nu_{i},~{}|\nu_{i}|>0}\prod_{i\geq
1}H_{\nu_{i}}(n_{i}),$

Pairing with $L_{\nu}$ we get a formula for $\langle
u_{\lambda},L_{\nu}\rangle$. By Lemma 10.10 we obtain:

######  Proposition 11.16.

There is an explicit formula for $L_{\nu}$.

$L_{\nu}=\exp(\sum_{i}a_{i}y_{i})\sum_{\lambda=n_{1}\geq
n_{2}\geq\dots}\frac{p_{\lambda}}{z_{\lambda}}\
\sum_{\nu=\sum_{i}\nu_{i},|\nu_{i}|>0}\prod_{i}H_{\nu_{i}}(n_{i}),$

where the last sum is over all finite tuples of integer partitions
$(\nu_{1},\nu_{2},\dots,\nu_{l})$ with $|\nu_{i}|>0$ and
$\nu=\sum_{i}\nu_{i}$.

######  Example 11.17.

Let $\nu$ be the partition $3\geq 2$. Then $L_{\nu}$ is
$\exp(\sum_{i}a_{i}y_{i})$ times

$\sum_{n\geq 1}\frac{p_{n}}{n}H_{3\geq 2}(n)+\sum_{n>m\geq
1}\frac{p_{n}p_{m}}{nm}(H_{3}(n)H_{2}(m)+H_{2}(n)H_{3}(m))+\sum_{n\geq
1}\left(\frac{p_{n}}{n}\right)^{2}H_{3}(n)H_{2}(n).$

####  11.4. Computing the operator $\pi_{k}$

Let $\lambda$ be an integer partition. We discuss how to compute
$\pi_{k}(p_{\nu}\exp(\sum_{i}a_{i}y_{k}))$.

We assume that $p_{\nu}\exp(\sum_{i}a_{i}y_{k})\in\mathcal{V}_{A,r}$, or
equivalently that $\operatorname{rank}(\nu)<r$. If the Young diagram of
$\lambda$ has $>k$ columns, then $\epsilon_{k}(p_{\nu})=0$ and thus
$\pi_{k}(p_{\nu}\exp(\sum_{i}a_{i}y_{k}))=0.$

Therefore, suppose that $\nu\in{\rm Part}(r,k)$. By Lemma 10.10 the
coefficients $b_{\lambda,\nu}$ in the expansion

$\pi_{k}(p_{\nu}\exp(\sum_{i}a_{i}y_{i}))=\sum_{\lambda}b_{\lambda,\nu}\frac{p_{\lambda}}{z_{\lambda}}\exp(\sum_{i}a_{i}y_{i}),$

are given by $\langle
u_{\lambda},\pi_{k}(p_{\nu}\exp(\sum_{i}a_{i}y_{k}))\rangle$. Let

(11.18) $f(u_{\lambda})=\sum_{\nu\in{\rm
Part}(r,k)}\overline{b}_{\lambda,\nu}f(u_{\nu})$

be the unique expansion $f(u_{\lambda})$ in $R$. By Proposition 10.11, since
$\pi_{k}(p_{\nu}\exp(\sum_{i}a_{i}y_{i}))\in\mathcal{F}_{\leq
k}\cap\mathcal{V}_{A,r}$, we have that $\langle
u_{\lambda},\pi_{k}(p_{\nu}\exp(\sum_{i}a_{i}y_{k}))\rangle$ only depends on
$f(u_{\lambda})$. And by the definition of $\pi_{k}$, we have that for every
$\psi\in\mathrm{P}\mathrm{a}\mathrm{r}\mathrm{t}(n,k)$ that

$\langle u_{\psi},\pi_{k}(p_{\nu}\exp(\sum_{i}a_{i}y_{i}))\rangle=\langle
u_{\psi},p_{\nu}\exp(\sum_{i}a_{i}y_{i})\rangle=\delta(\psi,\nu).$

Applying this to (11.18) we obtain

$b_{\lambda,\nu}=\overline{b}_{\lambda,\nu}.$

The coefficients $\overline{b}_{\lambda,\nu}$ can be computed from
$\\{\overline{b}_{n,\nu}\\}_{n\in\mathbb{N}}$ by multiplication. And Equation
(11.9) can be used to compute $\overline{b}_{n,\nu}$. However, the equation
for $\overline{b}_{\lambda,\nu}$ one obtains in this way is unwieldy. So we
will just carry out this computation in an example.

