# Realizations of Unisingular Representations by Hyperelliptic Jacobians

John Cullinan  Department of Mathematics, Bard College, Annandale-On-Hudson,
NY 12504, USA<EMAIL_ADDRESS>http://faculty.bard.edu/cullinan/

###### Abstract.

A representation of a finite group $G$ on a finite dimensional vector space
$V$ is called unisingular if every $g\in G$ has 1 as an eigenvalue in its
action on $V$. In this paper we show that certain unisingular representations
can be realized as mod 2 representations of hyperelliptic Jacobians over
$\mathbf{Q}$. We additionally identify new unisingular representations of the
symmetric and alternating groups.

##  1\. Introduction

###  1.1. Background and Motivation

In this note we look at a specific question at the intersection of Inverse
Galois Theory and the arithmetic of abelian varieties. Our goal here is to
raise the question of whether certain groups $G$ can be realized as Galois
over $\mathbf{Q}$, where we additionally specify the degree $d$ of the
generating polynomial that we seek and, furthermore, ask for the natural
symplectic embedding

$G\hookrightarrow{\rm
Perm}(d)\hookrightarrow\operatorname{Sp}_{2\lfloor(d-1)/2\rfloor}(2)$

to be irreducible (or absolutely irreducible).

Setting some terminology, if $\rho:G\to\operatorname{GL}(V)$ is a finite
dimensional representation of a finite group, we say that $\rho$ is
unisingular if $\det(1-\rho(g))=0$ for all $g\in G$. Alternatively, we say
that $\rho(G)$ is a fixed-point subgroup of $\operatorname{GL}(V)$.
Unisingular representations have been studied independently in the context of
Lie Theory (see, _e.g._ [17]) and in number theory (see, _e.g._ [5]) and,
recently, together [6]. In the number theoretic context, certain unisingular
representations can be used to describe point-counts on abelian varieties over
finite fields by interpreting the elements of $G$ as suitable Frobenius
elements in a certain fixed-point theorem.

Since sums of unisingular representations are unisingular, it makes sense to
focus on (and possibly classify all) irreducible unisingular representations.
This is still an enormous task, but when the vector space $V$ is defined over
a finite field $k$, some progress has been made. Two ways to approach this
problem are to 1) fix $\dim V$ and vary $k$, and 2) study specific classes of
groups and determine for which dimensions $\dim V$ and fields $k$ they have
unisingular representations. For example, 2-dimensional unisingular
representations, and 4- and 6-dimensional unisingular _symplectic_
representations over finite prime fields are necessarily reducible (see [9]
and [2]), while all 3-dimensional irreducible unisingular representations were
classified in [4]. For Lie-theoretic applications, we point the reader to [7]
and [17] as just some of many examples of that literature.

Let us now turn to the application that concerns us here. Let $A$ be an
abelian variety of dimension $g\geq 1$ defined over a number field $K$. If
$\mathfrak{p}\subseteq\mathcal{O}_{K}$ is a non-zero prime of good reduction
for $A$, then write $A_{\mathfrak{p}}$ for the reduction modulo $\mathfrak{p}$
of $A$. Then $A_{\mathfrak{p}}$ is a $g$-dimensional abelian variety defined
over the residue field
$\mathbf{F}_{\mathfrak{p}}:=\mathcal{O}_{K}/\mathfrak{p}$. Fix a rational
prime number $\ell$ and consider the following phenomenon. Suppose
$A_{\mathfrak{p}}$ has a point of order $\ell$ defined over
$\mathbf{F}_{\mathfrak{p}}$, for every good $\mathfrak{p}$, but no member of
the $K$-isogeny class of $A$ has a global point of order $\ell$. In other
words, suppose all but finitely many of the finite groups
$A_{\mathfrak{p}}(\mathbf{F}_{\mathfrak{p}})$ have order divisible by $\ell$,
but neither $A(K)$ nor any member $A^{\prime}$ of the $K$-isogeny class of $A$
has a point of order $\ell$ defined over $K$. By contrast, if _any_ member of
the $K$-isogeny class of $A$ had a point of prime order $\ell$ defined over
$K$, then all $A^{\prime}_{\mathfrak{p}}$ would have a point of order $\ell$
defined over $\mathbf{F}_{\mathfrak{p}}$ for all primes $\mathfrak{p}$ of good
reduction and _all_ $K$-isogenous $A^{\prime}$.

The first substantial work on this setup was done by Katz in [9] where he
showed that it amounts to understanding unisingular symplectic
representations. Specifically, the hypothesis that
$A_{\mathfrak{p}}(\mathbf{F}_{\mathfrak{p}})$ has a point of fixed order
$\ell$, for almost all primes $\mathfrak{p}$, means that
$\det(1-\overline{\rho}_{A,\ell}(g))=0$ for all
$g\in\operatorname{Gal}(\overline{K}/K)$, where

$\overline{\rho}_{A,\ell}:\operatorname{Gal}(\overline{K}/K)\to\operatorname{GSp}_{2g}(\mathbf{F}_{\ell})$

is the mod $\ell$ representation describing the action of Galois on the
$\ell$-torsion points of $A$. Subsequently, whether or not any members of the
$K$-isogeny class of $A$ have a global point of order $\ell$ is equivalent to
whether or not the Jordan-Hölder series (semisimplification) of the
representation $\overline{\rho}_{A,\ell}$ contains the trivial representation.
Duly motivated, we can now state our group-theoretic question as follows.

######  Question.

For all prime numbers $\ell$ and all positive integers $g$, can we classify
the irreducible fixed-point subgroups $G$ of
$\operatorname{GSp}_{2g}(\mathbf{F}_{\ell})$?

Such a classification would then give us a classification of mod $\ell$
representations attached to abelian varieties defined over a number field $K$
that have a point of order $\ell$ when reduced modulo all but finitely many
$\mathfrak{p}$, but neither they nor any member of their $K$-isogeny class has
a global point of order $\ell$.

For abelian varieties of dimensions 1 and 2 (so, symplectic representations of
dimensions 2 and 4, respectively), Katz showed that for any prime $\ell$, if
the mod $\ell$ representation is unisingular, then its Jordan-Hölder series
contains the trivial representation. This largely amounts to a group theory
argument showing that fixed-point subgroups $G$ of
$\operatorname{GL}_{2}(\mathbf{F}_{\ell})$ (resp.
$\operatorname{GSp}_{4}(\mathbf{F}_{\ell})$) have a trivial composition
factor; concretely, up to a global change of basis, the elements of $G$ fix a
common line.

