# On relational complexity and base size of finite primitive groups

Veronica Kelsey and Colva M. Roney-Dougal Acknowledgements: The authors would
like to thank Nick Gill for encouraging us to look at this problem and Peter
Cameron for his helpful remarks, which improved the paper.
Key words: permutation group; base size; relational complexity; computational
complexity
MSC2020: 20B15; 20B25; 20E32; 20-08

###### Abstract

In this paper we show that if $G$ is a primitive subgroup of $\mathrm{S}_{n}$
that is not large base, then any irredundant base for $G$ has size at most
$5\log n$. This is the first logarithmic bound on the size of an irredundant
base for such groups, and is best possible up to a small constant. As a
corollary, the relational complexity of $G$ is at most $5\log n+1$, and the
maximal size of a minimal base and the height are both at most $5\log n.$
Furthermore, we deduce that a base for $G$ of size at most $5\log n$ can be
computed in polynomial time.

##  1 Introduction

Let $\Omega$ be a finite set. A base for a subgroup $G$ of
$\mathrm{Sym}({\Omega})$ is a sequence
$\Lambda=(\omega_{1},\ldots,\omega_{l})$ of points of $\Omega$ such that
$G_{\Lambda}=G_{\omega_{1},\ldots,\omega_{l}}=1$. The minimum base size,
denoted $\mathrm{b}(G,\Omega)$ or just $\mathrm{b}(G)$ if the meaning is
clear, is the minimum length of a base for $G$. Base size has important
applications in computational group theory; see, for example, [20] for the
importance of a base and strong generating set.

In [16] Liebeck proved the landmark result that with the exception of one
family of groups, if $G$ is a primitive subgroup of
$\mathrm{S}_{n}=\mathrm{Sym}({\\{1,\ldots,n\\}})$ then $\mathrm{b}(G)<9\log
n$. The exceptional family are called large-base groups: product action or
almost simple groups whose socle is one or more copies of the alternating
group $\mathrm{A}_{r}$ acting on $k$-sets. In [18] Moscatiello and Roney-
Dougal improve this bound, and show that if $G$ is not large base then either
$G=\mathrm{M}_{24}$ in its 5-transitive action of degree 24, or
$\mathrm{b}(G)\leq\lceil\log n\rceil+1$. Here and throughout all $\log$s are
of base 2.

We say that a base $\Lambda=(\omega_{1},\ldots,\omega_{k})$ for a permutation
group $G$ is irredundant if

$G>G_{\omega_{1}}>G_{\omega_{1},\omega_{2}}>\cdots>G_{\omega_{1},\ldots,\omega_{l}}=1.$

If no irredundant base is longer than $\Lambda$, then $\Lambda$ is a maximal
irredundant base and we denote the length of $\Lambda$ by
$\mathrm{I}(G,\Omega)$ or $\mathrm{I}(G)$.

From Liebeck’s $9\log n$ bound on base size, a straightforward argument (see
Lemma 1.2) shows that if $G$ is a primitive non-large-base subgroup of
$\mathrm{S}_{n}$, then $\mathrm{I}(G)\leq 9\log^{2}n$. However, in [14] Gill,
Lodá and Spiga conjecture that for such groups $G$ there exists a constant $c$
such that $\mathrm{I}(G)\leq c\log n$. They show that for some families of
groups the conjecture holds with $c=7$. Our main result establishes this
conjecture, whilst also improving the constant.

######  Theorem 1.1.

Let $G$ be a primitive subgroup of $\mathrm{S}_{n}$. If $G$ is not large base,
then

$\mathrm{I}(G)<5\log n.$

There are infinitely many primitive groups for which the maximal irredundant
base size is greater than $\lceil\log n\rceil+1$. For example, if $d\geq 5$,
$G=\mathrm{PGL}_{d}(3)$ and $\Omega$ is the set of 1-spaces of
$\mathbb{F}_{3}^{d}$, then by Theorem 3.1
$\mathrm{I}(G,\Omega)=2d-1>\lceil\log n\rceil+1$. Hence, up to a small
constant the bounds in Theorem 1.1 are the best possible.

Relational complexity has been extensively studied in model theory, see for
example [15]. A rephrasing of the definition, to make it easier to work with
for permutation groups, was introduced more recently in [11]. For an excellent
discussion and more context, see [13]. Let $k,l\in\mathbb{N}$ with $k\leq l$,
and let
$\Lambda=(\lambda_{1},\ldots,\lambda_{l}),\Sigma=(\sigma_{1},\ldots,\sigma_{l})\in\Omega^{l}$.
We say that $\Lambda$ and $\Sigma$ are $k$-subtuple complete with respect to a
subgroup $G$ of $\mathrm{Sym}({\Omega})$, and write $\Lambda\sim_{k}\Sigma$,
if for every subset of $k$ indices $i_{1},\ldots,i_{k}$ there exists $g\in G$
such that
$(\lambda_{i_{1}}^{g},\ldots,\lambda_{i_{k}}^{g})=(\sigma_{i_{1}},\ldots,\sigma_{i_{k}}).$
The relational complexity of $G$, denoted $\mathrm{RC}(G)$, is the smallest
$k$ such that for all $l\geq k$ and all $\Lambda,\Sigma\in\Omega^{l}$, if
$\Lambda\sim_{k}\Sigma$ then $\Lambda\in\Sigma^{G}$. In [10] Cherlin gives
examples of groups with relational complexity 2, called binary groups, and
conjectures that this list is complete. In a dramatic breakthrough, Gill,
Liebeck and Spiga have just announced a proof of this conjecture, see [13].

Let $\Lambda$ be a base for a permutation group $G$. Then $\Lambda$ is minimal
if no proper subsequence of $\Lambda$ is a base. We denote the maximum size of
a minimal base by $\mathrm{B}(G)$. The height $\mathrm{H}(G)$ of $G$, is the
size of the largest subset $\Delta$ of $\Omega$ with the property that
$G_{(\Gamma)}\neq G_{(\Delta)}$ for each $\Gamma\subsetneq\Delta$. The
following key lemma relates all of these group statistics studied in this
paper.

######  Lemma 1.2.

_[14, Equation 1.1 and Lemma 2.1]_ Let $G$ be a subgroup of $\mathrm{S}_{n}$.
Then

$\mathrm{b}(G)\leq\mathrm{B}(G)\leq\mathrm{H}(G)\leq\mathrm{I}(G)\leq\mathrm{b}(G)\log
n,$

and

$\mathrm{RC}(G)\leq\mathrm{H}(G)+1.$

In [14], Gill, Lodá and Spiga prove that if $G\leq\mathrm{S}_{n}$ is primitive
and not large base, then $\mathrm{H}(G)<9\log n$ and so $\mathrm{RC}(G)<9\log
n+1$ and $\mathrm{B}(G)<9\log n$.

It will follow immediately from Theorem 1.1 and Lemma 1.2 that we can tighten
the all of these bounds.

######  Corollary 1.3.

Let $G$ be a primitive subgroup of $\mathrm{S}_{n}$. If $G$ is not large base
then

$\mathrm{RC}(G)<5\log n+1,\;\;\;\;\mathrm{B}(G)<5\log
n,\;\;\;\;\text{and}\;\;\;\;\mathrm{H}(G)<5\log n.$

Blaha proved in [1] that the problem of computing a minimal base for a
permutation group $G$ is NP-hard. Furthermore, he showed that the obvious
greedy algorithm to construct an irredundant base for $G$ produces one of size
$O(\mathrm{b}(G)\log\log n)$. Thus if $G$ is primitive and not large base, it
follows from Liebeck’s result that in polynomial time one can construct a base
of size $O(\log n\log\log n)$. Since an irredundant base size can be computed
in polynomial time (see for example [20]), we get the following corollary,
which improves this bound to the best possible result, up to a constant.

######  Corollary 1.4.

Let $G$ be a primitive subgroup of $\mathrm{S}_{n}$ which is not large base.
Then a base for $G$ of size at most $5\log n$ can be constructed in polynomial
time.

(We note that using the bound on $\mathrm{B}(G)$ from [14], a very slightly
more complicated argument would yield a similar result, but with $9\log n$ in
place of $5\log n$.)

The paper is structured as follows. In Section 2 we prove some preliminary
lemmas about $\mathrm{I}(G)$. In Section 3 we give upper and lower bounds on
the size of an irredundant base for $\mathrm{PGL}_{d}(q)$ acting on subspaces
of $\mathbb{F}_{q}^{d}$, which differ by only a small amount. In Section 4 we
prove a result which is a slight strengthening of Theorem 1.1 for almost
simple groups. Finally, in Section 5 we complete the proof of Theorem 1.1.

##  2 Preliminary bounds on group statistics

Here we collect various lemmas about bases, and about the connection between
$\mathrm{I}(G)$ and other group statistics.

For a subgroup $G$ of $\mathrm{Sym}({\Omega})$ and a fixed sequence
$(\omega_{1},\ldots,\omega_{l})$ of points from $\Omega$, we let
$G^{(i)}=G_{\omega_{1},\ldots,\omega_{i}}$ for $0\leq i\leq l$, so
$G^{(0)}=G$. Furthermore, the maximum length of a chain of subgroups in $G$ is
denoted by $\ell(G)$.

######  Lemma 2.1.

Let $G$ be a subgroup of $\mathrm{S}_{n}$.

  1. (i)

If $G$ is insoluble, then $\mathrm{I}(G)<\log|G|-1$.

  2. (ii)

If $G$ is transitive and $n\geq 5$, then $\mathrm{I}(G)\leq\log|G|-1.$

  3. (iii)

If $G$ is transitive and $b=\mathrm{b}(G)$, then $\mathrm{I}(G)\leq(b-1)\log
n+1.$

###### Proof.

Let $a$ be the number of prime divisors of $|G|$, counting multiplicity. Since
$G$ is insoluble there exists a prime greater than $2^{2}$ dividing $|G|$, and
so $|G|>2^{a+1}$. It is clear that $\mathrm{I}(G)\leq\ell(G)\leq a$, and so
Part (i) follows, and we assume from now on that $G$ is transitive.

Let ${l}=\mathrm{I}(G)$ with a corresponding base
$\Lambda=(\omega_{1},\ldots,\omega_{l})$. Since $G$ is transitive,
$[G^{(0)}:G^{(1)}]=n$ by the Orbit-Stabiliser Theorem. From
$[G^{(i-1)}:G^{(i)}]\geq 2$ for $2\leq i\leq{l}$, it follows that $|G|\geq
2^{l-1}n$. Hence if $n\geq 5$ then $|G|\geq 2^{l-1}\cdot 5>2^{l+1}$. Therefore
by taking logs Part (ii) follows.

Similarly, $|G|<n^{b}$, and so $2^{{l}-1}n\leq|G|\leq n^{b}$. Hence

$l-1+\log n=\log(2^{l-1}n)\leq\log|G|\leq b\log n,$

and so

${l}\leq b\log n-\log n+1=(b-1)\log n+1$

and Part (iii) follows. ∎

######  Lemma 2.2.

Let $G$ be a subgroup of $\mathrm{Sym}({\Omega})$, let $l\geq 1$ and let
$\Lambda=(\lambda_{1},\ldots,\lambda_{l})\in\Omega^{l}$. Then there exists a
subsequence $\Sigma$ of $\Lambda$ such that $\Sigma$ can be extended to an
irredundant base and $G_{\Sigma}=G_{\Lambda}$.

###### Proof.

The sequence $\Lambda$ cannot be extended to an irredundant base if and only
if there exists a subsequence $\lambda_{i},\ldots,\lambda_{i+j}$ of $\Lambda$
with $j\geq 1$ such that

$G^{(i)}=G^{(i+1)}=\cdots=G^{(i+j)}.$

Let $\Sigma$ be the subsequence of $\Lambda$ given by deleting all such
$\lambda_{i+1},\ldots,\lambda_{i+j}$. Since $G^{(i)}=G^{(i+j)}$ it follows
that $G_{\Lambda}=G_{\Sigma}$. ∎

The following describes the relationship between the irredundant base size of
a group and that of a subgroup.

######  Lemma 2.3.

Let $H$ and $G$ be subgroups of $\mathrm{S}_{n},$ with $H\leq G$. Then the
following hold.

  1. (i)

$\mathrm{I}(H)\leq\mathrm{I}(G)$.

  2. (ii)

If $H\trianglelefteq G$ then $\mathrm{I}(G)\leq\mathrm{I}(H)+\ell(G/H).$

  3. (iii)

If $H\trianglelefteq G$ and $[G:H]$ is prime, then
$\mathrm{I}(H)\leq\mathrm{I}(G)\leq\mathrm{I}(H)+1.$

###### Proof.

An irredundant base for $H\leq G$ can be extended to an irredundant base for
$G$, so Part (i) is clear. Part (ii) is [14, Lemma 2.8] and Part (iii) follows
immediately from Parts (i) and (ii). ∎

##  3 Groups with socle $\mathrm{PSL}_{d}(q)$ acting on subspaces

Throughout this section, let $q=p^{f}$ for $p$ a prime and $f\geq 1$, let $V$
be a $d$-dimensional vector space over $\mathbb{F}_{q},$ let
$\Omega=\mathcal{PG}_{m}(V)$ be the set of all $m$-dimensional subspaces of
$V$, and let $n=|\Omega|$. In this section we begin by proving Theorem 3.1
which bounds $\mathrm{I}(\mathrm{PGL}_{d}(q),\Omega)$ in terms of $d$ and $m$.
By finding lower bounds on $|\Omega|$ we then prove Proposition 3.6, which
bounds $\mathrm{I}(\mathrm{P}\Gamma\mathrm{L}_{d}(q),\Omega)$ in terms of
$|\Omega|$.

###  3.1 Bounds as a function of $d$ and $m$

In this subsection we prove the following theorem, which in the case $m=1$
recovers the lower bounds found by Lodá in [17].