######  Example 11.19.

Suppose that $k=2,r=3$ and $a_{1}=1,a_{i}=0$ for $i>1$. Then $c_{0}=1$,
$c_{1}=-1$ and $c_{i}=0$ for $i>1$. So $f(E_{1})=-f(u_{1})$,
$f(E_{2})=f(u_{2})/2-f(u_{1})^{2}/2+f(u_{1})$, and $f(E_{i})=0$ for $i>3$.
Then we have

$\frac{\sum_{m=1}^{k}f(E_{m})t^{m}}{\prod_{i\geq
1}(1-t^{i})^{a_{i}}}=-f(u_{1})t+\frac{(f(u_{2})+f(u_{1})^{2})t^{2}}{2(1-t)}.$

So by (11.9) we compute that $\sum_{n\geq 1}\frac{f(u_{n})}{n}t^{n}$ is

$f(u_{1})(-t)+f(u_{2})\frac{t^{2}}{2(1-t)}+f(u_{1}^{2})\left(t^{2}+\frac{t^{2}}{2(1-t)}\right)+f(u_{1}u_{2})\frac{-t^{2}}{2(1-t)}+f(u_{2}^{2})\frac{t^{4}}{4(1-t)^{2}}$

Thus for $n\geq 4$ we have that

$f(u_{n})=\frac{n}{2}f(u_{2})+\frac{n}{2}f(u_{1}^{2})-\frac{n}{2}f(u_{1}u_{2})+\frac{n(n+1)}{4}f(u_{2}^{2}).$

Which computes $\overline{b}_{n,\nu}$ for all $\nu$ and $n\geq 4$. To compute
$\overline{b}_{\lambda,\nu}$ for the remaining cases, $\lambda=n\geq m$,
simply multiply $f(u_{n})f(u_{m})$.

####  11.5. Proof of Theorem 1.17

We use a generating function argument to prove Theorem 1.17.

###### Proof of Theorem 1.17.

Fix an integer partition $\lambda$ and consider the generating fuction

$F_{\lambda}(t)=\sum_{k\geq
0}s_{(k,\lambda)}~{}t^{k+|\lambda|}\in\Lambda[[t]].$

The following identity appears in the theory of $\mathbf{FI}$ modules:

(11.20)
$F_{\lambda}(t)=q_{\lambda}(t)+\sum_{\nu~{}|~{}\lambda-\nu\in\mathrm{V}\mathrm{S}}(-1)^{|\lambda|-|\nu|}s_{\nu}t^{|\nu|}\sum_{m\geq
0}h_{m}t^{m},$

where the sum is over all $\nu$ such that the Young diagram of $\lambda$ minus
the Young diagram of $\nu$ is a vertical strip, and
$q_{\lambda}(t)\in\Lambda[t]$ has degree $\leq|\lambda|$ (in both $\Lambda$
and $t$). In [22, §5] these polynomials also go under the name $q_{\lambda}$,
although they use the coordinates $t_{i}:=p_{i}/i$ for the ring of symmetric
functions. (They show that this identity corresponds to a resolution in the
quotient of $\operatorname{Mod}\mathbf{FI}/\operatorname{Mod}\mathbf{FI}^{\rm
tors}$ and interpret $q_{\lambda}$ in terms of local cohomology in [22,
§7.4]).