For abelian threefolds (_i.e._ for the group
$\operatorname{GSp}_{6}(\mathbf{F}_{\ell})$) this is no longer the case for
odd $\ell$: there exist (reducible) unisingular representations whose Jordan-
Hölder series do not contain the trivial representation. However, when
$\ell=2$ all fixed-point subgroups of $\operatorname{Sp}_{6}(\mathbf{F}_{2})$
contain the trivial representation, as shown in [3]. In dimensions $\geq 4$ we
get our first examples of irreducible, symplectic, unisingular
representations. Namely, in [3] we raised the example, originally communicated
to us by Serre [15], that the Steinberg representation
$\mathsf{St}:\operatorname{SL}_{3}(\mathbf{F}_{2})\to\operatorname{Sp}_{8}(\mathbf{F}_{2})$
is both unisingular and absolutely irreducible. Using this example as a
starting point, in [6] we classified several new families of unisingular
symplectic representations, which we will recall below. Equipped with a
unisingular symplectic representation, we can turn to arithmetic and ask
whether or not we can construct an abelian variety $A$ with this specific
representation, _while minimizing the degree of the number field over which
$A$ is defined_.

Why do we ask for a minimal degree of definition? Fix $\ell$ and $g$ and
suppose that $G\subseteq\operatorname{GSp}_{2g}(\mathbf{F}_{\ell})$ is a
fixed-point subgroup. Let $A/\mathbf{Q}$ be an abelian variety such that the
mod $\ell$ representation

$\overline{\rho}_{A,\ell}:\operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to\operatorname{GSp}_{2g}(\mathbf{F}_{\ell})$

is surjective (such $A$ are known to exist by general principles). Then
consider the base-change to the fixed field of $G$:

$\xymatrix{&\mathbf{Q}(A[\ell])\ar@{-}^{\operatorname{GSp}_{2g}(\mathbf{F}_{\ell})}[dd]\ar@{-}_{G}[dl]\\\
K:=\mathbf{Q}(A[\ell])^{G}\ar@{-}[dr]\\\ \mathbf{Q}}$

This produces what we want: an abelian variety $A$, defined over a number
field $K$ with mod $\ell$ image isomorphic to $G$. However, the degree
$[K:\mathbf{Q}]$ is typically very large and such a construction does not
necessarily give a deeper understanding of the connection between unisingular
Galois representations and abelian varieties. For example, as we will see
below, we can construct a 4 dimensional abelian variety defined over
$\mathbf{Q}$ with an irreducible unisingular mod 2 representation of order
$2^{4}.3^{3}$. By comparison, the order of
$\operatorname{Sp}_{8}(\mathbf{F}_{2})$ is $2^{16}.3^{5}.5^{2}.7.17$. In
general, there may be some arithmetic obstruction to constructing abelian
varieties over $\mathbf{Q}$ with specified mod $\ell$ image (_e.g._ surjective
determinant), which is why we ask more generally for a minimal field of
definition, rather than asking for it over $\mathbf{Q}$.

When $\ell=2$, there is a natural source of potential examples: hyperelliptic
Jacobians. If $k$ is a field of characteristic different from 2, $C_{f}$ the
hyperelliptic curve of genus $g$ defined over $k$ by the equation

$y^{2}=f(x),$

and $J_{f}:=\operatorname{Jac}(C_{f})$ its Jacobian, then it is well known
that the 2-torsion field $k(J_{f}[2])$ has Galois group isomorphic to
$\operatorname{Gal}_{k}(f)$. As above, set $d=\deg(f)$ so that
$g=\lfloor\frac{d-1}{2}\rfloor$. Studying the image of the mod 2
representation for unisingularity amounts to understanding the Galois group
$\operatorname{Gal}_{k}(f)$ and its symplectic embedding

$\operatorname{Gal}_{k}(f)\to{\rm
Perm}(d)\to\operatorname{Sp}_{2g}(\mathbf{F}_{2}),$

where the first inclusion is basic Galois theory and the second is a standard
construction in permutation group theory (see [16, §2] for details). This
raises the next main question of the paper – the realization problem.

######  Question.

Can irreducible fixed-point subgroups of
$\operatorname{Sp}_{2g}(\mathbf{F}_{2})$ be realized as Galois groups over
$\mathbf{Q}$ via the 2-torsion subgroup of hyperelliptic Jacobians of genus
$g$?

If we could answer this question for a particular group $G$, then we would
have solved the realization problem for $G$ over $\mathbf{Q}$. This is a
problem of Inverse Galois Theory where we are additionally specifying the
degree of the polynomial generating the Galois group and asking for (absolute)
irreducibility of the symplectic representation of $G$. By restricting to the
prime $\ell=2$, we simplify the general problem of realization in two ways.
First, when $\ell=2$, the symplectic similitude group and isometry group
coincide:
$\operatorname{Sp}_{2g}(\mathbf{F}_{2})\simeq\operatorname{GSp}_{2g}(\mathbf{F}_{2})$.
This allows us to avoid surjectivity conditions on the determinant
representation when attempting to construct explicit examples. Secondly, when
$\ell=2$ there is a well understood mechanism for explicit computation in the
2-torsion subgroup. For odd $\ell$, it is only for very specific examples of
Jacobians that we can explicitly write down equations of the underlying curve
where we can also rigorously show that the image of the mod $\ell$
representation is exactly $G$ which, as the example above indicates, is
typically much smaller than the maximal image. By comparison, showing an image
is large typically amounts to computing Frobenius elements and showing that
the image meets enough conjugacy classes to generate a large subgroup of
$\operatorname{GSp}_{2g}(\mathbf{F}_{\ell})$.

###  1.2. Statement of Main Results

We are now ready to describe our main results. Let $f\in k[x]$ be an
irreducible polynomial of degree $d\geq 3$ with Galois group $G$. Let
$g=\lfloor(d-1)/2\rfloor$. We say that the hyperelliptic Jacobian $J_{f}$
unirealizes $G$ over $k$ if the composition $\varphi$ of the natural
permutation representation with its symplectic embedding (see [16] for the
details of this construction)

$\operatorname{Gal}_{k}(f)\simeq\underbrace{G\hookrightarrow
S_{d}\hookrightarrow\operatorname{Sp}_{2g}(\mathbf{F}_{2})}_{\varphi}$

is irreducible and unisingular; note that the irreducibility of the polynomial
$f$ does _not_ guarantee the irreducibility of the representation
$\varphi:{G}\to\operatorname{Sp}_{2g}(\mathbf{F}_{2})$.