######  Theorem 3.1.

Let $\mathrm{PGL}_{d}(q)$ act on $\Omega$. Then

$\mathrm{I}(\mathrm{PGL}_{d}(q))\leq(m+1)d-2m+1,$

and

$\mathrm{I}(\mathrm{PGL}_{d}(q))\geq\begin{cases}md-m^{2}+1&\text{ if }q=2,\\\
(m+1)d-m^{2}&\text{ if }q\neq 2.\end{cases}$

We begin by proving the upper bound in Theorem 3.1. Let $M=M(V)$ be the
algebra of all linear maps from $V$ to itself. Furthermore, let
$\omega_{0}=\langle\underline{0}\rangle$, let $l>1$ be an integer, let
$\Lambda=(\omega_{1},\omega_{2},\ldots,\omega_{l})\in\Omega^{l}$ and for
$0\leq k\leq l$, let

$M_{k}=\\{g\in M\;|\;\omega_{i}g\leq\omega_{i}\;\text{for}\;0\leq i\leq
k\\},\text{ so that }M_{0}=M.$

For $0\leq k\leq l-1$, it is easily verified that $M_{k+1}$ is a subspace of
$M_{k}$. Now assume in addition that

$M_{0}>M_{1}>\cdots>M_{l}=\mathbb{F}_{q}I$ (1)

with $l$ as large as possible. Fix a basis $\langle e_{1},\ldots,e_{d}\rangle$
of $V$ which first goes through $\omega_{1}\cap\omega_{2}$, then extends to a
basis of $\omega_{1}$, and then for each $k\geq 2$ extends successively to a
basis of $\langle\omega_{1},\ldots,\omega_{k}\rangle$. Therefore there exist
integers

$m=a_{1}\leq\cdots\leq a_{l}=d\;\text{ such that }\;\langle
e_{1},\ldots,e_{a_{k}}\rangle=\langle\omega_{1},\ldots,\omega_{k}\rangle.$

Since $\omega_{0}=\langle\underline{0}\rangle$, we may let $a_{0}=0$. From
now, on we identify $M$ with the algebra of $d\times d$ matrices over
$\mathbb{F}_{q}$ with respect to this basis.

We will show that $l\leq(m+1)d-2m+1$, from which the upper bound in Theorem
3.1 will follow. For $0\leq k\leq l-1$, let

$f_{k}=\dim(M_{k})-\dim(M_{k+1})$

and let $b_{k}=a_{k+1}-a_{k}$ so that $0\leq b_{k}\leq m$. In the following
lemmas we consider the possible values of $f_{k}$ based on $b_{k}$.

######  Lemma 3.2.

Let $f_{k}$ and $b_{k}$ be as above.

  1. (i)

The dimension of $M_{1}$ is $d^{2}-m(d-m),$ and so $f_{0}=m(d-m)$.

  2. (ii)

$f_{1}=b_{1}(d-b_{1})$.

###### Proof.

First consider Part (i). Since $\omega_{1}=\langle e_{1},\ldots,e_{m}\rangle$
it follows that $g=(g_{ij})\in M_{1}$ if and only if $e_{i}g\in\omega_{1}$ for
$1\leq i\leq m$. Equivalently, $g_{ij}=0$ for $1\leq i\leq m$ and $m+1\leq
j\leq d$. Hence $\dim(M_{1})=d^{2}-m(d-m)$, and the final claim follows from
$\dim(M_{0})=d^{2}$.

Now consider (ii). The subspace $M_{2}$ contains all matrices of shape

$\begin{pmatrix}x_{1}&0&0&0\\\ x_{2}&x_{3}&0&0\\\ x_{4}&0&x_{5}&0\\\
y_{1}&y_{2}&y_{3}&y_{4}\end{pmatrix}$

where $x_{1},$ $x_{3}$ and $x_{5}$ are square with $m-b_{1}$, $b_{1}$ and
$b_{1}$ rows respectively. Hence

$\displaystyle\dim(M_{2})$
$\displaystyle=(m-b_{1})^{2}+2b_{1}(m-b_{1})+2b_{1}^{2}+(d-m-b_{1})d$
$\displaystyle=d^{2}-m(d-m)-b_{1}(d-b_{1}),$

and the result follows from Part (i). ∎

######  Lemma 3.3.

Let $k\geq 2$. Then $f_{k}\geq\mathrm{max}\\{1,b_{k}(d-m)\\}$.

###### Proof.

For $0\leq k\leq l$ we define two subspaces of $M_{k}$, namely

$\displaystyle X_{k}$ $\displaystyle=\\{g\in M_{k}\;|\;e_{i}g=0\text{ for
}a_{k}+1\leq i\leq d\\}\text{ and}$ $\displaystyle Y_{k}$
$\displaystyle=\\{g\in M\;|\;e_{i}g=0\text{ for }1\leq i\leq a_{k}\\}.$

We begin by showing that

$M_{k}=X_{k}\oplus Y_{k}\;\;\;\text{and}\;\;\;\dim(Y_{k})=d(d-a_{k}).$ (2)

It is clear that $X_{k}\cap Y_{k}=\\{0_{M}\\}$. Let $g=(g_{ij})\in M_{k}$.
Then there exist $x=(x_{ij}),y=(y_{ij})\in M$ with $x_{ij}=g_{ij}$ and
$y_{ij}=0$ for $i\leq a_{k}$, and $x_{ij}=0$ and $y_{ij}=g_{ij}$ for $i\geq
a_{k}+1$. Then $g=x+y$ with $x\in X_{k}$ and $y\in Y_{k}$, hence
$M_{k}=X_{k}\oplus Y_{k}$. Since $g\in Y_{k}$ if and only if $g_{ij}=0$ for
$i\leq a_{k}$, it follows that $\dim({Y_{k}})=d(d-a_{k})$. Hence (2) holds.

Our assumption that $M_{k}>M_{k+1}$ implies that $f_{k}\geq 1$, so we may
assume that $b_{k}\geq 1$. By (2),

$\displaystyle f_{k}$ $\displaystyle=\dim(M_{k})-\dim(M_{k+1})$
$\displaystyle=\big{(}\dim(X_{k})+\dim(Y_{k})\big{)}-\big{(}\dim(X_{k+1})+\dim(Y_{k+1})\big{)}$
$\displaystyle=\dim(X_{k})-\dim(X_{k+1})+d(d-a_{k})-d(d-a_{k+1})$
$\displaystyle=\dim(X_{k})-\dim(X_{k+1})+b_{k}d.$

We now bound $\dim(X_{k})-\dim(X_{k+1})$. By choice of basis
$\omega_{k+1}=\langle
u_{1},\ldots,u_{m-b_{k}},e_{a_{k}+1},\ldots,e_{a_{k}+b_{k}}\rangle$ for some
$u_{1},\ldots,u_{m-b_{k}}\in\langle\omega_{1},\ldots,\omega_{k}\rangle$. Hence
if $v\in\\{e_{a_{k}+1},\ldots,e_{a_{k}+b_{k}}\\}$ then $\langle v\rangle
M_{k+1}\leq\omega_{k+1}$. Therefore $\langle vM_{k+1}\rangle$ has dimension at
most $m$, and so $\dim(X_{k+1})\leq\dim(X_{k})+b_{k}m$. Hence

$f_{k}=\dim(X_{k})-\dim(X_{k+1})+b_{k}d\geq-b_{k}m+b_{k}d=b_{k}(d-m).$

∎

Proof of upper bound of Theorem 3.1. We shall show that $l\leq(m+1)d-2m+1$,
from which the result will follow, since
$\mathrm{I}(\mathrm{PGL}_{d}(q),\Omega)=\mathrm{I}(\mathrm{GL}_{d}(q),\Omega),$
and $\mathrm{GL}_{d}(q)$ is a subgroup of $M$.

For $0\leq b\leq m$, let

$C_{b}=\big{\\{}k\in\\{0,\ldots,l-1\\}\;\big{|}\;b_{k}=b\big{\\}}.$

and let $c_{b}=|C_{b}|$. Then

$l=\sum_{b=0}^{m}c_{b}.$ (3)

Since $a_{l}=d$ and $a_{0}=0$ it follows that

$d=a_{l}-a_{0}=\sum_{k=0}^{l-1}(a_{k+1}-a_{k})=\sum_{k=0}^{l-1}b_{k}=\sum_{b=0}^{m}bc_{b}=\sum_{b=1}^{m}bc_{b}.$
(4)

Since $a_{1}=m$ and $a_{0}=0$ it follows that $b_{0}=m$, so $0\in C_{m},$ and
$c_{m}\geq 1$. Since $\omega_{1}\neq\omega_{2}$ it follows that $b_{1}\neq 0$,
and $1\in C_{b_{1}}$, so

$c_{b_{1}}\geq 1\;\;\;\;\;\text{and}\;\;\;\;\;c_{m}\geq 1+\delta_{mb_{1}}.$
(5)

Lemmas 3.2 and 3.3 yield

$f_{0}=m(d-m),\;f_{1}=b_{1}(d-b_{1})=b_{1}(m-b_{1})+b_{1}(d-m),\;\text{and}\;f_{k}\geq\mathrm{max}\\{1,b_{k}(d-m)\\}\text{
for $k\geq 2.$}$ (6)

Since $M_{0}=M$ and $M_{l}=\mathbb{F}_{q}I$ it follows from the definition of
$f_{k}$ that

$\displaystyle d^{2}-1$ $\displaystyle=\dim(M_{0})-\dim(M_{l})$
$\displaystyle=\sum_{k=0}^{l-1}\Big{(}\dim(M_{k})-\dim(M_{k+1})\Big{)}$
$\displaystyle=\sum_{k=0}^{l-1}f_{k}$ $\displaystyle=\sum_{k\in
C_{0}}f_{k}+f_{1}+\sum_{k\in C_{b_{1}}\backslash\\{1\\}}f_{k}+\sum_{k\notin
C_{0}\cup C_{b_{1}}}f_{k}$ $\displaystyle\geq\sum_{k\in
C_{0}}1+b_{1}(m-b_{1})+b_{1}(d-m)+\sum_{k\in
C_{b_{1}}\backslash\\{1\\}}b_{1}(d-m)+\sum_{k\notin C_{0}\cup
C_{b_{1}}}b_{k}(d-m)\;\;\;\;\;\;\text{ by }\eqref{7}$
$\displaystyle=c_{0}+b_{1}(m-b_{1})+\sum_{k\in
C_{b_{1}}}b_{1}(d-m)+\sum_{k\notin C_{0}\cup C_{b_{1}}}b_{k}(d-m)$
$\displaystyle=c_{0}+b_{1}(m-b_{1})+\sum_{k\notin C_{0}}b_{k}(d-m)$
$\displaystyle=c_{0}+b_{1}(m-b_{1})+(d-m)\sum_{b=1}^{m}bc_{b}$
$\displaystyle=c_{0}+b_{1}(m-b_{1})+(d-m)d\;\;\;\;\;\;\text{ by
\eqref{c0c1}.}$

By rearranging we find that

$c_{0}\leq md-b_{1}(m-b_{1})-1.$ (7)

We bound $\mathrm{I}(G)$ by maximizing $l=\sum_{b=0}^{m}c_{b}$ subject only to
Equations (4), (5) and (7). By (4) an upper bound on $\sum_{b=0}^{m}c_{b}$ is
given by maximizing $c_{0}$, maximizing $c_{b}$ for $b$ small and minimizing
$c_{b}$ for $b$ large. Hence we substitute $c_{0}=md-b_{1}(m-b_{1})-1$ by (7),
substitute $c_{b}=0$ for $b\notin\\{0,1,b_{1},m\\}$, and maximise $c_{1}$ and
minimise $c_{m}$ subject to (5).

First let $m=1$. Since $b_{1}\neq 0$ it follows that $b_{1}=1$, and hence
$c_{1}=d$ by (4). Now let $m\geq 2$. Then there are three possibilities for
$b_{1}$. If $b_{1}=m$ then to minimise $c_{m}$ subject to (5) let $c_{m}=2$,
and so (4) yields $c_{1}=d-2m$. If $b_{1}=1$ then $c_{m}=1$, and (4) yields
$c_{1}=d-m$. Otherwise $c_{m}=c_{b_{1}}=1$, and (4) yields $c_{1}=d-m-b_{1}$.
Hence in all cases

$|C_{1}\cup C_{b_{1}}\cup C_{m}|=2+d-m-b_{1}.$

Therefore

$\sum_{b=0}^{m}c_{b}\leq\big{(}{m}d-b_{1}(m-b_{1})-1\big{)}+2+d-{m}-b_{1}=({m}+1)d-m+1-b_{1}(m-b_{1}+1).$

Hence if $\sum_{b=0}^{m}c_{b}$ is maximal then $b_{1}(m-b_{1}+1)$ is minimal
subject to $1\leq b_{1}\leq m$. Therefore $b_{1}$ is $1$ or $m$, and so

$\sum_{b=0}^{m}c_{b}\leq({m}+1)d-2{m}+1.$

The result now follows from (3). ∎

We now consider the lower bounds in Theorem 3.1.

Proof of lower bound of Theorem 3.1. Let $G=\mathrm{GL}_{n}(q)$. Here we give
a sequence of $m$-spaces of $V$ such that each successive point stabilizer in
$G$ is a proper subgroup of its predecessor. Its length is therefore a lower
bound on $\mathrm{I}(\mathrm{PGL}_{n}(q),\Omega)$.

For $1\leq k\leq md-m^{2}+d$, we define the following three variables

$r_{k}=\Big{\lfloor}\frac{k-2}{m}\Big{\rfloor}+m+1,\;\;\;\;\;\;\;s_{k}=m-(k-2\bmod
m),\;\;\;\;\;\;\;\text{ and }\;t_{k}=k-md+m^{2}.$

Notice that if $m+2\leq k\leq md-m^{2}+1$, then

$m+2\leq r_{k}\leq d\;\;\;\;\text{and}\;\;\;\;1\leq s_{k}\leq m,$ (8)

whilst $t_{k}\leq d$ for all $k$ and

$2\leq t_{k}\leq m+1\;\;\;\;\;\;\text{ if and only if
}\;\;\;\;\;\;md-m^{2}+2\leq k\leq md-m^{2}+m+1.$

Hence the following sets $W_{k}$ of $m$ linearly independent vectors of $V$
are well defined.