Now an integer partition $A$ and $r,k\in\mathbb{N}$ with $k\geq|A|$. We use
the notation $R,E_{\lambda}$ from above, relative to these choices if
parameters. From here on we consider the image of our generating functions
under the specialization $\Lambda[[t]]\to R[[t]].$ We have that

$s_{\nu}t^{|\nu|}\sum_{m\geq
0}h_{m}t^{m}=\frac{s_{\nu}t^{|\nu|}}{1+\sum_{m\geq
1}(-1)^{m}e_{m}t^{m}}=\frac{s_{\nu}t^{|\nu|}}{\prod_{i}(1-t^{i})^{a_{i}}-\sum_{m=1}^{k}E_{m}t^{m}}.$

factoring out $\prod_{i}(1-t^{i})^{a_{i}}$ and applying a geometric series
expansion we have that this equals

$\frac{s_{\nu}t^{|\nu|}}{\prod_{i}(1-t^{i})^{a_{i}}}\sum_{n=0}^{r-1}\left(\frac{\sum_{m=1}^{k}E_{m}t^{m}}{\prod_{i}(1-t^{i})^{a_{i}}}\right)^{n}.$

For any $u\in\mathcal{V}_{A,r}\cap\mathcal{F}_{\leq k}$, applying
$\langle-,u\rangle$ to this expression produces a rational function in
$\mathbb{Z}[[t]]$. Thus $\sum_{m\geq 0}\langle s_{\nu}h_{m},u\rangle
t^{m+|\nu|}$ is rational and therefore so is $\sum_{m\geq|\nu|}\langle
s_{(m,\nu)},u\rangle t^{m+|\nu|}$ by (11.20). The denominator of this function
is of degree $\leq r|A|$ and its roots are roots of unity of order $\leq|A|$.

Finally if $M$ is an ${\mathbf{FS}}{{}^{\mathrm{op}}}$ module of type $(d,s)$,
then $\operatorname{ch}(M)$ is a finite sum of
$u_{A,r,k}\in\mathcal{V}_{A,r}\cap\mathcal{F}_{\leq k}$ for $|A|\leq d$ and
$r\leq s$. So the generating function $\sum_{m\geq|\nu|}\langle
s_{(m,\nu)},\operatorname{ch}(M)\rangle t^{m+|\nu|}$ is rational and its
denominator has degree $\leq sd$, and the roots of the denominator are roots
of unity of order $\leq d$. ∎

######  Remark 11.21.

In practice, the strategy of the proof of Theorem 1.17 can be used to make
computations. For example, fix $A$ an integer partition, and
$d,k\in\mathbb{N}$ with $d>|A|$. Then the proof of Theorem 1.17 yields a
formula for $\sum_{n}\langle h_{n},L_{\lambda}\rangle t^{n}$. Namely it is

$\sum_{n}\langle h_{n},L_{\lambda}\rangle
t^{n}=\frac{t^{|\lambda|}\operatorname{rank}(\lambda)!}{\lambda!\left(\prod_{i}(1-t^{i})^{a_{i}}\right)^{\operatorname{rank}(\lambda)+1}}.$

∎

####  11.6. Relationship with $\mathbf{FI}_{d}$ module characters

Sam–Snowden [24] characterized the Hilbert series of modules over the category
$\mathbf{FI}_{d}$. They express these characters as polynomials in the
elements $\sigma_{n}:=\sum_{k\geq 0}{n\choose k}h_{n}$, and observe that
$\sigma_{n}$ takes the form $q_{n}(T_{1},T_{2},\dots)\exp(y_{1})$ where
$q_{n}$ is a polynomial in $T_{k}:=\sum_{n\geq 0}{n\choose k}\frac{p_{n}}{n}.$
In fact, $q_{n}$ is the $n^{\rm th}$ complete ordinary Bell polynomial. For
$\lambda=1^{m_{1}}2^{m_{2}}\dots$, define
$\sigma^{\lambda}:=\prod_{i}\sigma_{i}^{m_{i}}$. Then if $k\in\mathbb{N}$ and
$\lambda$ satsifies $\operatorname{rank}(\lambda)\leq k$ we have that

$\sigma^{\lambda}\sigma_{0}^{k-\operatorname{rank}(\lambda)}\in\mathcal{F}_{\leq
k}\cap\mathcal{V}_{1^{k},|\lambda|+1}$

Sam–Snowden show that these elements are linearly independent. Thus as
$\lambda$ ranges over partitions with $\leq k$ rows and $|\lambda|<r$, we
obtain a collection of linearly independent elements in $\mathcal{F}_{\leq
k}\cap\mathcal{V}_{1^{k},r}.$ Since the dimension of this space is the number
of partitions with $\leq k$ rows and $<r$ columns, which is strictly larger
than the number of
$\sigma^{\lambda}\sigma_{0}^{k-\operatorname{rank}(\lambda)},$ this collection
of elements does not span.