The smallest $g$ for which there exist any irreducible fixed-point subgroups
$G\subseteq\operatorname{Sp}_{2g}(\mathbf{F}_{2})$ is $g=4$ and the maximal
fixed-point subgroups of $\operatorname{Sp}_{8}(\mathbf{F}_{2})$ are ${\rm
L}_{3}(2):2$ and $\operatorname{AGL}_{2}(3)$ [6]. Note that ${\rm
L}_{3}(2):2\simeq\operatorname{PGL}_{2}(7)$ and recall from [6, Thm. 1.7] that
$\operatorname{PGL}_{2}(q)$ has an irreducible unisingular representation into
$\operatorname{Sp}_{q+1}(\mathbf{F}_{2})$ when, simultaneously, $q$ is odd,
$4|(q+1)$, and $3|(q-1)$ . Thus, the group $\operatorname{PGL}_{2}(7)$ is the
smallest of an infinite family of fixed-point subgroups of sympectic groups.
(The two smallest groups in this family are $\operatorname{PGL}_{2}(7)$ and
$\operatorname{PGL}_{2}(19)$.) Our first result is on the unirealizations of
these smallest examples.

######  Theorem 1.2.1.

  1. (1)

There exist infinitely many polynomials $f\in\mathbf{Q}[x]$ of degree 9 such
that $J_{f}$ unirealizes $\operatorname{AGL}_{2}(3)$ over $\mathbf{Q}$.

  2. (2)

The 8-dimensional, irreducible, unisingular, $\mathbf{F}_{2}$-representation
of ${\rm L}_{3}(2):2$ (Steinberg representation) does not occur as the mod 2
representation of a genus 4 hyperelliptic Jacobian.

Even though the proof of Item (2) is not difficult – the group ${\rm
L}_{3}(2):2$ is not a transitive subgroup of $S_{9}$ or $S_{10}$ – it raises
the more interesting question of whether or not there exists any abelian
fourfold over $\mathbf{Q}$ with this mod 2 representation. We point the reader
to Example 3.0.3 below for an explicit instance of item (1) above.

In the second part of the paper we identify new classes of characteristic zero
unisingular representations. Here we focus on the symmetric and alternating
groups and show that some of their Specht modules are unisingular over
$\mathbf{Z}$. Not all Specht modules for $S_{n}$ are unisingular over
$\mathbf{C}$; for example, the standard representation $\mathcal{S}^{(n-1,1)}$
of $S_{n}$ is not unisingular once $n>1$, for any $n$, since the $n$-cycles
have characteristic polynomial $(x^{n}-1)/(x-1)$, which does not vanish at
$x=1$. Nevertheless, we can prove the following.

######  Theorem 1.2.2.

The Specht modules $\mathcal{S}^{(n-2,1,1)}$ and $\mathcal{S}^{(n-2,2)}$ are
unisingular for $S_{n}$.

Computation suggests that many more classes of Specht modules
$\mathcal{S}^{\lambda}$ are unisingular, but we will not investigate them
further except to state that to determine precisely which ones are unisingular
would be an interesting topic for further exploration. We similarly expect
that many classes of Specht modules can be proved _not_ to be unisingular and
put forth the following conjecture, based on evidence we supply in Section 5.

######  Conjecture 1.2.3.

Let $n\geq 5$ be odd. Then the class of elements of order $2n-4$ in the
conjugate representation
$\mathcal{S}^{(n-2,2)^{\prime}}=\mathcal{S}^{(2,2,1,\dots,1)}$ does not afford
1 as an eigenvalue. Therefore the Specht module
$\mathcal{S}^{(n-2,2)^{\prime}}$ is not unisingular.

Specht modules are irreducible over $\mathbf{C}$ and defined over
$\mathbf{Z}$, so it makes sense to reduce them modulo any prime $\ell$, and in
particular $\ell=2$. However, they might not remain irreducible upon
reduction. For example, when $n=5$, the module $\mathcal{S}^{(3,1,1)}$ is
unisingular of dimension 6, and irreducible in characteristic 0. Modulo 2, the
representation is still (clearly) unisingular, but decomposes into a sum of
two 1-dimensional (trivial) factors and a 4-dimensional factor.

So this raises the question – which of the unisingular Specht modules of
Theorem 1.2.2 can be realized by hyperelliptic Jacobians? In order to achieve
this, we would require

  1. (1)

an even-dimensional, unisingular, irreducible representation
$\mathcal{S}^{\lambda}$ of $S_{n}$; write
$\dim\mathcal{S}^{\lambda}=f^{\lambda}=2d$, and

  2. (2)

$\mathcal{S}^{\lambda}$ to be irreducible modulo 2 (see [8, Main Theorem] for
precise conditions on when this occurs), and

  3. (3)

$f(x)\in\mathbf{Q}[x]$ an irreducible polynomial of degree $2d+1$ or $2d+2$
with Galois group $S_{n}$, and

  4. (4)

the permutation representation $S_{n}\to{\rm Perm}(2d+1\text{ or
}2)\to\operatorname{Sp}_{2d}(\mathbf{F}_{2})$ to coincide with the reduction
modulo 2 of $\mathcal{S}^{\lambda}$.

In fact, we expect that these items are _never_ simultaneously satisfied, and
give some evidence for this expectation at the end of the paper. If so, this
would mean that while unirealizations of the Specht modules may exist over
$\mathbf{Q}$, even for Jacobians, they cannot be realized by _hyperelliptic_
Jacobians.

However, this gives a potential window into determining minimal fields of
definitions for unirealizing these Specht modules and other unisingular
representations by hyperelliptic Jacobians. Namely, if we consider group
extensions $S_{n}.Q$ and field diagrams

$\xymatrix{L\ar@{-}^{S_{n}}[d]\\\ K\ar@{-}^{Q}[d]\\\ \mathbf{Q}}$

and ask for polynomials $f(x)$ with Galois group $S_{n}.Q$ of the appropriate
degree to unirealize $S_{n}$ by $J_{f}$ over the base change to $K$. We will
not pursue this question in this paper and will be content to simply determine
new families of unisingular representations of the symmetric group.