$W_{k}=\begin{cases}\big{\\{}e_{i}\;|\;i\in\\{1,\ldots,m+1\\}\backslash\\{m+2-k\\}\big{\\}}&\text{
for }1\leq k\leq m+1,\\\
\big{\\{}e_{i}\;|\;i\in\\{1,\ldots,m,r_{k}\\}\backslash\\{s_{k}\\}\big{\\}}&\text{
for }m+2\leq k\leq md-m^{2}+1,\\\
\big{\\{}e_{1}+e_{t_{k}},e_{i}\;|\;i\in\\{2,\ldots,m+1\\}\backslash\\{t_{k}\\}\big{\\}}&\text{
for }md-m^{2}+2\leq k\leq md-m^{2}+m+1,\\\
\big{\\{}e_{1}+e_{t_{k}},e_{i}\;|\;i\in\\{2,\ldots,m\\}\big{\\}}&\text{ for
}md-m^{2}+m+2\leq k\leq md-m^{2}+d.\end{cases}$

Let $\omega_{k}=\langle W_{k}\rangle\in\Omega$ and let
$G^{(k)}=G_{\omega_{1},\ldots,\omega_{k}}$. For $1\leq x,y\leq d$ let $T(x,y)$
be the matrix $I+E_{x,y}$ and let $\mathrm{Supp}_{x}(W_{k})$ be the set of
vectors in $W_{k}$ which are non-zero in position $x$. Recall that

$e_{i}T(x,y)=\begin{cases}e_{i}+e_{y}&\text{if }i=x,\\\
e_{i}&\text{otherwise.}\end{cases}$

Hence if a vector $v$ is zero in position $x$, then $vT(x,y)=v$. Therefore
$\omega_{k}T(x,y)=\omega_{k}$ if and only if
$\mathrm{Supp}_{x}(W_{k})T(x,y)\subseteq\omega_{k}$. In particular, if
$\mathrm{Supp}_{x}(W_{k})=\emptyset$ then $\omega_{k}T(x,y)=\omega_{k}$.
Furthermore, $T(x,y)\in G$ unless $q=2$ and $x=y$.

It is clear that $G>G^{(1)}$, so let $k\in\\{2,\ldots,md-m^{2}+1\\}$ and let
$j\leq k$. We shall show that there exist $x$ and $y$ such that
$\omega_{k}T(x,y)\neq\omega_{k}$ and $\omega_{j}T(x,y)=\omega_{j}$ for all
$j<k$. Hence $T(x,y)\in G^{(k-1)}\backslash G^{(k)}$ and so
$G^{(k-1)}>G^{(k)}$.

First consider $k\in\\{2,\ldots,m+1\\},$ and let $T=T(m+1,m+2-k)$. Then
$\mathrm{Supp}_{m+1}(W_{1})=\emptyset$, and for $1<j\leq k$

$\mathrm{Supp}_{m+1}(W_{j})T=\\{e_{m+1}\\}T=\\{e_{m+1}+e_{m+2-k}\\}.$

Hence $\mathrm{Supp}_{m+1}(W_{j})T\subseteq\omega_{j}$ if and only if $j\neq
k$. Therefore $\omega_{j}T=\omega_{j}$ for $j<k$, and
$\omega_{k}T\neq\omega_{k}.$

Next consider $k\in\\{m+2,\ldots,md-m^{2}+1\\}$. Hence (8) holds, and so we
may let $T$ be the matrix $T(r_{k},s_{k})$. If $j\leq m+1$ or if $r_{j}\neq
r_{k}$, then $\mathrm{Supp}_{r_{k}}(W_{j})=\emptyset$ and so
$\omega_{j}T=\omega_{j}.$ Therefore assume that $j\geq m+2$ and $r_{j}=r_{k}$.
Then

$\mathrm{Supp}_{r_{k}}(W_{j})T=\\{e_{r_{k}}\\}T=\\{e_{r_{k}}+e_{s_{k}}\\}.$

Since $(r_{j},s_{j})=(r_{k},s_{k})$ if and only if $j=k$, it follows that
$\mathrm{Supp}_{r_{k}}(W_{j})T\subseteq\omega_{j}$ if and only if $j\neq k$.
Therefore $\omega_{j}T=\omega_{j}$ for $j<k$, and $\omega_{k}T\neq\omega_{k}.$
Hence $G^{(k-1)}>G^{(k)}$ for $1\leq k\leq md-m^{2}+1$, and so if $q=2$ then
the result follows.

It remains to consider $q>2$ and $k\geq md-m^{2}+2$. Let $T=T(t_{k},t_{k})$
and let
$u\in\\{e_{i},e_{1}+e_{i}\;|\;1\leq i\leq d\\}$. Then

$uT=\begin{cases}e_{1}+2e_{t_{k}}&\text{ if }u=e_{1}+e_{t_{k}},\\\ 2u&\text{
if }u=e_{t_{k}},\\\ u&\text{ otherwise.}\end{cases}$

If $1\leq j\leq md-m^{2}+1$ then $W_{j}\subseteq\\{e_{1},\ldots,e_{d}\\}$, and
if $md+m^{2}+1<j<k$ then
$W_{j}\subseteq\\{e_{1}+e_{t_{j}},e_{1},\ldots,e_{d}\\}$ with $t_{j}\neq
t_{k}$. Hence, if $j<k$ then
$\mathrm{Supp}_{t_{k}}(W_{j})T\subseteq\omega_{j}$, and so
$\omega_{j}T=\omega_{j}$. Since $e_{1}+e_{t_{k}}\in\omega_{k}$ but
$e_{1}+2e_{t_{k}}\notin\omega_{k}$ it follows that
$\omega_{k}T(t_{k},t_{k})\neq\omega_{k}.$ Hence $G^{(k-1)}>G^{(k)}$ for $1\leq
k\leq md-m^{2}+d$, and so the result follows. ∎

######  Remark 3.4.

The interested reader may wish to check, using the notation of the previous
proof, that the following holds. Let $\Lambda=(\omega_{i})_{2\leq i\leq
md-m^{2}+1}$ if $q=2$, and let $\Lambda=(\omega_{i})_{m+1\leq i\leq
md-m^{2}+d}$ if $q>2$. Then $\Lambda$ is a minimal base for the action of
$\mathrm{PGL}_{d}(q)$ on $\mathcal{PG}_{m}(q)$. Hence

$\mathrm{B}(\mathrm{PGL}_{d}(q),\mathcal{PG}_{m}(q))\geq\begin{cases}md-m^{2}&\text{if
$q=2$,}\\\ (m+1)d-m^{2}-m&\text{if $q\neq 2$.}\end{cases}$

###  3.2 Upper bounds as a function of $|\Omega|$

Our main result in this subsection is Proposition 3.6, which bounds
$\mathrm{I}(\mathrm{PGL}_{d}(q),\Omega)$ as a function of $n=|\Omega|$, rather
than of $m$, $d$ and $q$. We begin by bounding the size of
$\Omega=\mathcal{PG}_{m}(\mathbb{F}_{q}^{d})$.

######  Lemma 3.5.

Let $n(d,m,q)=|\mathcal{PG}_{m}(\mathbb{F}_{q}^{d})|$. Then

$\log|\Omega|=\log\big{(}n(d,m,q)\big{)}>\begin{cases}\frac{d^{2}}{4}+\frac{1}{2}&\text{if
$q=2$ and $m=\frac{d}{2}\geq 2$,}\\\ {m(d-m)}\log q&\text{for all $m$ and
$q$.}\end{cases}$

###### Proof.

If either $q\neq 2$ or $m\neq\frac{d}{2}$ then the result holds by [17, Lemma
4.2.8], so assume otherwise and let $k=\frac{d}{2}+1=m+1\geq 3$. The statement
is then equivalent to $n(2k-2,k-1,2)>2^{(k-1)^{2}+\frac{1}{2}}$.

We now induct on $k$. Since $n(4,2,2)=35>2^{2^{2}+\frac{1}{2}}$, the result
holds for $k=3$. Now

$\displaystyle n(2k,k,2)$
$\displaystyle=\frac{(2^{2k}-1)(2^{2k-1}-1)(2^{2k-2}-1)\cdots(2^{k+1}-1)}{(2^{k}-1)(2^{k-1}-1)(2^{k-2}-1)\cdots(2-1)}$
$\displaystyle=\frac{(2^{2k}-1)(2^{2k-1}-1)}{(2^{k}-1)^{2}}\cdot\frac{(2^{2k-2}-1)\cdots(2^{k+1}-1)(2^{k}-1)}{(2^{k-1}-1)(2^{k-2}-1)\cdots(2-1)}$
$\displaystyle=\frac{(2^{2k}-1)(2^{2k-1}-1)}{(2^{k}-1)^{2}}\cdot
n(2k-2,k-1,2)$
$\displaystyle\geq\frac{(2^{2k}-1)(2^{2k-1}-1)}{(2^{k}-1)^{2}}\cdot
2^{(k-1)^{2}+\frac{1}{2}},\;\;\;\;\text{by induction.}$

It is easily verified that

$(2^{k}+1)(2^{2k-1}-1)=2^{3k-1}+2^{2k-1}-2^{k}-1>2^{3k-1}-2^{2k-1}=2^{2k-1}(2^{k}-1).$

Hence

$\frac{(2^{2k}-1)(2^{2k-1}-1)}{(2^{k}-1)^{2}}2^{(k-1)^{2}}=\frac{(2^{k}+1)(2^{2k-1}-1)}{(2^{k}-1)}2^{(k-1)^{2}}>\frac{2^{2k-1}(2^{k}-1)}{(2^{k}-1)}2^{(k-1)^{2}}=2^{k^{2}},$

and the result follows. ∎

Recall that $q=p^{f}$ with $p$ prime, and
$\Omega=\mathcal{PG}_{m}(\mathbb{F}_{q}^{d})$, with $n=|\Omega|$.

######  Proposition 3.6.

Let $G=\mathrm{P}\Gamma\mathrm{L}_{d}(q)$ and assume that $m\leq\frac{d}{2}$.
Then

$\mathrm{I}(G,\Omega)\leq\begin{cases}2(d-1)+1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq
2\log n+1&\text{ if $m=1$ and $q=2,$}\\\ \frac{4}{3}(d-1)\log q+1+\log
f\leq\frac{4}{3}\log n+1+\log f&\text{ if $m=1$ and $q\geq 3,$}\\\
{\frac{d^{2}}{2}+1}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\leq
2\log n&\text{ if $m=\frac{d}{2}\geq 2$ and $q=2$,}\\\ 2m(d-m)\log q+\log
f\;\;\leq 2\log n+\log f&\text{ otherwise.}\end{cases}$

###### Proof.

Since $G=\mathrm{PGL}_{n}(q)\rtimes C_{f}$, Lemma 2.3(ii) and Theorem 3.1
imply that

$\mathrm{I}(G)=\mathrm{I}(\mathrm{PGL}_{n}(q))+\ell(C_{f})\leq(m+1)d-2m+1+\log
f.$ (9)

First let $m=1$, so that $\mathrm{I}(G)\leq 2(d-1)+1+\log f$. By Lemma 3.5
$(d-1)\log q<\log n$. Hence the result is immediate for $q=2$, and follows
from $\log q>\frac{3}{2}$ for $q\geq 3$.

Now let $m=\frac{d}{2}\geq 2$ so that $\mathrm{I}(G)\leq\frac{d^{2}}{2}+1+\log
f$. If $q=2$ then $\frac{d^{2}}{4}+\frac{1}{2}<\log n$ by Lemma 3.5, and so
the result follows. If $q\geq 3$, then it follows from $d\geq 4$ that
$1\leq\frac{d^{2}}{4}$, and so

$\frac{d^{2}}{2}+1\leq\frac{3d^{2}}{4}<\frac{d^{2}}{2}\log q=2m(d-m)\log q.$

Therefore

$\mathrm{I}(G)\leq 2m(d-m)\log q+\log f<2\log n+\log f,$

by Lemma 3.5. Finally consider $1<m<\frac{d}{2}$. Then $1\leq d-2m$, and so

$d-2m+1\leq 2(d-2m)\leq m(d-2m).$

Hence by (9)

$\mathrm{I}(G)-\log f\leq md+d-2m+1\leq md+m(d-2m)=2m(d-m)\leq 2m(d-m)\log q.$

Therefore, $\mathrm{I}(G)\leq 2m(d-m)\log q+\log f\leq 2\log n+\log f,$ by
Lemma 3.5. ∎

##  4 Almost simple groups

In this section we prove Theorem 1.1 for almost simple groups. More precisely,
we prove the following result.

######  Theorem 4.1.

Let $G$ be an almost simple primitive subgroup of $\mathrm{S}_{n}$. If $G$ is
not large base then

$\mathrm{I}(G,\Omega)<5\log n-1.$

We begin with two definitions which we shall use to divide this section into
cases.

######  Definition 4.2.

Let $G$ be almost simple with socle $G_{0}$, a classical group with natural
module $V$. A subgroup $H$ of $G$ not containing $G_{0}$ is a subspace
subgroup if for each maximal subgroup $M$ of $G_{0}$ containing $H\cap G_{0}$
one of the following holds.

  1. (i)

$M=G_{U}$ for some proper non-zero subspace $U$ of $V$, where if
$G_{0}\neq\mathrm{PSL}(V)$ then $U$ is either totally singular or non-
degenerate, or if $G$ is orthogonal and $p=2$ a non-singular 1-space.

  2. (ii)

$G_{0}=\mathrm{Sp}_{d}(2^{f})$ and $M\cap G_{0}=\mathrm{GO}^{\pm}_{d}(2^{f})$.

A transitive action of $G$ is a subspace action if the point stabilizer is a
subspace subgroup of $G$.

######  Definition 4.3.

Let $G$ be almost simple with socle $G_{0}$. A transitive action of $G$ on
$\Omega$ is standard if up to equivalence of actions one of the following
holds, and is non-standard otherwise.

  1. (i)

$G_{0}=\mathrm{A}_{r}$ and $\Omega$ is an orbit of subsets or partitions of
$\\{1,\ldots,r\\}$.

  2. (ii)

$G$ is a classical group in a subspace action.

This section is split into three subsections. The first considers
$G_{0}=\mathrm{PSL}_{d}(q)$ acting on subspaces and pairs of subspaces. In the
second, we deal with the case of $G$ another classical group in a subspace
action. Finally in the third we prove Theorem 4.1.

###  4.1 $G_{0}=\mathrm{PSL}_{d}(q)$

Let $G$ be almost simple with socle $\mathrm{PSL}_{d}(q)$, in a subspace
action on a set $\Omega$. We shall use the results in Section 3 to bound
$\mathrm{I}(G,\Omega)$.