Sam–Snowden compute the character of an $\mathbf{FI}_{d}$ module to be a sum
of $s_{\nu}\sigma^{\lambda}\sigma_{0}^{k-\operatorname{rank}(\lambda)}$. We do
not know if it is possible to use such elements to construct a basis of
$\mathcal{F}_{\leq k^{\prime}}\cap\mathcal{V}_{1^{k},r^{\prime}}$ for some
$r^{\prime},k^{\prime}$.

###  12\. Further Questions

We list some questions related to our work. Many of these questions stem from
the fact that we still do not understand the representation theory of
${\mathbf{FS}}{{}^{\mathrm{op}}}$ modules.

####  12.1. Behavior of the type under kernels and cokernels

For applications, it would be useful to have more sophisticated methods of
proving that a specific ${\mathbf{FS}}{{}^{\mathrm{op}}}$ module is of class
$(d,s)$.

There are two numerical invariants associated to an $\mathbf{FI}$ module $M$,
the stable degree $\delta(M)$ and the local degree $h^{\rm max}(M)$,
introduced and used to great effect in [8]. The type of an
${\mathbf{FS}}{{}^{\mathrm{op}}}$ module is roughly analogous to these
invariants. Given a morphism of ${\mathbf{FS}}{{}^{\mathrm{op}}}$ modules
$f:N_{1}\to N_{2}$, is it possible to bound the type of $\ker f$ and
$\operatorname{coker}f$ in terms of the type of $N_{1}$ and $N_{2}$ (or other
invariants)?

####  12.2. Grothendieck group of ${\mathbf{FS}}{{}^{\mathrm{op}}}$ modules

When do two finitely generated ${\mathbf{FS}}{{}^{\mathrm{op}}}$ modules have
the same character? In other words, what is the kernel of the map of
Grothendieck groups:
$\mathrm{K}_{0}({\mathbf{FS}}{{}^{\mathrm{op}}})\to\prod_{n}\mathrm{K}_{0}(\mathrm{S}_{n})$?

####  12.3. Are ${\mathbf{FS}}{{}^{\mathrm{op}}}$ module characters cycle
bounded?

We say that a symmetric function is _cycle bounded_ if it is a linear
combination of finitely many functions of the form

$\exp(\sum_{i\geq 1}a_{i}y_{i})\sum_{\lambda\text{ integer
partition}}c_{\lambda}p_{\lambda},$

where only finitely many $c_{\lambda}$ are non-zero and
$(a_{1},a_{2},\dots)\in\mathbb{N}^{\oplus\infty}$. Is the character of every
finitely generated ${\mathbf{FS}}{{}^{\mathrm{op}}}$ module cycle bounded? If
not, what is an example of a finitely generated
${\mathbf{FS}}{{}^{\mathrm{op}}}$ module that is not cycle bounded?

We note that projective ${\mathbf{FS}}{{}^{\mathrm{op}}}$ modules are cycle
bounded, and cycle bounded characters are closed under sums, induction product
and Kronecker product. And if $t$ is cycle bounded, then $s_{\lambda}[t]$ is
also cycle bounded. On the other hand, Sam–Snowden show in [24] that when
$d>1$ there are $\mathbf{FI}_{d}$ modules that are not cycle-bounded.

####  12.4. Characters of ${\mathbf{FS}}{{}^{\mathrm{op}}}$ modules in
characteristic $>0$

What are the consequences of Theorem 1.2 for fields of positive
characteristic? Is it possible to carry out a similar combinatorial analysis
for the characters of ${\mathbf{FS}}{{}^{\mathrm{op}}}$ modules in this case?

####  12.5. Other combinatorial categories

The arguments we use to prove Theorem 1.2 are mostly general. The specific
combinatorics of ${\mathbf{FS}}{{}^{\mathrm{op}}}$ only enters in §6. We are
interested in extensions of the method of proof of Theorem 1.2, to other
categories studied by Sam–Snowden [23]. See for instance Conjecture 3.22. For
which combinatorial categories does the method of proof of Theorem 1.2 work?

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