###  1.3. Notation and Conventions

We have already introduced much of the notation we will use in the paper. In
general, we use Atlas [1] notation for finite groups, _e.g._ ${\rm L}_{n}(q)$
for the finite group $\operatorname{PSL}_{n}(\mathbf{F}_{q})$. We will make
extensive use of the LMFDB [10] database for explicit polynomial and/or field
extensions. In each instance we provide a link to the LMFDB to the relevant
object. Similarly we rely on the online Atlas at
http://brauer.maths.qmul.ac.uk/Atlas/v3/ for explicit matrix representations
of large degree.

###  1.4. Acknowledgments

We would like to thank Gunter Malle and Alexandre Zalesski for helpful
conversations. We also thank Drew Sutherland for access to a version of Magma
allowing us to perform some of the lengthy eigenvalue computations in Section
5.

##  2\. Permutation Representations and Jacobians

For a detailed background on the 2-torsion of Jacobians, we refer the reader
to [16, §1.2]; in this section we will just give a brief sketch of how one
associates to a hyperelliptic curve the 2-torsion subgroup of its Jacobian.
All of this holds in far more generality, but in this paper we will work
exclusively over $\mathbf{Q}$ for simplicity.

Let $C_{f}/\mathbf{Q}$ be the hyperelliptic curve defined by the squarefree
polynomial $f(x)$ of degree $d\geq 3$:

$C_{f}:y^{2}=f(x).$

Then $C_{f}$ has genus $g=\lfloor(d-1)/2\rfloor$ and its Jacobian
$J_{f}:=\operatorname{Jac}(C_{f})$ is an abelian variety of dimension $g$,
also defined over $\mathbf{Q}$. Let $K_{f}$ be the splitting field over
$\mathbf{Q}$ of $f$ and $\operatorname{Gal}_{\mathbf{Q}}(f)$ its Galois group;
we allow for intransitive Galois groups for the moment.

If $A$ is a principally polarized abelian variety of dimension $g$ defined
over $\mathbf{Q}$, then the 2-torsion subgroup $A[2]$ of $A$ has the structure
$A[2]\simeq(\mathbf{Z}/2)^{2g}$ as a Galois module, yielding the Galois
representation (the mod 2 representation)

$\overline{\rho}_{A,2}:\operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to\operatorname{GL}_{2g}(\mathbf{F}_{2}).$

The Weil pairing is an alternating, Galois-equivariant pairing on $A[2]$,
whence the image of $\overline{\rho}_{A,2}$ lies in
$\operatorname{Sp}_{2g}(\mathbf{F}_{2})$.

Via divisors, $C_{f}$ embeds into $J_{f}$ and the 2-torsion subgroup of
$J_{f}$ is identified with the Weierstrass points of $C_{f}$. This
construction shows us that

$\operatorname{im}\overline{\rho}_{J_{f},2}\simeq\operatorname{Gal}(K_{f}/\mathbf{Q})\subseteq
S_{d}\subseteq\operatorname{Sp}_{2g}(\mathbf{F}_{2}).$

By comparing orders, we see that once $g\geq 3$ the mod 2 representation of a
hyperelliptic Jacobian cannot be surjective onto
$\operatorname{Sp}_{2g}(\mathbf{F}_{2})$. For the explicit details on how one
goes from the natural permutation representation on the roots of $f$ to the
symplectic embedding, see [16, Prop. 2.1].

Therefore, if we are interested in constructing interesting (_e.g._ absolutely
simple) examples of abelian varieties over $\mathbf{Q}$ which have an even
number of points modulo all but finitely many $p$, but that no member of its
$\mathbf{Q}$-isogeny class has a rational point of order 2, then one approach
would be to follow the procedure from the Introduction:

  1. (1)

identify groups $G$ with irreducible, unisingular, symplectic
($2g$-dimensional), representations $\rho$ over $\mathbf{F}_{2}$, and

  2. (2)

determine irreducible polynomials $f\in\mathbf{Q}[x]$ of degree $d=2g+1$ or
$2g+2$ with $\operatorname{Gal}_{\mathbf{Q}}(f)\simeq G$, such that the
composition

$G\to S_{d}\hookrightarrow\operatorname{Sp}_{2g}(\mathbf{F}_{2})$

of the permutation representation with the natural inclusion into the
symplectic group $\operatorname{Sp}_{2g}(\mathbf{F}_{2})$ coincides with
$\rho$.

If we can find a suitable polynomial $f$ that achieves both items, then recall
from the Introduction that we say $J_{f}$ unirealizes $G$. One cautionary
remark is in order: it does not immediately follow that if $f$ is an
irreducible polynomial then the $2g$-dimensional symplectic representation of
$G$ is irreducible. For example, in [18, Prop. 2.6], Zarhin shows that if $n$
is _prime_ , and 2 is a primitive root modulo $n$, and $f(x)$ is irreducible
of degree $n$, then the associated mod 2 representation on the Jacobian
$J_{f}$ is irreducible. (If the permutation representation is, additionally,
2-transitive, then the mod 2 representation is absolutely irreducible.) While
one may be able to prove more general relationships between the irreducibility
of $f$ and the irreducibility of the associated symplectic representation, we
will only give _ad hoc_ reasoning that our examples yield irreducible mod 2
representations.

##  3\. Unirealization of $\operatorname{AGL}_{2}(3)$

Our first result concerns the smallest-sized groups in the families of
unisingular representations introduced in [6].

######  Theorem 3.0.1.

  1. (1)

There exist infinitely many polynomials $f\in\mathbf{Q}[x]$ of degree 9 such
that the hyperelliptic Jacobian $J_{f}$ unirealizes
$\operatorname{AGL}_{2}(3)$.

  2. (2)

The 8-dimensional, irreducible, unisingular, $\mathbf{F}_{2}$-representation
of ${\rm L}_{3}(2):2$ does not occur as the mod 2 representation of a genus 4
hyperelliptic Jacobian.