We begin with a preliminary lemma.

######  Lemma 4.4.

Let $1\leq m\leq\frac{d}{2}$, let $p$ be a prime, let $f\geq 1$ and let
$q=p^{f}$. Then

$m(d-m)\log q\geq\begin{cases}\log f+1&\text{ if }m=1,\\\ 3\log f+4&\text{ if
}m\geq 2.\end{cases}$

###### Proof.

Let $m=1$. Then

$m(d-m)\log q=(d-1)\log q=(d-1)f\log p\geq f\geq\log f+1.$

Now let $m\geq 2$, so that $m(d-m)\geq 4$. Then

$m(d-m)\log q\geq 4\log q\geq 3\log q+1=3f\log p+1\geq 3f+1\geq 3(\log
f+1)+1=3\log f+4$

as required. ∎

######  Proposition 4.5.

Let $G$ be almost simple with socle $\mathrm{PSL}_{d}(q)$ acting on
$\Omega=\mathcal{PG}_{m}(V),$ and let $n=|\Omega|$. Then

$\mathrm{I}(G)<3\log n.$

###### Proof.

If $m=1$ then $G\leq\mathrm{P}\Gamma\mathrm{L}_{d}(q)$, so
$\mathrm{I}(G)\leq\mathrm{I}(\mathrm{P}\Gamma\mathrm{L}_{d}(q))$ by Lemma
2.3(i). Otherwise $G\cap\mathrm{P}\Gamma\mathrm{L}_{d}(q)$ has index at most 2
in $G$, so by Lemma 2.3(i) and (iii)

$\mathrm{I}(G)\leq\mathrm{I}(G\cap\mathrm{P}\Gamma\mathrm{L}_{d}(q))+1\leq\mathrm{I}(\mathrm{P}\Gamma\mathrm{L}_{d}(q))+1.$

Therefore we can bound $\mathrm{I}(G)$ by our bound for
$\mathrm{I}(\mathrm{P}\Gamma\mathrm{L}_{d}(q))$ when $m=1$, and by one more
than that when $m>1$. Thus Proposition 3.6 yields $\mathrm{I}(G)\leq 2\log
n+\log f+1$. Combining Lemmas 3.5 and 4.4 gives $\log f+1\leq m(d-m)\log
q<\log n$, hence the result holds. ∎

We now consider the action of $G$ on the following subsets of
$\mathcal{PG}_{m}(V)\times\mathcal{PG}_{d-m}(V)$, with $m<\frac{d}{2}:$

$\displaystyle\Omega^{\oplus}=$
$\displaystyle\Big{\\{}\\{U,W\\}\;\Big{|}\;U,W\leq V,\;\dim U=m,\;\dim
W=d-m\text{ with }U\oplus W=V\Big{\\}},$ $\displaystyle\Omega^{\leq}=$
$\displaystyle\Big{\\{}\\{U,W\\}\;\Big{|}\;U,W\leq V,\;\dim U=m,\;\dim
W=d-m\text{ with }U\leq W\Big{\\}}.$

Note that in both cases we require $d\geq 3$.

######  Lemma 4.6.

Let $G$ be almost simple with socle $\mathrm{PSL}_{d}(q)$, let
$H=G\cap\mathrm{P}\Gamma\mathrm{L}_{d}(q)$ and let $\Omega$ be either
$\Omega^{\oplus}$ or $\Omega^{\leq}$. Then

$\mathrm{I}(G,\Omega)\leq 2\mathrm{I}(H,\mathcal{PG}_{m}(V))+1.$

###### Proof.

We first show that

$\mathrm{I}(H,\Omega)\leq\mathrm{I}(H,\mathcal{PG}_{m}(V))+\mathrm{I}(H,\mathcal{PG}_{d-m}(V)).$
(10)

Let $l=\mathrm{I}(H,\Omega)$ and let
$\Lambda=\big{(}\\{U_{1},W_{1}\\},\ldots,\\{U_{l},W_{l}\\}\big{)}$ be a
corresponding base, where $\dim(U_{i})=m$ for all $i$. Let
$\Pi=(U_{1},\ldots,U_{l})$ and let $\Sigma=(W_{1},\ldots,W_{l})$. Then by
Lemma 2.2 $\Pi$ and $\Sigma$ contain subsequences which can be extended to
irredundant bases for the action of $H$ on $\mathcal{PG}_{m}(V)$ and
$\mathcal{PG}_{d-m}(V)$ respectively.

Let $\Pi^{\prime}$ be the subsequence of $\Pi$ which contains $U_{i}$ if and
only if $H_{U_{1},\ldots,U_{i-1}}>H_{U_{1},\ldots,U_{i-1},U_{i}}$. Then
$\Pi^{\prime}$ can be extended to an irredundant base for the action of $H$ on
$\mathcal{PG}_{m}(V)$. Let $k=|\Pi^{\prime}|$, so
$k\leq\mathrm{I}(H,\mathcal{PG}_{m}(V)).$

Let $\Sigma^{\prime}=(W_{j_{1}},\ldots,W_{j_{(l-k)}})$ be the subsequence of
$\Sigma$ which contains $W_{i}$ if and only if
$H_{U_{1},\ldots,U_{i-1}}=H_{U_{1},\ldots,U_{i-1},U_{i}}$. Assume, for a
contradiction, that $\Sigma^{\prime}$ cannot be extended to an irredundant
base for the action of $H$ on $\mathcal{PG}_{d-m}(V)$. Since $H$ is
irreducible, $H>H_{W_{j_{1}}}$. Therefore there exists $s\geq 2$ such that

$H_{W_{j_{1}},W_{j_{2}},\ldots,W_{j_{(s-1)}}}=H_{W_{j_{1}},W_{j_{2}},\ldots,W_{j_{(s-1)}},W_{j_{s}}}.$

Let $i=j_{s}$. Then intersecting both sides of the above expression with
$H_{W_{1},\ldots,W_{i-1}}$ gives

$H_{W_{1},\ldots,W_{i-1}}=H_{W_{1},\ldots,W_{i-1},W_{i}}.$ (11)

Since $W_{i}\in\Sigma^{\prime}$ it follows that

$H_{U_{1},\ldots,U_{i-1}}=H_{U_{1},\ldots,U_{i-1},U_{i}}.$ (12)

Elements of $H=G\cap\mathrm{P}\Gamma\mathrm{L}_{d}(q)$ cannot map $U_{i}$ to
$W_{i}$. Therefore (11) and (12) imply that

$H_{\\{U_{1},W_{1}\\},\ldots,\\{U_{i-1},W_{i-1}\\}}=H_{\\{U_{1},W_{1}\\},\ldots,\\{U_{i-1},W_{i-1}\\},\\{U_{i},W_{i}\\}},$

a contradiction since $\Lambda$ is irredundant. Hence
$l-k\leq\mathrm{I}(H,\mathcal{PG}_{n-m}(V))$, and so (10) holds.

The subgroups of $\mathrm{Sym}({\mathcal{PG}_{m}(V)})$ and
$\mathrm{Sym}({\mathcal{PG}_{n-m}(V)})$ representing the actions of $H$ are
permutation isomorphic. Therefore (10) implies that $\mathrm{I}(H,\Omega)\leq
2\mathrm{I}(H,\mathcal{PG}_{m}(V))$. Since $H$ has index at most 2 in $G$, the
result follows from Lemma 2.3(iii). ∎

######  Lemma 4.7.

Let $\Omega$ be either $\Omega^{\oplus}$ or $\Omega^{\leq}$, and let
$n=|\Omega|$. Let $G$ be an almost simple subgroup of $\mathrm{Sym}({\Omega})$
with socle $\mathrm{PSL}_{d}(q)$. Then

$\mathrm{I}(G)<5(\log n-1).$

###### Proof.

Let $H=G\cap\mathrm{P}\Gamma\mathrm{L}_{d}(q)$, then by Proposition 3.6 and
Lemma 4.6

$\mathrm{I}(G)\leq
2\mathrm{I}(H,\mathcal{PG}_{m})+1\leq\begin{cases}4(d-1)+3&\text{ if
}m=1\text{ and }q=2,\\\ \frac{8}{3}(d-1)\log q+2\log f+3&\text{ if }m=1\text{
and }q\geq 3,\\\ 4{m(d-m)}\log q+2\log f+1&\text{ otherwise}.\end{cases}$ (13)

Since $n\geq 2|\mathcal{PG}_{m}(V)|$, Lemma 3.5 gives

$m(d-m)\log q<\log\frac{n}{2}=\log n-1.$ (14)

Recall that $d\geq 3$. First let $m=1$. If $(d,q)=(3,2)$, then
$n\in\\{21,28\\}$. Therefore by (13) it follows that $\mathrm{I}(G)\leq
11<5(\log n-1)$. Hence if $q=2$ then we may assume that $d\geq 4$, and so by
(13) and (14)

$\mathrm{I}(G)\leq 4(d-1)+3\leq 5(d-1)<5(\log n-1).$

Still with $m=1$, let $q\geq 3$. Then

$\begin{array}[]{rll}\mathrm{I}(G)&\leq\frac{8}{3}(d-1)\log q+2\log f+3&\text{
by \eqref{14},}\\\ &\leq\frac{8}{3}(d-1)\log q+2(d-1)\log q+1&\text{ by Lemma
\ref{logfbound},}\\\ &<5(d-1)\log q&\text{ since $1<\frac{1}{3}(d-1)\log q$,
}\\\ &<5(\log n-1)&\text{ by \eqref{2t}.}\end{array}$

Finally, let $m\geq 2$. Then $2\log f+1\leq m(d-m)\log q$ by Lemma 4.4. Hence
by (13) and (14) it follows that

$\mathrm{I}(G)\leq 4{m(d-m)}\log q+2\log f+1\leq 5{m(d-m)}\log q<5(\log n-1).$

∎

###  4.2 $G_{0}$ another classical group

######  Lemma 4.8.

Let $G$ be almost simple with socle $G_{0}=\mathrm{P}\Omega_{8}^{+}(q),$
acting faithfully and primitively on a set $\Omega$ of size $n$. Then

$\mathrm{I}(G)<5\log n-1.$

###### Proof.

Let $q\geq 3$. Then the reader may check that $6f<q^{2}$, and so by [14,
(6.19)]

$|G|<6fq^{28}\leq q^{30}.$

If $q=2$, then $|G|\leq 6|G_{0}|<q^{30}$ also. Hence by Lemma 2.1(ii), since
$n>4$,

$\mathrm{I}(G)\leq\log q^{30}-1=5\log q^{6}-1<5\log n-1,$

by [14, (6.20)]. ∎

######  Proposition 4.9.

Let $G$ be almost simple with socle $G_{0}$, a classical group with natural
module $V$. Assume that $G_{0}\neq\mathrm{PSL}(V)$ and
$G_{0}\neq\mathrm{P}\Omega_{8}^{+}(q)$. Let $0<m<d$, let $\Omega$ be a
$G$-orbit of totally isotropic, totally singular, or non-degenerate subspaces
of $V$ of dimension $m$, and let $n=|\Omega|$. Then

$\mathrm{I}(G,\Omega)<5\log n-1.$

###### Proof.

First let $G_{0}=\mathrm{P}\Omega_{d}^{+}(q)$ and $m=\frac{d}{2}$. Then $d\geq
10$, and so $2d^{2}-12d-16>0$. Hence $10d^{2}-20d>8d^{2}-8d+16$ and it follows
that $\frac{d^{2}}{8}-\frac{d}{4}>\frac{d^{2}}{10}-\frac{d}{10}+\frac{1}{5}$.
By [5, Table 4.12 ]

$n=\prod_{i=1}^{\frac{d}{2}-1}(q^{i}+1)>\prod_{i=1}^{\frac{d}{2}-1}q^{i}=q^{\frac{d^{2}}{8}-\frac{d}{4}}>q^{\frac{d^{2}}{10}-\frac{d}{10}+\frac{1}{5}}.$
(15)

Hence

$\begin{array}[]{lll}\mathrm{I}(G)&\leq\log|G|-1&\text{ by Lemma
\ref{fands}\ref{sneaky},}\\\
&\leq\log\Big{(}q^{\frac{d^{2}}{2}-\frac{d}{2}+1}\Big{)}-1&\text{ by
\cite[cite]{[\@@bibref{}{GLS}{}{}, p25]}},\\\
&=5\log\Big{(}q^{\frac{d^{2}}{10}-\frac{d}{10}+\frac{1}{5}}\Big{)}-1\\\
&<5\log n-1&\text{ by \eqref{prod}.}\end{array}$

Therefore we may assume for the rest of the proof that
$G_{0}\neq\mathrm{P\Omega}_{d}^{+}(q)$, so by [14, Lemma 7.14]

$\frac{1}{2}m(d-m)\log q<\log n.$ (16)

Since $\Omega$ is a $G$-orbit of subspaces, if $G_{0}=\mathrm{PSp}_{4}(q)$
then $G$ does not induce the graph isomorphism by [2, Table 8.14]. Hence since
$G_{0}\neq\mathrm{P}\Omega_{8}^{+}(q)$ and
$\Omega\subseteq\mathcal{PG}_{m}(V)$, we may assume that
$G\leq\mathrm{P}\Gamma\mathrm{L}_{d}(q)$. Then Lemma 2.3(i) implies that

$\mathrm{I}(G)\leq\mathrm{I}\big{(}\mathrm{P}\Gamma\mathrm{L}_{d}(q),\mathcal{PG}_{m}(V)\big{)},$

and so in particular the bounds from Proposition 3.6 apply.