###### Proof.

For item 1, we refer to the constructions of Malle in [11, §7]. There he
exhibits two different 2-parameter families of rational polynomials with
Galois group $\operatorname{AGL}_{2}(3)$. For infinitely many specializations
of both parameters, these polynomials are irreducible of degree 9 and have
Galois group $\operatorname{AGL}_{2}(3)$ over $\mathbf{Q}$. The group
$\operatorname{AGL}_{2}(3)$ is maximal in $S_{9}$ by the O’Nan-Scott Theorem,
and the inclusion $\operatorname{AGL}_{2}(3)\hookrightarrow
S_{9}\to\operatorname{Sp}_{8}(2)$ is absolutely irreducible and unisingular by
[6, Thm. 4.1]. Since $\operatorname{Sp}_{8}(2)$ has a unique (up to conjugacy)
subgroup isomorphic to $S_{9}$ and $S_{9}$ has a unique (up to conjugacy)
subgroup isomorphic to $\operatorname{AGL}_{2}(3)$, the specializations of
Malle’s polynomials give precisely this fixed-point subgroup of
$\operatorname{Sp}_{8}(2)$. By the process outlined above, there exists a
hyperelliptic Jacobian of genus 4 defined over $\mathbf{Q}$ unirealizing
$\operatorname{AGL}_{2}(3)$.

For item 2, we simply observe that ${\rm
L}_{3}(2):2\simeq\operatorname{PGL}_{2}(7)$ is not a transitive subgroup of
$S_{9}$ or $S_{10}$, and so is not the Galois group of an irreducible degree 9
or 10 polynomial over $\mathbf{Q}$. ∎

For number theoretic constructions it is often desirable to have examples
ramified at very few, small, primes. For the family

$\displaystyle g_{a,t}(x)$
$\displaystyle=x^{9}+(6a-21)x^{8}+(15a^{2}-102a+171)x^{7}+(18a^{3}-195a^{2}+648a-675)x^{6}$
$\displaystyle\
+(9a^{4}-168a^{3}+903a^{2}-1890a+1296)x^{5}+(-51a^{4}+516a^{3}-1773a^{2}+2430a$
$\displaystyle\
-972)x^{4}+(-a^{6}+6a^{5}+81a^{4}-604a^{3}+1350a^{2}-972a+t)x^{3}+(3a^{6}-12a^{5}$
$\displaystyle\
-57a^{4}+282a^{3}-324a^{2}-3t)x^{2}+(-3a^{6}+6a^{5}+18a^{4}-36a^{3})x+a^{6},$

of Malle [11] with Galois group $\operatorname{AGL}_{2}(3)$, we have

(3.0.2)
$\displaystyle\operatorname{disc}(g_{a,t}(x))=-2^{8}3^{9}t^{4}a^{6}r(a,t)^{3},$

where

$\displaystyle r(a,t)$
$\displaystyle=a^{12}-12a^{11}+96a^{10}-520a^{9}+2166a^{8}-6960a^{7}-2a^{6}t+17524a^{6}$
$\displaystyle\
+12a^{5}t-34200a^{5}+48a^{4}t+49329a^{4}-272a^{3}t-49572a^{3}+342a^{2}t$
$\displaystyle\ +26244a^{2}-108at+t^{2}+108t$

is irreducible in $\mathbf{Q}[a,t]$. By (3.0.2) it is only at specializations
$(a,t)$ that are powers of 2 and 3 that $\operatorname{disc}(g_{a,t}(x))$ can
be solely divisible by the primes 2 and 3. Finding such rational
specializations amounts to determining the rational points on the algebraic
curves defined by

$r(a,t)=\pm 2^{\alpha}3^{\beta}.$

For a fixed value of $\pm 2^{\alpha}3^{\beta}$, Riemann-Hurwitz shows that the
curve has genus $>$ 1, hence by Faltings will have finitely many rational
points. At these rational points we get interesting cases of hyperelliptic
Jacobians. We are content to highlight an illustrative example.

######  Example 3.0.3.

The polynomial

$g_{1,-32}(x)=x^{9}-15x^{8}+84x^{7}-204x^{6}+150x^{5}+150x^{4}-172x^{3}-12x^{2}-15x+1$

defines a number field isomorphic to the one with LMFDB label
9.3.82556485632.1 with class number 1 and discriminant $-\,2^{22}\cdot 3^{9}$.
The genus 4 hyperelliptic Jacobian defined by $y^{2}=g_{1,-32}(x)$ is ramified
only at the primes 2 and 3, and unirealizes $\operatorname{AGL}_{2}(3)$.

In particular, $J_{f}$ enjoys the properties of being defined over
$\mathbf{Q}$, having an even number of points over $\mathbf{F}_{p}$ for all
$p\geq 5$, and no member of the $\mathbf{Q}$-isogeny class of $J_{f}$ has a
global 2-torsion point.

To conclude this example, we mention that in the paper [6] we showed that the
only subgroups $G$ of $\operatorname{AGL}_{2}(3)$ that act irreducibly on the
space $\mathbf{F}_{2}^{8}$ are

${\rm A}\Gamma{\rm L}_{1}(9),\
\operatorname{AGL}_{1}(9),\operatorname{PSU}_{3}(2),\text{and}\ {\rm
ASL}_{2}(3),$

of orders 144, 72, 72, and 216, respectively. For example, the specialization
$g_{1,1}(x)$ has Galois group ${\rm A}\Gamma{\rm L}_{1}(9)$, showing that the
hyperelliptic Jacobian defined by $g_{1,1}(x)$ unirealizes an even smaller
irreducible fixed-point subgroup of $\operatorname{Sp}_{8}(2)$ than
$\operatorname{AGL}_{2}(3)$.

######  Remark 3.0.4.

The next-smallest group in the unisingular family of
$\operatorname{PGL}_{2}(q)$ is $\operatorname{PGL}_{2}(19)$. We point out that
there are five irreducible polynomials of degree 20 with Galois group
$\operatorname{PGL}_{2}(19)$ over $\mathbf{Q}$ currently indexed in the LMFDB:

  * •

20.2.1339432011824591977742384617725412507648.1

  * •

20.2.2288697943298220011370049010943499874304.1

  * •

20.0.14595833152648338774900083206236247150592.1

  * •

20.0.180218116213002899035577493449527785489956864.1

  * •

20.2.206007596521214410095208558252435839890349094339.1

While the 20-dimensional $\mathbf{F}_{2}$-representation of
$\operatorname{PGL}_{2}(19)$ was shown to be unisingular in [6, Thm. 1.7],
these polynomials define 9-dimensional hyperelliptic Jacobians. Hence, these
symplectic embeddings are $\operatorname{PGL}_{2}(19)\hookrightarrow
S_{18}\to\operatorname{Sp}_{18}(\mathbf{F}_{2})$. The group
$\operatorname{PGL}_{2}(19)$ has three 2-modular irreducible representations
defined over $\mathbf{F}_{2}$, of degrees 1, 18, and 20. The 18-dimensional
representation is not unisingular since the characteristic polynomial of the
elements of order 19 is cyclotomic of degree 18, which does not have 1 as a
root in $\mathbf{F}_{2}$.

We would therefore require a degree 21 or 22 polynomial with Galois group
$\operatorname{PGL}_{2}(19)$ in order to unirealize
$\operatorname{PGL}_{2}(19)$. However, $\operatorname{PGL}_{2}(19)$ is not a
transitive subgroup of $S_{21}$ or $S_{22}$. Thus,
$\operatorname{PGL}_{2}(19)$ cannot be unirealized by a hyperelliptic Jacobian
over $\mathbf{Q}$. This illustrates some of the difficulties involved with
constructing hyperelliptic unirealizations.

##  4\. Unisingular Representations of the Symmetric Group

Now we turn to the symmetric and alternating groups and prove that certain
families of Specht modules are always unisingular, over any field. While this
may be well known to experts, we were unable to find this specific result in
the literature.

Let $n$ be a positive integer, $S_{n}$ the symmetric group on $n$ letters,
$\lambda$ a partition of $n$, and $\mathcal{S}^{\lambda}$ the associated
Specht module for $S_{n}$. Since $\mathcal{S}^{\lambda}$ is defined over
$\mathbf{Z}$, it makes sense to reduce $\mathcal{S}^{\lambda}$ modulo $p$ and
consider it as a (possibly reducible) representation of $S_{n}$ over
$\mathbf{F}_{p}$.

We now focus on two partitions of $n$ in particular: $(n-2,1,1)$ and
$(n-2,2)$. It is well known that

$\dim\mathcal{S}^{(n-2,2)}=n(n-3)/2\qquad\text{and}\qquad\dim\mathcal{S}^{(n-2,1,1)}=\dim\mathcal{S}^{(n-2,2)}+1.$

Our main theorem, which we quote from the Introduction, is on the
unisingularity of these Specht modules.

######  Theorem 4.0.1.

The Specht modules $\mathcal{S}^{(n-2,1,1)}$ and $\mathcal{S}^{(n-2,2)}$ are
unisingular for $S_{n}$.

If $n$ is odd, then we conjecture below that only one class of elements fails
to have 1 as an eigenvalue in the conjugate Specht module
$\mathcal{S}^{(n-2,2)^{\prime}}=\mathcal{S}^{(n-2,2)}\otimes\mathcal{S}^{(1,\dots,1)}$.
Moreover, restricting these representations to $A_{n}$ yields irreducible,
unisingular representations since the class in question consists of odd
permutations.

In the proof below we use the standard notions of tableaux, tabloids, and
polytabloids, as developed comprehensively in [14]. We adopt their notation as
well: if $t$ is a tableau of shape $\lambda$, write ${\\{t\\}}$ and $e_{t}$
for the associated tabloid and polytabloid, respectively. We remind the reader
that a _derangement_ of a set of cardinality $m\geq 2$ is a permutation of
that set with no fixed points. We also remind the reader that if $e_{t}$ is a
polytabloid and $\pi\in S_{n}$, then $\pi e_{t}=e_{\pi t}$.

Before getting into the details, we sketch the idea of the proof, which is
simple: for each $\sigma\in S_{n}$ we identify an explicit element of
$\mathcal{S}^{\lambda}$ fixed by $\sigma$. For many classes of elements, this
is straightforward since if $\sigma$ is a derangement of $m\leq n-3$ or $n-4$
elements, then put the fixed elements into the unique column of a tableau of
shape $(n-2,1,1)$, or into the first two columns of a tableau of shape
$(n-2,2)$. For derangements of larger cardinalities we construct a fixed
element as follows.

Let $t$ be a tableau of shape $\lambda$ and let $\sigma\in S_{n}$ be of order
$k$. Define

$E_{\sigma,t}=\sum_{j=0}^{k-1}e_{\sigma^{j}t}.$

It is clear that $\sigma$ fixes $E_{\sigma,t}$ since $\sigma e_{t}=e_{\sigma
t}$ and the action is linear, but it is possible that $E_{\sigma,t}$ may be
the zero polytabloid. For example, in the Specht module
$\mathcal{S}^{(n-1,1)}$, if $\sigma$ is an $n$-cycle and $t$ is any tableau,
then $E_{\sigma,t}=0$; consider $\sigma=(12\cdots n)$ and $t$ the tableau

$\begin{tabular}[]{cccc}$1$&$2$&$\cdots$&$n-1$\\\ $n$\end{tabular}.$

Then

$\sigma^{j}t=\begin{tabular}[]{cccc}$j+1$&$j+2$&$\cdots$&$j-1$\\\
$j$\end{tabular},$

so

$\displaystyle
e_{\sigma^{j}t}=\begin{tabular}[]{cccc}\hline\cr$j+1$&$j+2$&$\cdots$&$j-1$\\\
\hline\cr\text@underline{ \ \ $j$ \ \
}\end{tabular}-\begin{tabular}[]{cccc}\hline\cr$j$&$j+2$&$\cdots$&$j-1$\\\
\hline\cr\text@underline{ \ \ $j+1$ \ \
}\end{tabular}:=\mathbf{j}-(\mathbf{j+1}).$

Therefore

$E_{\sigma,t}=\sum_{j=0}^{n-1}e_{\sigma^{j}t}=\mathbf{n}-\mathbf{1}+\sum_{j=1}^{n-1}\mathbf{j}-(\mathbf{j+1})=0,$

and so is not a candidate for a nontrivial fixed vector. Much of our proof
below will amount to showing that $E_{\sigma,t}\neq 0$ for various classes of
$\sigma$.