We begin with $m=1$. If $q=2$ then we split into two cases. If $d\leq 4$ then
by Proposition 3.6

$\mathrm{I}(G)\leq 2(d-1)+1\leq 7<5\log n-1.$

If instead $d\geq 5,$ so that $\frac{1}{2}(d-1)\geq 2$, then by Proposition
3.6 and (16)

$\mathrm{I}(G)\leq 2(d-1)+1\leq 2(d-1)+\frac{1}{2}(d-1)-1<5\log n-1.$

To complete the case of $m=1$, let $q\geq 3$. Since
$G_{0}\neq\mathrm{PSL}_{d}(q)$ we may assume that $d\geq 3$ and so it can be
verified that $\frac{6}{7f}(2+\log f)+1<d$. Hence

$2+\log f<\frac{7}{6}f(d-1)\leq\frac{7}{6}f(d-1)\log p=\frac{7}{6}(d-1)\log
q.$

Therefore it follows from Proposition 3.6 and (16) that

$\mathrm{I}(G)\leq\frac{4}{3}(d-1)\log q+\log f+1<\frac{5}{2}(d-1)\log
q-1<5\log n-1.$

Now let $m=\frac{d}{2}$ and $q=2$. Then by Proposition 3.6 and (16)

$\mathrm{I}(G)\leq\frac{d^{2}}{2}+1=4\Big{(}\frac{1}{2}m(d-m)\Big{)}+1<4\log
n+1<5\log n-1,$

since $n>4$.

Hence we may assume that $m\geq 2$, and that if $m=\frac{d}{2}$ then $q\geq
3$. Therefore

$\begin{array}[]{lll}\mathrm{I}(G)&\leq 2m(d-m)\log q+\log f&\text{ by
Proposition \ref{g01},}\\\ &\leq 2m(d-m)\log q+\frac{1}{3}m(d-m)\log
q-\frac{4}{3}&\text{ by Lemma \ref{logfbound},}\\\ &<\frac{14}{3}\log
n-\frac{4}{3}&\text{ by \eqref{smallom},}\\\ &<5\log n-1.\end{array}$

∎

###  4.3 Proof of Theorem 4.1

We begin by proving Theorem 4.1 for non-standard actions.

######  Proposition 4.10.

Let $G$ be an almost simple, primitive non-standard subgroup of
$\mathrm{Sym}({\Omega})$ and let $n=|\Omega|$. Then

$\mathrm{I}(G,\Omega)\leq 4\log n+1.$

###### Proof.

By a landmark result of Burness and others [4, 5, 6, 8], either
$(G,\Omega)=(M_{24},\\{1,\ldots,24\\})$ or $\mathrm{b}(G,\Omega)\leq 6$. By
[14, p10], $\mathrm{I}(\mathrm{M}_{24},\\{1,\ldots,24\\})=7<2\log 24$. If
$\mathrm{b}(G)\leq 5$, then the result follows by Lemma 2.1(iii). Hence we may
assume that $\mathrm{b}(G,\Omega)=6$.

Let $G$ have point stabilizer $H$. By a further result of Burness [3, Theorem
1], either

$(G,H)\in\big{\\{}(\mathrm{M}_{23},\mathrm{M}_{22}),(\mathrm{Co}_{3},\mathrm{McL}.2),(\mathrm{Co}_{2},\mathrm{U}_{6}(2).2),(\mathrm{Fi}_{22}.2,2.\mathrm{U}_{6}(2).2)\big{\\}}\text{
or}$ (17)
$(\mathrm{Soc}(G),H)\in\big{\\{}(E_{7}(q),P_{7}),(E_{6}(q),P_{1}),(E_{6}(q),P_{6})\big{\\}}.$
(18)

Therefore $H$ is insoluble, and so $\mathrm{I}(H)+1<\log|H|$ by Lemma 2.1(i).
Hence if we can prove in each case that $|H|<[G:H]^{4}$ then the result will
follow, since

$\mathrm{I}(G)=\mathrm{I}(H)+1<\log|H|<\log[G:H]^{4}=4\log n.$

Since $\mathrm{M}_{23}$ is the point stabilizer of $\mathrm{M}_{24}$ it
follows that

$\mathrm{I}(\mathrm{M}_{23},\\{1,\ldots,23\\})=\mathrm{I}(\mathrm{M}_{24},\\{1,\ldots,23,24\\})-1=6<2\log
23.$

If $G$ is $\mathrm{Co}_{3}$, $\mathrm{Co}_{2}$ or $\mathrm{Fi}_{22}.2$, then
using [12] it is easily verified that $|H|<[G:H]^{4}$, so the result follows.

Therefore assume that $(\mathrm{Soc}(G),H)$ is as in (18). Let $m(G)$ be the
smallest degree of a faithful transitive permutation representation of $G$. If
$|G|<m(G)^{5}$, then

$|H|=\frac{|G|}{[G:H]}<\frac{m(G)^{5}}{[G:H]}\leq[G:H]^{4},$

and so the result will follow.

First let $G_{0}=E_{6}(q)$. By [12]

$|E_{6}(q)|=\frac{q^{36}(q^{12}-1)(q^{9}-1)(q^{8}-1)(q^{6}-1)(q^{5}-1)(q^{2}-1)}{(3,q-1)}$

and $|\mathrm{Out}(E_{6}(q))|\leq 2f(3,q-1)\leq q(3,q-1)$. Hence

$|G|\leq
q^{37}(q^{12}-1)(q^{9}-1)(q^{8}-1)(q^{6}-1)(q^{5}-1)(q^{2}-1)<q^{37+12+9+8+6+5+2}=q^{79}.$

By [21, p2]

$m(G)\geq
m(G_{0})\geq\frac{(q^{9}-1)(q^{8}+q^{4}+1)}{q-1}=(q^{8}+q^{7}+\cdots+q+1)(q^{8}+q^{4}+1)>q^{16}.$

Hence $|G|<q^{79}<q^{80}<m(G)^{5}$ as required.

Now let $G_{0}=E_{7}(q)$. By [12]

$|E_{7}(q)|=\frac{q^{63}(q^{18}-1)(q^{14}-1)(q^{12}-1)(q^{10}-1)(q^{8}-1)(q^{6}-1)(q^{2}-1)}{(2,q-1)}$

and $|\mathrm{Out}(E_{7}(q))|=(2,q-1)f<(2,q-1)q$. Hence

$|G|\leq
q^{64}(q^{18}-1)(q^{14}-1)(q^{12}-1)(q^{10}-1)(q^{8}-1)(q^{6}-1)(q^{2}-1)<q^{64+18+14+12+10+8+6+2}=q^{134}.$

By [21, p5]

$m(G)=\frac{(q^{14}-1)(q^{9}+1)(q^{5}+1)}{q-1}=(q^{13}+q^{12}+\cdots+q+1)(q^{9}+1)(q^{5}+1)>q^{13+9+5}=q^{27}.$

Hence $|G|<q^{134}<q^{135}<m(G)^{5}$. ∎

We note that this bound could be improved if the groups with minimal base size
5 were classified.

Proof of Theorem 4.1. If the action of $G$ on $\Omega$ is non-standard then
the result follows by Proposition 4.10. Hence we may assume that $G$ is
standard.

If $G$ is alternating and not large base, then $\Omega$ is a set of
partitions. Hence $\mathrm{I}(G,\Omega)<2\log|\Omega|$ by [14, Lemma 6.6].

Therefore $G$ is classical, and the action of $G$ on $\Omega$ is a subspace
action. If $G$ is as in Case (ii) of Definition 4.2, then
$\mathrm{I}(G,\Omega)<\frac{11}{3}\log|\Omega|$ by [14, Lemma 6.7]. If
$G_{0}=\mathrm{PSL}_{d}(q)$ and $\Omega$ is a set of subspaces, or a set of
pairs of subspaces, of $V$, then the result holds by Proposition 4.5 and Lemma
4.7 respectively. If $G_{0}\neq\mathrm{PSL}_{d}(q)$ and $\Omega$ is a set of
subspaces, then the result follows by Proposition 4.9. Hence by [7, 5.4] we
may assume that either $G_{0}=\mathrm{P}\Omega_{8}^{+}(q)$ and $G$ contains a
triality automorphism; or $G_{0}=\textrm{Sp}_{4}(2^{f})^{\prime}$ and $G$
contains a graph automorphism. In the former case the result holds by Lemma
4.8. In the latter $\mathrm{I}(G,\Omega)<\frac{11}{3}\log|\Omega|$ by [14,
Lemma 6.12]. ∎

##  5 Proof of Theorem 1.1

Here we use the form and notation of the O’Nan Scott Theorem from [19]. We
begin by considering groups of type PA, and then we prove Theorem 1.1.

######  Lemma 5.1.

Let $G$ be a subgroup of $\mathrm{S}_{n}$ of type PA that is not large base.
Then

$\mathrm{I}(G)<5\log n.$

###### Proof.

Since $G$ is of type PA there exists an integer $r\geq 2$, a finite set
$\Delta$ and an almost simple subgroup $H$ of $\mathrm{Sym}({\Delta})$ such
that $G\leq H\wr\mathrm{S}_{r}$. Since $G$ is not large base, neither is $H$.
Let $s=|\Delta|$, so that $n=s^{r}$ with $s\geq 5$. Then

$\begin{array}[]{rll}\mathrm{I}(G,\Omega)&\leq\mathrm{I}(H^{r},\Delta^{r})+\ell(\mathrm{S}_{r})&\text{
by Lemma \ref{3comb}\ref{subgroup} and \ref{normal},}\\\
&\leq\mathrm{I}(H^{r},\Delta^{r})+\frac{3}{2}r&\text{ by
\cite[cite]{[\@@bibref{}{PC}{}{}]},}\\\ &\leq
r(\mathrm{I}(H,\Delta)-1)+1+\frac{3}{2}r&\text{ by
\cite[cite]{[\@@bibref{}{GLS}{}{}, Lemma 2.6]},}\\\ &<r(5\log
s-2)+1+\frac{3}{2}r&\text{ by Theorem \ref{ASlem},}\\\ &<5\log
s^{r}-\frac{1}{2}r+1\\\ &\leq 5\log n&\text{ since }r\geq 2.\end{array}$

∎

We can now prove Theorem 1.1.

Proof of Theorem 1.1 Let $G$ be a primitive group which is not large base. If
$G$ is almost simple, then the result holds by Theorem 4.1. If $G$ is of type
PA then the result holds by Lemma 5.1. For all other $G$, the result holds by
[14, Props 3.1, 4.1 and 5.1] ∎

## References

  * [1] K.D. Blaha. Minimum bases for permutation groups: the greedy approximation. J. Algorithms 13 (1992), no. 2, 297–306.
  * [2] J.N. Bray, D.F Holt, C.M. Roney-Dougal. The maximal subgroups of the low-dimensional finite classical groups. Lond. Math. Soc. Lect. Note Ser. 407. Cambridge University Press, Cambridge, 2013.
  * [3] T. Burness. On base sizes for almost simple primitive groups. J. Algebra 516 (2018), 38–74.
  * [4] T. Burness. On base sizes for actions of finite classical groups. J. Lond. Math. Soc. (2) 75 (2007), no. 3, 545–562.
  * [5] T. Burness and M. Giudici. Classical groups, derangements and primes. Volume 25 of Aust. Math. Soc. Lect. Ser. Cambridge: Cambridge University Press, 2016.
  * [6] T. Burness, M. Liebeck and A. Shalev. Base sizes for simple groups and a conjecture of Cameron. Proc. Lond. Math. Soc. (3) 98 (2009), no. 1, 116–162.
  * [7] T. Burness, M. Liebeck and A. Shalev. Generation and random generation: from simple groups to maximal subgroups. Adv. Math. 248 (2013), 59-95.
  * [8] T. Burness, E. O’Brien and R. Wilson. Base sizes for sporadic simple groups. Israel J. Math. 177 (2010), 307–333.
  * [9] P. Cameron, R. Solomon and A. Turull. Chains of subgroups in symmetric groups. J. Algebra 127 (1989), no. 2, 340–352.
  * [10] G. Cherlin. Sporadic homogeneous structures. The Gelfand Mathematical Seminars, 1996–1999, 15–48, Gelfand Math. Sem., Birkhäuser Boston, Boston, MA, 2000.
  * [11] G. Cherlin, G. Martin and D. Saracino. Arities of permutation groups: wreath products and k-sets. J. Combin. Theory Ser. A 74 (1996), no. 2, 249–286.
  * [12] J. H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker and R.A. Wilson. $\mathbb{ATLAS}$ of finite groups. Maximal subgroups and ordinary characters for simple groups. Oxford University Press, Eynsham, 1985.
  * [13] N. Gill, M. Liebeck and P. Spiga. Cherlin’s conjecture on finite primitive binary permutation groups arXiv:2106.05154.
  * [14] N. Gill, B. Lodá and P. Spiga. On the height and relational complexity of a finite permutation group. Nagoya Math. J, to appear.
  * [15] A. Lachlan. On countable stable structures which are homogeneous for a finite relational language. Israel J. Math. 49 (1984), no. 1-3, 69-153.
  * [16] M. Liebeck. On minimal degrees and base sizes of primitive permutation groups. Arch. Math. (Basel) 43 (1984), no. 1, 11-15.
  * [17] B. Lodá. The height and relational complexity of finite primitive permutation groups. PhD Thesis, University of South Wales, 2020.
  * [18] M. Moscatiello and C.M. Roney-Dougal. Base size of primitive permutation groups. Monatsh. Math, to appear.
  * [19] C.E. Praeger. The inclusion problem for finite primitive permutation groups. Proc. Lond. Math. Soc. (3), 60(1):68-88, (1990).
  * [20] C.C. Sims. Computation with finitely presented groups. Encyclopedia of Mathematics and its Applications, 48. Cambridge University Press, Cambridge, 1994.
  * [21] A.V. Vasilyev, Minimal permutation representations of finite simple exceptional groups of types $E_{6}$, $E_{7}$ and $E_{8}$. Alg. and Logic 36, 302-310 (1997).

Veronica Kelsey & Colva M. Roney-Dougal: Mathematical Institute, Univ. St
Andrews, KY16 9SS, UK
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>


SJBsjb JWNjwn BSbs JOjo DBdb

[1]

[1]This document contains results of a research project funded in part by
Space Telescope Science Institute (AR 17031). Experiments were funded in part
by Europlanet travel grant 22-EPN3-107.