Notation. For ease of exposition, if $t$ is the tableau

$t:=\begin{tabular}[]{cccc}$a$&$d$&$\cdots$&$e$\\\ $b$\\\ $c$\end{tabular},$

then we write the associated tabloid $\\{t\\}$ as
$\left\\{\begin{smallmatrix}a\\\ -\\\ b\\\ -\\\ c\end{smallmatrix}\right\\}$
and polytabloid $e_{t}$ as $\left[\begin{smallmatrix}a\\\ -\\\ b\\\ -\\\
c\end{smallmatrix}\right]$, so that

$\left[\begin{smallmatrix}a\\\ -\\\ b\\\ -\\\
c\end{smallmatrix}\right]=\left\\{\begin{smallmatrix}a\\\ -\\\ b\\\ -\\\
c\end{smallmatrix}\right\\}+\left\\{\begin{smallmatrix}b\\\ -\\\ c\\\ -\\\
a\end{smallmatrix}\right\\}+\left\\{\begin{smallmatrix}c\\\ -\\\ a\\\ -\\\
b\end{smallmatrix}\right\\}-\left\\{\begin{smallmatrix}a\\\ -\\\ c\\\ -\\\
b\end{smallmatrix}\right\\}-\left\\{\begin{smallmatrix}c\\\ -\\\ b\\\ -\\\
a\end{smallmatrix}\right\\}-\left\\{\begin{smallmatrix}b\\\ -\\\ a\\\ -\\\
c\end{smallmatrix}\right\\}.$

Similarly, if $s$ is the tableau

$s:=\begin{tabular}[]{cccc}$a$&$b$&$\cdots$&$e$\\\ $c$&$d$\end{tabular},$

then we write $\\{s\\}=\left\\{\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right\\}$ and $e_{s}=\left[\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right]=\left\\{\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right\\}-\left\\{\begin{smallmatrix}c&b\\\
a&d\end{smallmatrix}\right\\}-\left\\{\begin{smallmatrix}a&d\\\
c&b\end{smallmatrix}\right\\}+\left\\{\begin{smallmatrix}c&d\\\
a&b\end{smallmatrix}\right\\}$.

###### Proof of Theorem 1.2.2, $\lambda=(n-2,1,1)$.

Let $X=\\{1,\dots,n\\}$ and identify ${\rm Perm}_{X}$ with $S_{n}$. If
$Y\subseteq X$ has cardinality $m$, then we may naturally identify ${\rm
Perm}_{Y}$ with a subgroup of ${\rm Perm}_{X}$, isomorphic to $S_{m}$. Let
$Y=\\{y_{1},\dots,y_{m}\\}\subseteq X$ have cardinality $m\leq n-3$. Write
$X\setminus Y=\\{x_{1},\dots,x_{n-m}\\}$. Let $\sigma\in{\rm Perm}_{Y}$ and
consider the tableau

$t:=\begin{tabular}[]{ccccccc}$x_{1}$&$x_{4}$&$\cdots$&$x_{n-m}$&$y_{1}$&$\cdots$&$y_{m}$\\\
$x_{2}$\\\ $x_{3}$\end{tabular}.$

Then $\sigma e_{t}=e_{\sigma t}=e_{t}$, with the latter equality due to
$\sigma$ fixing the $x_{i}$. Hence, permutations fixing 3 or more letters have
fixed vectors in the Specht module $\mathcal{S}^{(n-2,1,1)}$.

We break the rest of the proof up according to whether $\sigma$ is a
derangement on $n-2$, $n-1$, or $n$ letters. For notational ease, if $\sigma$
is a derangement of the subset $Y=\\{y_{1},\dots,y_{m}\\}$ then we relabel its
elements and take $Y=\\{1,2,\dots,m\\}$. For each class of derangements
$\sigma$, we will define a tableau $t$ and then show that the sum
$E_{\sigma,t}$ of polytabloids is nonzero. To do this, we use Garnir calculus
to write the constituents of $E_{\sigma,t}$ as sums of standard polytabloids.
Since the standard polytabloids form a basis for the Specht module, we will,
in each case, conclude that $E_{\sigma,t}\neq 0$ by a dimension count. We give
explicit details in one case, and then will be content to simply list examples
of fixed vectors in a table, since the corresponding details are nearly
identical in each case.

Case 1: $|Y|=n-2$ Let $\sigma\in{\rm Perm}_{Y}$ and write $\sigma$ as a
product of disjoint cycles. If the smallest cycle is a transposition, write
$\sigma=(ab)\pi$, where $\pi$ is a derangement of the $n-4$ letters
$c,\dots,d$. We take $1$ and $2$ to be fixed by $\sigma$. Let

$t:=\begin{tabular}[]{ccccc}$1$&$c$&$\cdots$&$d$&$b$\\\ $2$\\\
$a$\end{tabular}.$

Let $m$ be the order of $\sigma$ (which is even). Then it is straightforward
to work out that

$E_{\sigma,t}=\sum_{j=0}^{m-1}e_{\sigma^{j}t}=\frac{m}{2}\left(\left[\begin{smallmatrix}1\\\
-\\\ 2\\\ -\\\ a\end{smallmatrix}\right]+\left[\begin{smallmatrix}1\\\ -\\\
2\\\ -\\\ b\end{smallmatrix}\right]\right),$

which is nonzero because it is an independent sum of standard (basis)
polytabloids.

If the smallest cycle in the decomposition of $\sigma$ is a 3-cycle, then
write $\sigma=(abc)\pi$, where $\pi$ is a derangement of the $n-5$ letters
$d,\dots,e$. Let $f$ and $g$ be fixed by $\sigma$. Let $t$ be the tableau

$t:=\begin{tabular}[]{cccccc}$a$&$d$&$\cdots$&$e$&$f$&$g$\\\ $b$\\\
$c$\end{tabular}.$

Then $\sigma e_{t}=e_{\sigma t}=e_{t}$.

Next, suppose that the smallest factor in the cycle decomposition of $\sigma$
is a $k$-cycle, with $4\leq k\leq\lfloor\frac{n+1}{2}\rfloor$. Without loss of
generality, we take the letters 1 and 2 to be fixed and $\sigma=(3\cdots
k+2)\pi$, where $\pi$ is a derangement of the remaining letters, and

$t:=\begin{tabular}[]{ccccccc}$1$&$4$&$\cdots$&$k+2$&$k+3$&$\cdots$&$n$\\\
$2$\\\ $3$\end{tabular}.$

We let $m$ denote the order of $\sigma$. It is routine to verify that

$\sigma^{j}t=\begin{tabular}[]{ccccccc}$1$&$j+4$&$\cdots$&$j+2$&$\pi^{j}(k+3)$&$\cdots$&$\pi^{j}(n)$\\\
$2$\\\ $j+3$\end{tabular}.$

The polytabloid $e_{\sigma^{j}t}$ has no column descents and is row-equivalent
to the standard polytabloid $\left[\begin{smallmatrix}1\\\ -\\\ 2\\\ -\\\
j+3\end{smallmatrix}\right]$. Therefore,

$E_{\sigma,t}=\left(\frac{m}{k}\right)\sum_{j=0}^{k-1}\left[\begin{smallmatrix}1\\\
-\\\ 2\\\ -\\\ j+3\end{smallmatrix}\right],$

as a sum of basis polytabloids is non-zero and, by construction, fixed by
$\sigma$.