[type=editor, orcid=0000-0003-2110-8152]

[1]

Conceptualization, Methodology, Software, Investigation, Writing - Original
Draft

1]organization=Auburn University, addressline=Department of Physics,
city=Auburn, postcode=AL, 36849, country=USA

2]organization=Comenius University in Bratislava, addressline=Department of
Experimental Physics, Faculty of Mathematics, Physics and Informatics, city =
Bratislava, country = Slovakia [orcid=0000-0003-2152-6987] Writing - Original
Draft, Conceptualization [orcid=0009-0007-2894-3085] Investigation, Resources
[orcid=0000-0002-4309-6060] Investigation, Resources
[orcid=0000-0002-2668-7248] Conceptualization, Writing - Review & Editing

[cor1]Corresponding author

# Clearing the Air: Updated CS Fluorescence Efficiencies Don’t Solve the
NUV/Sub-mm Cometary CS Abundance Discrepancy

S.J. Bromley<EMAIL_ADDRESS>[  [  J. Wm. Noonan  B. Stachová  J. Országh
D. Bodewits

###### Abstract

Observations of carbon monosulfide (CS), a possible photodissociation fragment
of CS2, has a long history serving as a remote proxy for atomic sulfur, and
more broadly, one of the sulfur reservoirs in cometary bodies. Interpretations
of CS fluorescence emissions in ultraviolet observations of comets have, to-
date, relied on a murky and ill-documented lineage of calculations for CS
fluorescence efficiencies that can be difficult to assess. We report new
fluorescence efficiencies of the CS radical, utilizing a rovibrational
structure with vibrational states up to $v=8$ and rotational states up to
$N=100$. A new set of band transition rates is produced through benchmarking
comparisons to laboratory electron impact experiments, and subsequently
utilized in the fluorescence calculations. We confirm historical reports of
the ultraviolet $(0,0)$ band fluorescence efficiency, and conclude that blends
and an expanded rovibrational structure have little impact on the fluorescence
properties. Combined, the present results affirm the accuracy of the
historical record of sulfur abundances derived via ultraviolet CS emissions in
cometary observations with e.g., IUE and HST. Benchmark comparisons to IUE
observations of comet Bradfield show favorable agreement with the theoretical
models. We expand the model to the time-domain, and find that thermalization
of the ground state and/or ‘hot’ initial distributions of CS lead to noticable
changes in the shape of the $(0,0)$ band. The impact on existing sulfur
abundances derived via ultraviolet CS observations, and new connections to
sulfur reservoirs in protoplanetary disks are discussed. The model code and
computed fluorescence efficiencies are made publicly available.

###### keywords:

Comets, Spectroscopy,

##  1 Introduction

The astrochemical pathways leading to the formation of sulfur and sulfur-
bearing species remain poorly understood. The abundance of atomic sulfur in
the interstellar medium is nearly 1000 times larger than the abundances
measured in molecular clouds and protoplanetary disks (Tieftrunk et al., 1994;
Le Gal et al., 2019; Kama et al., 2019). One of the few sulfur-bearing
molecules that has been detected in molecular clouds, protoplanetary disks,
and our own solar system’s remnant planetesimals, like comets, is carbon
monosulfide, or CS (Tieftrunk et al., 1994; Le Gal et al., 2019; Teague et
al., 2018; Noonan et al., 2023, and sources therein). In astrophysical context
the molecule is an important tracer of the C/O ratio within protoplanetary
disks when observed in tandem with SO in the sub-mm (Keyte et al., 2023).
While CS is commonplace in the planetary formation process, establishing its
role and significance in the sulfur ecosystem remains a key goal for the
field.

Observations of CS in astrophysical environments are typically made through
two main methods: sub-millimeter (sub-mm) ro-vibrational transitions in the
ground X${}^{1}\Sigma$ state and near-ultraviolet (NUV) fluorescence in the
A${}^{1}\Pi$-X${}^{1}\Sigma$ system. These techniques have been applied to
comets within our solar system to investigate potential precursor molecules,
such as CS2, OCS, and H2CS. However, discrepancies often arise between the CS
abundances determined from NUV and sub-mm observations of the same comet
(Noonan et al., 2023; Biver et al., 2023). These variations complicate direct
comparisons across different wavelengths and require an evaluation of the
excitation efficiencies employed in calculating both the total molecular
column densities and production rates. For sub-mm emissions, these can also be
affected by collisions with other molecules, as well as excitation by thermal
continua such as e.g. emission from the nucleus.

CS was first detected in comet C/1975 V1 (West) in 1975 (Smith et al., 1980)
and to derive its general abundance, the authors calculated the fluorescence
efficiency (or $g$-factor) of the NUV $(0,0)$ band, presumably with only one
vibrational level in each electronic state, at a heliocentric/geocentric
distance of 0.52 au. The following years saw investigations of the A-X band
structure to derive a Boltzmann temperature of 70 K (Jackson et al., 1982),
and found that the ratios of the $(0,0)$, $(1,0)$, and $(0,1)$ bands of the
A-X system are well described by resonant fluorescence of solar radiation. The
spatial alignment of S and CS emissions observed in comet C/1979 Y1
(Bradfield), as discussed in Sanzovo et al. (1993), suggested CS2 as a parent
molecule of CS due to its short lifetime and corresponding small scale length.
Following this, Sanzovo et al. (1993) recalculated the $g$-factor for the CS
A-X band and determined production rates for 15 comets. Subsequent papers
referred to a fluorescence efficiency provided through a private communication
with few details on how it was derived (e.g., Noll et al. 1995; Stern et al.
1998; Noonan et al. 2023). These derived $g$-factors are summarized in Table
1. Even for models that used the same solar reference spectrum, there is a
variation of more than an order magnitude between the derived fluorescence
efficiencies, and the effect of the comet’s orbital velocity with respect to
the Sun on the fluorescence efficiency (Swing’s effect) has not been
evaluated. As noted in Noonan et al. (2023), improved fluorescence
efficiencies for the NUV bands of CS may contribute to a resolution of the
NUV/sub-mm CS abundance discrepancies. To enhance the accuracy of CS column
density measurements derived from NUV observations, it is necessary to
rigorously evaluate the fluorescence efficiencies of the
A${}^{1}\Pi$-X${}^{1}\Sigma$ bands.

Publication | Heliocentric | Solar Spectral | CS A${}^{1}\Pi$-X${}^{1}\Sigma$
---|---|---|---
| Distance (au) | Reference |  $(0,0)$ $g$-factor (pho.-1 mol.-1 s-1)
Smith et al. (1980) | 0.52 | Malitson et al. (1960) | 3.8 $\times$ 10-3
Jackson et al. (1982) | 1.0 | Kohl et al. (1978) | 7 $\times$ 10-4
Sanzovo et al. (1993) | 0.52 | Kohl et al. (1978); A’Hearn et al. (1983) | 3.9 $\times$ 10-3
Noll et al. (1995) | 1.0 | A’Hearn et al. (1983) | 5.8 $\times$ 10-4
Stern et al. (1998) | 1.0 | A’Hearn et al. (1983) | 5 $\times$ 10-4
Table 1: CS A${}^{1}\Pi$-X${}^{1}\Sigma$ $g$-factors from 1980 - Present

In this paper we perform new computational modeling of the fluorescing CS
A${}^{1}\Pi$-X${}^{1}\Sigma$ system using the FlorPy code (Bromley et al.,
2024), utilizing an updated set of transition rates benchmarked against an
electron impact spectrum produced from e+CS2 collisions. We review our
modeling methodology, benchmarking experiment, and discuss for the first time
the Swings effect for the CS A${}^{1}\Pi$-X${}^{1}\Sigma$ system. We conclude
by reviewing astrophysical observations that can benefit from the improved
model and review open questions as to the parentage of the CS radical.

##  2 Fluorescence Modeling

###  2.1 Theoretical Approach

Once an atom or molecule is produced in a cometary coma, the emitter begins to
absorb and re-emit sunlight. Assuming a species is formed in the lowest energy
(ground) state, absorptions of solar photons occur and re-distribute electron
population amongst its energy levels. The level populations $n_{i}$ can be
computed from a series of coupled equations written as

$\frac{dn_{i}}{dt}=-n_{i}\sum^{N}_{i\neq{k}}(A_{i\rightarrow{k}}+{\rho}B_{i\rightarrow{k}})+\sum^{N}_{j\neq{i}}n_{j}(A_{j\rightarrow{i}}+{\rho}B_{j\rightarrow{i}})$
(1)

where $A$ is the Einstein $A$ coefficient (the ‘transition rate’), $B$ are the
Einstein coefficients for absorption and stimulated emission, and $\rho$ is
the radiation field. We include $N$ individual levels in our model, resulting
in $N$ equations and $N-1$ unknowns. After sufficient time, the level
populations reach a steady-state called fluorescence equilibrium where
$dn/dt=0$, and the system is completed by the replacement of one equation with
the normalization condition $\sum_{i}n_{i}=1$ (Bromley et al., 2021; Magnani &
A’Hearn, 1986; Bromley et al., 2024). The system of equations can be re-cast
into matrix form, whose solution $n=A^{-1}\times{B}$ is tractable by numerous
computational methods. From the level populations, the fluorescence
efficiencies, or $g$-factors, are computed (in units of J s-1 mol.-1) as

$g_{ji}=\frac{hc}{\lambda_{ji}}n_{j}A_{j\rightarrow{i}}$ (2)

where $hc/\lambda_{ji}$ is the photon energy of the transition, $n_{j}$ is the
equilibrium population of the upper level involved in the transition, and
$A_{j\rightarrow{i}}$ is the transition rate. $g$-factors are also commonly
reported in units of photons s-1 mol.-1, computed with the omission of
$hc/\lambda$ in Eq. 2.

The fluorescence rate equations (Eq. 1) describe the level populations,
assuming an environment dominated by photoabsorption and radiative decays. For
atoms or molecule produced close to the nucleus, in the collisional part of a
cometary coma, the species may thermalize to some degree with the neutral
coma. For example, Jackson et al. (1982) found that the distribution of
rotational states in CS, as inferred from high-resolution spectra of the NUV
$(0,0)$ band, implied some degree of thermalization. This complication may
impact the band luminosities of CS and contribute to the ongoing discrepancy
between NUV and sub-mm-derived CS abundances.

The thermalization of the X${}^{1}\Sigma$ ground electronic state can be
readily incorporated into Eq. 1. For a given rotational temperature
$T_{\textrm{rot}}$, the (relative) level populations within X${}^{1}\Sigma$
can be computed as a Boltzmann distribution characterized by
$T_{\textrm{rot}}$. Eq. 1 thus reduces to $N-m$ linearly coupled equations,
where $N$ is the total number of levels, and $m$ is the number of known
(constrained) levels described by $T_{\textrm{rot}}$. The matrix solution to
the resulting $A\textrm{x}=B$ functional form, where $A$ is now an
$m\times{m}$ matrix, can be similarly solved, where the known $n_{j}{\rho}B$
terms are incorporated into matrix $B$ in the ($N-m$) system of equations. In
the present manuscript, we solve both the collision-free (Eq. 1) and
partially-thermalized cases to investigate whether such effects dramatically
alter the fluorescence properties of CS.

For the purposes of computing fluorescence efficiencies of CS, we utilize the
Python-based fluorescence code ‘FlorPy’ described in Bromley et al. (2021,
2024). In practice, the rate equations are agnostic to the details of the
atomic or molecular structure, and rely solely on transitions connecting
levels described by their energies and total angular momenta $J$. Transitions
themselves are described by a wavelength (or equivalently, an energy) with an
associated transition rate in units of s-1. Einstein B coefficients are
computed from the Einstein $A$ coefficients. We utilize here the solar
spectrum described in Bromley et al. (2024), where the fluxes pumping the
A${}^{1}\Pi$-X${}^{1}\Sigma$ transitions of CS are from the re-calibration
efforts described in Coddington et al. (2021), with the original high-
resolution spectrum deriving from Hall & Anderson (1991). Pumping of the
infrared transitions of CS is computed from high-resolution infrared spectra
generated with the Planetary Spectrum Generator (PSG; Villanueva et al. 2018).
The infrared solar spectrum utilized in the PSG derives from a series of high-
resolution Fourier Transform Spectrometer measurements conducted by the ACE
mission (Hase et al., 2010).

In Bromley et al. (2024) we presented a pipeline that was developed to produce
fluorescence emission models of diatomic molecules, with particular
application to CO+. This pipeline relied on existing molecular constants and
employed the PGOPHER program (Western, 2017) to model the vibronic structure
of the cation. We apply an analagous approach here to construct a line list of
CS using molecular constants reported in Bergeman & Cossart (1981) and the
PGOPHER program (Western, 2017) for the X${}^{1}\Sigma$ ground state and the
first excited state A${}^{1}\Pi$. Within each electronic state, vibrational
levels up to $v_{\textrm{max}}=8$ and rotational levels up to
$N_{\textrm{max}}=100$ are retained. Within each vibrational level, we utilize
the experimentally-derived constants for molecular rotation $(B)$ and the
centrifugal distortion constants $D$. As noted by Bergeman & Cossart (1981),
the A${}^{1}\Pi$-X${}^{1}\Sigma$ bands of CS are subject to substantial
perturbations from interactions with other electronic states. In most reported
laboratory and astronomical observations of CS in the ultraviolet, the
spectral resolution does not permit clear separation between rotational lines,
owing to the small rotational constants ($B\sim 0.8$ cm-1). For simplicity, we
also neglect perturbations as their impacts on the wavelengths are
substantially smaller then that of the rotational constant $B$ without
affecting the accuracy of the computed fluorescence efficiencies.

Previous calculations on diatomics relevant to cometary science demonstrated
that including both rovibronic transitions between electronic states and
rovibrational transitions within electronic states are required to accurately
model the level populations (Magnani & A’Hearn, 1986; Schleicher, 2010;
Bromley et al., 2024). In the present calculations, rovibronic transitions
between all included vibrational levels of the X${}^{1}\Sigma$ and
A${}^{1}\Pi$ electronic states are retained, as well as vibrational and
rotational transitions within the X${}^{1}\Sigma$ ground state. Einstein $A$
coefficients for the transitions of CS are sourced from the calculations of
Wei Xing & Sun (2020), and benchmarked against experimental measurements as
described below. Energy level and transition line lists, which are used as
inputs to the fluorescence model, are created from parsing the outputs of the
PGOPHER calculations, which are built on the constants described previously
and the newly-benchmarked transition rates, described further below.

###  2.2 Experimental Benchmarking

In the rate equations of fluorescence equilibrium, fluorescence efficiencies
and their resulting band luminosities (produced by sums over each band’s
transitions) depend on the transition rates. Further, fluorescence
efficiencies and band luminosities are proportional to the Einstein $A$
coefficient (Eq. 2). Wei Xing & Sun (2020) noted significant discrepancies
between computed lifetimes of CS vibrational levels and those reported from
experimental efforts. To improve the accuracy of the computed data utilized in
our models we re-normalized the theoretical lifetimes to the experimentally-
measured lifetimes reported by Mahon et al. (1997), who used laser-induced
fluorescence to selectively excite bands of CS and measure the resulting
radiative lifetimes. For $v>4$, for which no measured radiative lifetimes are
available, a linear fit ($R^{2}=0.997$) to the $v<4$ lifetimes of Mahon et al.
(1997) is used to extrapolate out to higher $v$. The linear fit is used to
predict experimental lifetimes for $v>4$, which are $\sim$30% larger than
those originally reported in Wei Xing & Sun (2020).

Figure 1: Comparison of the theoretical model (blue; summed over model
components (red)) and electron impact spectrum produced by 100 eV electrons
incident on CS2 gas (black). Emission bands of the CS
A${}^{1}\Pi$-X${}^{1}\Sigma$ system are labeled by
$(v^{\prime},v^{\prime\prime})$, where $v^{\prime}$ and $v^{\prime\prime}$ are
the upper and lower vibrational numbers, respectively. The modeled spectrum is
generated from benchmarked transition rates and vibrational populations
derived from the strongest feature associated with each A${}^{1}\Pi$
vibrational level. The electron impact spectrum is corrected for the
sensitivity of the optical elements, and a minor wavelength shift is applied
to the theoretical spectrum to account for wavelength calibration errors.

For a given vibrational level labeled by $v^{\prime}$, the branching ratios to
lower vibrational levels, labeled by $v^{\prime\prime}$, are described by
Franck-Condon factors. While these values are reported in various theoretical
efforts, we opted to benchmark the relative transition rates (Franck-Condon
factors) extracted from the tables of Wei Xing & Sun (2020) against
experimental measurements. For this purpose, the emission spectrum of CS was
recorded from crossed-beam electron impact experiments on CS2 gas at the
Laboratory of Electron Induced Fluorescence at Comenius University in
Bratislava, Slovak Republic in Summer 2023. Details of the experimental
apparatus are publicly available (see Danko et al. (2013); Bodewits et al.
(2019); Bromley et al. (2024), and references therein). In short, an energy-
monochromatized electron beam is crossed perpendicularly with a neutral beam
of CS2 gas inside a vacuum chamber (base pressure $\sim 10^{-8}$ mbar).
Experiments were conducted in the single-collision regime with chamber
pressures during gas injection not exceeding $10^{-4}$ mbar. Photons from
decaying collision products (CS, CS2, and CS${}_{2}^{+}$) were recorded by a
Czerny-Turner monochromator equipped with a UV-sensitive photomultiplier tube.
The optical response of the apparatus was established by previous measurements
of a known system, H2 $a^{3}\Sigma_{g}^{+}$-$b^{3}\Sigma_{u}^{+}$, and the
thermal blackbody produced by a heated tungsten filament. An example of the
resulting laboratory spectrum is shown in Figure 1 for a collision energy of
100 eV.

At electron energies above the threshold of CS (A${}^{1}\Pi$) formation from
CS2 ($\sim$9.27 eV), a large number of CS A${}^{1}\Pi$-X${}^{1}\Sigma$ bands
are excited between 250 – 280 nm. Our observed relative intensities of the
many CS bands agree well with those reported in Ajello & Srivastava (1981). In
order to benchmark the vibrational transition rates reported in Wei Xing & Sun
(2020), an iterative procedure was applied (described in the Appendix of
Bromley et al. 2024). First, the relative intensities of all bands were
extracted by either an integral over the spectral extent of each band, or in
the case of strong blends (the case for most bands of CS), was manually
adjusted to match the laboratory spectrum. To refine the handling of the many
blended features, vibrational level populations are derived from the strongest
bands of each A${}^{1}\Pi$ vibrational level, and manually adjusted to match
the observed spectrum. The relative transition rates are then normalized to
the lifetimes described above. We note that the rotational structure of the
bands is not resolved, though the lifetimes appear weakly sensitive to the
rotational quanta $J$ (Wei Xing & Sun, 2020). Further, for the purposes of the
present benchmarking, a rotational temperature of $\sim$300 K was assumed. It
is possible that the rotational distribution is non-thermal, but a more
rigorous derivation of the rotational populations cannot be derived at the
present spectral resolution. Lastly, the level populations are re-derived
using the updated transition rates, and the process was repeated until
sufficient agreement with the laboratory spectrum was achieved. In total, this
process leads to changes of order 20 – 30% in transition rates of the
strongest bands, and up to factors of $\sim$2 in weak transitions. Figure 1
shows a comparison of the laboratory spectrum at 100 eV electron energy and
our final, benchmarked model built on the re-calibrated transition rates. The
agreement with the experimental spectrum is excellent and lends confidence to
our use of the benchmarked transition rates in computing fluorescence
efficiencies.

##  3 Results

###  3.1 Band Luminosities

$v^{\prime}$ $\backslash$ $v^{\prime\prime}$ | 0 | 1 | 2
---|---|---|---
| 257.77 | 266.51 | 275.77
0 | 5.02e-22 | 5.55e-23 | 1.55e-24
| 6.52e-04 | 7.44e-05 | 2.16e-06
| 250.94 | 259.21 | 267.96
1 | 1.76e-23 | 4.28e-23 | 1.29e-23
| 2.23e-05 | 5.59e-05 | 1.74e-05
| 244.58 | 252.44 | 260.72
2 | 8.39e-26 | 1.75e-24 | 1.68e-24
| 1.03e-07 | 2.22e-06 | 2.20e-06
Table 2: Fluorescence Efficiencies of the CS (A${}^{1}\Pi$-X${}^{1}\Sigma$)
bands at $r_{\textrm{h}}=1$ au and $v_{\textrm{h}}=0$ km s-1. Fluorescence
efficiencies are provided as a function of upper ($v^{\prime}$; rows) and
lower ($v^{\prime\prime}$; columns) vibrational quantum numbers, respectively.
For each band, the following values are provided (top to bottom): vacuum
wavelength (nm) for the R(0) line within the band, the band luminosities
(summed over all transitions) in units of Joules s-1 molecule-1, and band
luminosities in units of photons s-1 mol.-1. Band luminosities for all bands
(including $v^{\prime},v^{\prime\prime}>2$) are provided as supplementary
files.

Previous fluorescence efficiencies of the A${}^{1}\Pi$-X${}^{1}\Sigma$ (0,0)
band of CS range from $1\times 10^{-4}$ to $5.8\times 10^{-4}$ photons s-1
mol.-1 at a heliocentric distance of 1 au (Smith et al., 1980; Jackson et al.,
1982; Noll et al., 1995; Stern et al., 1998) (see Table 1). As a first test of
our model, we computed fluorescence efficiencies of the CS ultraviolet bands
at a heliocentric distance of $r_{\textrm{h}}$ = 1 au and a heliocentric
velocity of $v_{\textrm{h}}$ = 0 km s-1. The resulting band luminosities,
summed over the transitions in each band, are provided in Table 2. The
luminosity of the $(0,0)$ band, $6.52\times 10^{-4}$ pho. s-1 mol.-1, is in
good agreement with the value reported in Noll et al. (1995). Both Noll et al.
(1995) and the present work utilize the molecular constants reported in
Bergeman & Cossart (1981). However, the extent of the rovibrational structures
utilized in the former is not clear. We find that bands from $v>2$ (not shown)
have negligible band luminosities and are omitted from Table 2. The present
values affirm that neither the updated solar spectrum, nor the large
rovibrational level structure utilized here, have introduced large changes to
the baseline g-factors.

As most UV comet observations have utilized low spectral resolution ($R\sim
1000$), the topic of blends must be addressed. In the laboratory, the strong
$(0,0)$ band is blended with the neighboring $(1,1)$ band. In fluorescence,
the equilibrium populations of the X${}^{1}\Sigma(v=1)$ rotational levels are
small compared to $(v=0)$. Consequently, the fluorescence band luminosity of
the $(1,1)$ band is less than 1/10th that of the $(0,0)$ band, and the overlap
between the bands is negligible. It can be concluded that any observed flux
attributed to the CS $(0,0)$ band in comet observations is largely free of
blends.

Figure 2: Present CS fluorescence model versus low-dispersion spectra of comet
Bradfield (observation ID LWR 6008), collected on 10 Jan. 1980 at
$r_{\textrm{h}}$ = 0.711 au (A’Hearn & Feldman, 1980). The NUV CS bands (left
to right) $(1,0)$, $(0,0)$, and $(0,1)$ show good relative agreement with the
fluorescence model. Figure 3: Present CS fluorescence models versus high-
dispersion spectra of comet Bradfield (observation ID LWR 6757), collected on
24 Jan. 1980 at $r_{\textrm{h}}$ = 0.925 au (Jackson et al., 1982). The shape
of the $(0,0)$ band is well described by the fluorescence model with fixed
relative ground state populations described by $T=70$ K (Jackson et al.,
1982).

Lastly, we consider the thermalization of the X${}^{1}\Sigma$ ground state. As
CS is produced in the inner coma, it is plausible that the ground state
populations are described by a Boltzmann distribution (Jackson et al., 1982).
The NUV spectrum produced from pumping of a thermalized ground state
population is likely to excite a wider range of transitions than pure
fluorescence alone. We pursued calculations of band luminosities assuming
ground state thermalization temperatures of 30, 70, 100, and 300 Kelvin. The
resulting luminosities of the $(0,0)$ band differ by at most 4%, though the
band shape can change dramatically with ground state temperature. Thus, for
the purposes of deriving CS abundances, pure-fluorescence band luminosities
are of sufficient accuracy.

Figure 4: Computed spectrum of CS A${}^{1}\Pi$-X${}^{1}\Sigma~{}(0,0)$ from
two time-dependent pure-fluorescence models with different initial conditions.
Orbital conditions of $r_{\textrm{h}}=1$ au and $v_{\textrm{h}}=0$ km s-1 are
assumed. The two models utilize different initial conditions, where the levels
at $t=0$ are given by Boltzmann distributions characterized by $T=50$ K (black
curve) and $T=1000$ K (red curve). Transition-resolved $g$-factors are
convolved with Gaussians with 0.02 nm FWHM to produce the presented profiles.

###  3.2 Cometary Spectra Benchmarks

As a benchmark test for these models, we apply them to both high and low
dispersion spectra of comet Bradfield, collected with the International
Ultraviolet Explorer (IUE). Figure 2 shows a comparison of the low-resolution
IUE spectrum (LWR 6008, 1980 Jan. 10) of A’Hearn & Feldman (1980) at
$r_{\textrm{h}}=0.711$ AU. Good agreement is observed between the cometary CS
and the modeled spectra of the NUV $(0,0)$, $(0,1)$, and $(1,0)$ bands.

High-dispersion observations of Comet Bradfield, described in Jackson et al.
(1982) were collected on 24 Jan. 1980. Figure 3 shows a comparison of the IUE
spectra and our spectral modeling, assuming a ground state (X${}^{1}\Sigma$)
temperature of 70 K. The agreement with the observed band shape is greatly
improved in the thermalized ground state model when compared to pure
fluorescence alone. An apparent deficiency in flux, red-ward of maximum
intensity, remains apparent. A minor wavelength shift of +0.015 nm is also
applied to the theoretical spectrum, assumed to correct for inaccuracies in
the wavelength calibration of the high-dispersion IEU spectra. However, this
shift may potentially indicate additional effects currently excluded from the
present models.

CS is formed in the inner coma, where collisions are still important
contributors to the excitation of CS. Molecular dissociation also tends to
populate higher rotational states within the fragment molecule. Krishna Swamy
& Tarafdar (1993) developed a theoretical emission model which explicitly
included collisions between CS and H2O in the rate equations. They computed
the level populations time-dependently, assuming a constant outflow velocity,
at conditions expected for comets Halley, Bradfield, and Giacobini-Zinner, as
well as exploring different initial X${}^{1}\Sigma$ rotational populations
expected from e.g. formation of CS from dissociation reactions. Comparisons to
high-resolution spectra of 1P/Halley showed difficulties in matching the
observed profiles. It is likely that the spectrum of CS, integrated over the
inner coma, is sampling a distribution of CS that is still developing in time,
likely from a non-thermal distribution(s) of initial X${}^{1}\Sigma$
rotational states. However, complications such as e.g. inaccurate molecular
data or pointing/tracking difficulties during the IUE observations may also
contribute to this problem (Krishna Swamy & Tarafdar, 1993). This appears to
be less severe in comet Bradfield, with good agreement achieved between the
observed low- and high-dispersion spectra and the present models which assume
a thermal distribution for X${}^{1}\Sigma$. Additional models may be required
to further explain the disagreements with high-resolution spectra of CS in
1P/Halley. These complications are discussed further below in the context of
time-dependent fluorescence models.

###  3.3 Heliocentric Distance, Velocity, and Time-Dependent Effects

Once produced, CS has a comparable photochemical lifetime to H2O, of order
$10^{5}$ seconds against photoionization at 1 au (Meier & A’Hearn, 1997; Heays
et al., 2017). The use of equilibrium fluorescence efficiencies in deriving
accurate abundances from observed fluxes necessitates the validity of
fluorescence equilibrium. We checked the validity of assuming fluorescence
equilibrium by computing a time-dependent solution of the fluorescence rate
equations (Eq. 1).

We utilize the time-dependent matrix method described previously in Bromley et
al. (2024), assuming first that the CS is initially formed in the lowest
energy state. From time-dependent calculations of the populations and
$g$-factors, we find that the band luminosity of the $(0,0)$ evolves from a
value $\sim 10\%$ greater than its equilibrium value at $t=0$, and
asymptotically approaches within $5\%$ of the equilibrium fluorescence
efficiency within $t=10^{3}-10^{4}$ seconds at 1 au. This suggests that any
observed emission from CS initially produced in the ground rovibrational state
is practically indistinguishable from a population of CS in fluorescence
equilibrium.

CS2, the often-attributed parent of CS (Noonan et al., 2023) produces CS in
highly excited vibrational and rotational states upon photodissociation
(McCrary et al., 1985). Krishna Swamy & Tarafdar (1993) found that such
distributions relax toward fluorescence equilibrium much slower than CS
initially formed in the ground rovibrational state. We explored similar
possibilities in the present time-dependent models, assuming initial
conditions described by Boltzmann distributions with a range of temperatures
from 50 to 1000 K.

Figure 4 shows the time-evolution of the two extremes, where each population
of CS is formed with X${}^{1}\Sigma$ levels described by a Boltzmann
temperature of 50 K and 1000 K. The population of CS formed with $T=50$ K
relaxes to fluorescence equilibrium rapidly. The population of CS formed at
$T=1000$ K, for which the most-populated X${}^{1}\Sigma$ peaks around $J=20$,
evolves to the same equilibrium over longer timescales of order $10^{4}$
seconds. The distribution of CS formed initially at 1000 K has apparent
separation between the R branch (blueward of 257.7 nm) and the Q ($\sim$257.8
nm) and P (redward of 257.8 nm) branches. With increasing time, the two
branches merge, with a noticeable peak increasing in intensity and shifting
from the wavelength position of the P branch towards that seen in fluorescence
equilibrium. If such a distribution were initially formed and evolving in the
inner coma, one would expect a shift in the wavelength position in the peak
flux, as was observed for comet Bradfield (Figs. 2-3).

Krishna Swamy & Tarafdar (1993) also noted a shift in the wavelength position
of the peak flux in their time- and density-dependent models. As noted
previously, an apparent wavelength shift of order 0.015 nm is apparent between
the present fluorescence model and comparisons to the observed high-resolution
IUE spectrum of comet Bradfield (Fig. 3). As this shift is several times
larger than the spectral pixel width, it is likely that the wavelength shift
is a real, physical effect, rather than a calibration error. From Krishna
Swamy & Tarafdar (1993), these shifts may be caused by an out-of-equilibrium
distribution of CS, which is being excited by both collisional and radiative
processes in the inner coma. Visual inspection of Fig. 4, which shows
instantaneous $g$-factors ($g(t)=hc/\lambda\times{n(t)}A$), exhibits a similar
trend. When incorporated into density-dependent 3-dimensional models of
cometary comae, integration over the appropriate line of sight may yield
profiles similar to that observed in e.g. comets Halley and Bradfield, where
CS in the inner coma is likely not in fluorescence equilibrium. While existing
models assuming a thermalized ground state, such as those presented in Fig. 3
find good relative agreement with the observed spectra (assuming an arbitrary
wavelength shift), additional work may be necessary to accurately describe the
shape of the CS $(0,0)$ band. Further, additional work may be needed to
understand whether the observed wavelength shifts confer information about as-
yet-unidentified parent molecules through the initial distribution of
rovibrational states populated upon photodissociation to CS.

As noted in the case of CO+ (Bromley et al., 2024), the variations in
equilibrium band shapes are driven through competition of absorption rates
between electronic states and radiative rovibrational transition rates within
the ground electronic state. In the case of CS, the infrared transitions are
relatively weak, with the strongest IR band, the
X${}^{1}\Sigma$-X${}^{1}\Sigma$ ($1,0)$ band between 7.8 - 7.9 $\mu$m, having
a band luminositiy $\sim{1}/100^{\textrm{th}}$ that of the
A${}^{1}\Pi$-X${}^{1}\Sigma$ $(0,0)$ band. Thus CS is a weak infrared emitter.
These trends are consistent with the weak dependence of the band shape on
heliocentric distance (Fig. 6), discussed further below.

The absorption rates in the fluorescence rate equations (Eq. 1) depend on both
heliocentric velocity and distance. Band luminosities may be insensitive to
heliocentric distance while the distribution of features within a band can be
affected dramatically, as was observed in e.g. CO+ (Bromley et al., 2024).
Large variations in fluorescence efficiency— exceeding factors of two—with
respect to heliocentric velocity have been observed in diatomics such as OH
(Schleicher & A’Hearn, 1988) and CN (Schleicher, 2010). However, reports on
how CS band luminosities vary with heliocentric velocity are currently lacking
and we therefore explored the potential for similar trends in CS. Figure 5
shows the band luminosities of the two strongest A${}^{1}\Pi$-X${}^{1}\Sigma$
bands of CS as a function of heliocentric velocity. The band luminosity of the
strong $(0,0)$ band is relatively insensitive to heliocentric velocities for
$|v_{\textrm{h}}|<20$ km s-1. At large heliocentric velocities
$|v_{\textrm{h}}|>20$ km s-1, variations in excess of 10% are apparent. The
neighboring weak $(1,1)$ band appears insensitive to velocity, changing by at
most 3% over the present velocity space of -50 to +50 km km s-1, a reasonable
parameter space for solar system comets.

Figure 5: Heliocentric velocity dependencies of the two strongest ultraviolet
A${}^{1}\Pi$-X${}^{1}\Sigma$ bands of CS. Each band is normalized to a value
of 1 at $v_{\textrm{h}}=0$ km s-1.

Similarly, previous studies have commonly adjusted the fluorescence
efficiencies of CS from $r_{\textrm{h}}=1$ au by a factor of
$1/r_{\textrm{h}}^{2}$. In some cases, such as CN (Schleicher, 2010), the band
luminosities are sensitive to both heliocentric distance and velocity. As
discussed previously, the NUV bands of CS are weakly sensitive to heliocentric
velocity. To quantify the extent to which scaling $g$-factors from 1 au to
$r_{\textrm{h}}$ is valid, we explicitly calculated the fluorescence
efficiencies up to $r_{\textrm{h}}=10$ au. Within the range of typical
heliocentric distances ($<$ few au) where UV observations have captured CS
ultraviolet emission, the band luminosities changed by at most 2% between
calculations re-scaled from 1 au, and those that explicitly account for the
distance dependence of the absorption rates in the rate equations (Eq. 1). As
shown in Fig. 6, changes in heliocentric distance can have a modest impact on
the distribution of lines within a band. However, at the low spectral
resolution with which CS has historically been observed ($\sim$0.5Å FWHM), the
effect on the shape of the bands is negligible. In short, re-scaling of CS
band luminosities from 1 au by a factor of $1/r_{\textrm{h}}^{2}$ for the
purposes of deriving column densities and/or abundances is justified.

Figure 6: Synthetic fluorescence spectra of the CS ultraviolet $(0,0)$ band at
$r_{\textrm{h}}=0.1,1,10$ au at high (0.01 nm FWHM, solid lines) and low (0.5
nm FWHM, dashed lines) spectral resolution. The vertical axis is proportional
to J s-1 mol-1, with all models normalized to a maximum value of 1 to
highlight changes to band shape.

##  4 Astrophysical Implications

The continued stability of the CS A${}^{1}\Pi$-X${}^{1}\Sigma$ fluorescence
efficiencies, despite marked improvements in solar spectral references, an
expanded rovibronic structure, and experimentally-benchmarked transition
rates, confirms that the NUV-derived CS column densities in cometary comae
from 1980 onward are accurate, even when the Swings effect was not explicitly
accounted for. In turn, we are able to proceed in scrutinizing the cometary
sulfur abundance catalog with renewed confidence in the measured column
densities and production rates of CS, although the source of the molecule
remains unclear (Meier & A’Hearn, 1997; Roth et al., 2021; Noonan et al.,
2023). We note that our higher resolution calculations and ability to reliably
calculate the Swings effect have implications for observations taken beyond
the solar system, and in particular for protoplanetary disks.

As stated earlier, CS has been observed in protoplanetary disks frequently in
the sub-mm, typically via the $J=2\rightarrow 1$ rotational transition (see
for example Kama et al. 2019). As such, these observations probe the cold mid-
plane of protoplanetary disks. NUV fluorescence of the CS molecule is
extremely unlikely in this region due to the optically thick dust, preventing
incident stellar radiation from exciting CS. Any CS fluorescence in the UV
observed in protoplanetary disks would be linked to the upper layers of the
disk, in the illuminated photodissociation region (Keyte et al., 2023).
Despite many observations of protoplanetary disks encompassing the wavelength
range of the CS A${}^{1}\Pi$-X${}^{1}\Sigma^{+}$ region as part of the ULYSSES
survey111https://ullyses.stsci.edu/ (e.g., Arulanantham et al. 2018; Espaillat
et al. 2022; Skinner & Audard 2022) we were unable to find any studies that
confirmed the presence of this band. With the improved models for the CS
$(0,0)$ band presented in this paper it may be possible to directly compare
the abundances of CS in the midplane to CS that has been transported up into
the PDR and is fluorescing. A similar strategy to capture the temperature
structure in the disk has been employed by observing both the CO
A${}^{1}\Pi$-X${}^{1}\Sigma^{+}$ band and the CO rovibrational lines at 4.7-5
$\mu$m (Arulanantham et al., 2018). Detections of the CS
A${}^{1}\Pi$-X${}^{1}\Sigma$ band coincident with the sub-mm rotational
emissions would provide insight into the variability of the S/O abundance as a
function of vertical location in the disk. A systematic re-analysis of NUV
observations of T-Tauri stars obtained with space telescopes like HST and IUE
could produce such results.

This work was initiated following the identification of discrepancies in
sulfur abundances derived from sub-mm and UV observations of comets, as
discussed in Noonan et al. (2023). A possible source of these discrepancies,
inaccurate fluorescence efficiencies for the NUV bands of CS, has now been
ruled out. The present $g$-factors, utilizing a greatly expanded solar
spectrum, energy level structure, and experimentally-benchmarked transition
rates, recovers comparable $g$-factors to previous studies of the molecule.
However, additional factors may contribute to the ongoing disrepancies between
NUV- and sub-mm-derived CS abundances:

  * •

The current state-of-the-art photochemical lifetime data for CS is purely
theoretical (e.g. Heays et al. 2017), and large errors in the
photodissociation and/or photoionization lifetimes may lead to either enhanced
or substantially reduced CS abundances.

  * •

Submillimeter observations probe the very innermost coma where collisional
effects are important; these may severely complicate a fluorescence efficiency
calculation for the rotational transitions of CS.

  * •

As noted in Noonan et al. (2023), CS2 alone can not explain the abundance of
CS inferred from UV observations. Prompt emission of CS following dissociation
from possibly unidentified parent species could contribute to either NUV or
sub-mm emission features, complicating a derivation of the CS abundances.
Further, prompt emission populates a different array of rovibrational states,
with typically higher rotational states compared to solar fluroescence (see
e.g. the case of OH in A’Hearn et al. 2015). These highly-excited states
evolve differently in time compared to molecules formed in the ground
rovibrational state, which can present a substantial challenge for accurate
spectral modeling. High-resolution UV comet spectra, capable of spectrally
separating the rotational transitions within the $(0,0)$ band would directly
test this possibility, and could theoretically be carried out using the G230H
Echelle grating for the HST’s STIS instrument (R$\sim$114,000). However, the
low throughput and narrow slit size would require that the target be both
unusually bright and have a well-characterized orbit to minimize acquisition
errors.

##  5 Conclusions

The historical record of CS A${}^{1}\Pi$-X${}^{1}\Sigma$ fluorescence
efficiencies covers a range of reported values, with minimal description of
the molecular and solar data used in their production. As such, the confidence
in existing column densities and abundances of CS derived via UV observations
of the $(0,0)$ was unclear. We reported new fluorescence efficiency
calculations of the CS radical utilizing a large rovibronic structure and a
new set of experimentally benchmarked transition rates. The present $g$-factor
for the UV A-X $(0,0)$ band at $\sim$2580 Å, $6.52\times 10^{-4}$ photons
sec.-1 mol.-1, is consistent with previous literature and suggests changes in
derived CS abundances of $\leq 11\%$.

Concluding, we find that the extent of the rovibronic structure utilized in
the calculations has a negligible impact on the fluorescence efficiency of the
strongest systems. We report, for the first time, explicit calculation of the
Swings effect for CS, and find that the bands are weakly sensitive to the
heliocentric velocity of the emitting environment. In addition, the band shape
changes negligibly as a function of $r_{\textrm{h}}$, and scaling of
$g$-factors at 1 au to a given heliocentric distance via
$1/r_{\textrm{h}}^{2}$ is valid. Thermalization of the X${}^{1}\Sigma$ ground
state leads to observable changes in band shape, as realized in archived IUE
comet spectra. However, this effect changes band the luminosities by only a
few per cent compared to the pure-fluorescence models, and does not impact any
previously derived CS abundances.

From time-dependent propagation of the fluorescence equations, we find that CS
initially formed in the ground rovibrational state quickly achieves
fluorescence equilibrium, suggesting that the use of $g$-factors computed with
the assumption of fluorescence equilibrium has not introduced large errors in
existing abundance derivations. However, time-dependent models with a ‘hot’
distribution of CS, formed initially at $T=1000$ K evolves much differently in
time and produces similar band luminosities. The time-dependent modeling
suggests that observed wavelength shifts between the theoretical models and
IUE cometary spectra are consistent with CS being formed in a variety of
rovibrational states. Additional modeling, incorporating collisional (density-
dependent) effects may be necessary to fully explain the observed band shapes
of the CS $(0,0)$ band in the NUV. Further, more refined modeling may provide
insights into the possible parents of CS through an understanding of the
initial rovibrational states of CS populated by photodissociation of parent
molecules. A possible observing schema with HST is noted which could provide
sufficient spectral resolution to investigate possible CS prompt emission.

We describe possible avenues for investigating sulfur reservoirs in
protoplanetary disks via UV fluorescence of the CS ultraviolet A-X $(0,0)$
band. Such analyses may shed light on the temperature structure in the disk
via comparisons of archival sub-mm and UV observations in the midplane and
photodissociation regions. Lastly, we note that inaccurate molecular data for
photodissociation or photoionization of CS, collisional effects, or prompt
emission from unidentified parent species may dramatically complicate the
interpretation of CS abundances derived from NUV or sub-mm observations of
comets.

We hope that the calculations presented here will be used to further
investigate the discrepancies between NUV- and sub-mm-derived sulfur
abundances in cometary observations. The fluorescence model, input files,
fluorescence efficiencies, and related supplementary files are provided on the
Zenodo service (https://zenodo.org/records/12555642).

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