Finally, if $\sigma$ is an $(n-2)$-cycle then by relabeling the elements it
suffices to assume $\sigma=(1\cdots n-2)$. Let $t$ be the tableau

$t:=\begin{tabular}[]{cccccc}$1$&$2$&$\cdots$&$n-2$\\\ $n-1$\\\
$n$\end{tabular}.$

Then for $j=0,\dots n-3$ we have

$\sigma^{j}t=\begin{tabular}[]{ccccccc}$j+1$&$j+2$&$\cdots$&$n-2$&1&$\cdots$&$j$\\\
$n-1$\\\ $n$\end{tabular}.$

By Garnir calculus, and by replacing polytabloids with their row- and column-
equivalents, it is straightforward to verify that for $j=1,\dots,n-3$ we have

$e_{\sigma^{j}t}=\left[\begin{smallmatrix}j+1\\\ -\\\ n-1\\\ -\\\
n\end{smallmatrix}\right]=\left[\begin{smallmatrix}1\\\ -\\\ n-1\\\ -\\\
n\end{smallmatrix}\right]-\left[\begin{smallmatrix}1\\\ -\\\ j+1\\\ -\\\
n\end{smallmatrix}\right]+\left[\begin{smallmatrix}1\\\ -\\\ j+1\\\ -\\\
n-1\end{smallmatrix}\right],$

expressing $e_{\sigma^{j}t}$ as a sum of standard polytabloids. Summing over
$j$, we obtain

$\displaystyle E_{\sigma,t}$ $\displaystyle=\sum_{j=0}^{n-3}e_{\sigma^{j}t}$
$\displaystyle=\left[\begin{smallmatrix}1\\\ -\\\ n-1\\\ -\\\
n\end{smallmatrix}\right]+\sum_{j=1}^{n-3}\left[\begin{smallmatrix}1\\\ -\\\
n-1\\\ -\\\ n\end{smallmatrix}\right]-\left[\begin{smallmatrix}1\\\ -\\\
j+1\\\ -\\\ n\end{smallmatrix}\right]+\left[\begin{smallmatrix}1\\\ -\\\
j+1\\\ -\\\ n-1\end{smallmatrix}\right]$
$\displaystyle=(n-2)\left[\begin{smallmatrix}1\\\ -\\\ n-1\\\ -\\\
n\end{smallmatrix}\right]-\sum_{j=1}^{n-3}\left[\begin{smallmatrix}1\\\ -\\\
j+1\\\ -\\\
n\end{smallmatrix}\right]+\sum_{j=1}^{n-3}\left[\begin{smallmatrix}1\\\ -\\\
j+1\\\ -\\\ n-1\end{smallmatrix}\right].$

This is a sum of $3n-8$ basis vectors in a space of dimension $(n-1)(n-2)/2$.
Once $n\geq 6$ this is an independent sum by comparing dimensions. When $n=5$,
the representation is easily verified to be unisingular anyway.

Cases 2 & 3: $|Y|\in\\{n-1,n\\}$ In the following table we give examples of
fixed vectors for each class of permutation $\sigma$. The cases themselves
break up into four subcases, depending on the factorization of $\sigma$ into
disjoint permutations. The four subcases depend on the size of the smallest
cycle type occurring in the factorization. Therefore, we consider when
$\sigma$ is a single cycle; when $\sigma$ factors as a transposition and a
derangement of the remaining letters; a 3-cycle and a derangement; and a
$k$-cycle and a derangement, where $k\geq 4$. For ease of exposition, we
denote by

$(a\cdots b)[c\dots d],$

the product of a cycle $(a\cdots b)$ and a derangement of the remaining
letters $[c\cdots d]$, which may or may not be a cycle. We denote by $m$ the
order of $\sigma\in S_{n}$. In the table below, we note that each
$E_{\sigma,t}$ is written as a sum of standard (basis) polytabloids of
dimension $\leq\dim\mathcal{S}^{(n-2,1,1)}$, and is therefore nonzero. We
label any fixed elements in  red. This completes the proof, since for each
class of permutation it suffices (by relabeling elements) to consider a single
element and show that it has a non-zero fixed vector in the vector space
$\mathcal{S}^{(n-2,1,1)}$.

$|Y|$ | $\sigma$ |  Tableau $t$ | Fixed Vector
---|---|---|---
$n-1$ | $(12\cdots(n-1))$ |  | $1$ | $2$ | $\cdots$ | $n-2$
---|---|---|---
$n-1$ |  |  |
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}n}$ |  |  |
$E_{\sigma,t}=-\sum_{j=2}^{n-2}\left[\begin{smallmatrix}1\\\ -\\\ j\\\ -\\\
j+1\end{smallmatrix}\right]$
$n-1$ |  $((n-1)n)[2\cdots(n-3)]$, |  | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1}$ | $2$ | $\cdots$ | $n-3$ | $n-1$
---|---|---|---|---
$n-2$ |  |  |  |
$n$ |  |  |  |
$E_{\sigma,t}=\sum_{j=0\atop j\text{ even}}^{m-1}\left[\begin{smallmatrix}1\\\
-\\\ \sigma^{j}(n-3)\\\ -\\\ n\end{smallmatrix}\right]$
|  |  | $+\sum_{j=0\atop j\text{ odd}}^{m-1}\left[\begin{smallmatrix}1\\\ -\\\ \sigma^{j}(n-3)\\\ -\\\ n-1\end{smallmatrix}\right]$
$n-1$ | $(abc)[d\cdots e]$ |  | $a$ | $d$ | $\cdots$ | $e$ | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}f}$
---|---|---|---|---
$b$ |  |  |  |
$c$ |  |  |  |
$e_{t}$
$n-1$ | $(23\cdots k+1)[(k+2)\cdots n]$ |  | ${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1}$ | $2$ | $\cdots$ | $k-1$ | $\cdots$ | $n$
---|---|---|---|---|---
$k$ |  |  |  |  |
$k+1$ |  |  |  |  |
$E_{\sigma,t}=\left(\frac{m}{k}\right)\left(\left[\begin{smallmatrix}1\\\ -\\\
k\\\ -\\\ k+1\end{smallmatrix}\right]-\left[\begin{smallmatrix}1\\\ -\\\ 2\\\
-\\\ k\end{smallmatrix}\right]+\sum_{j=2}^{k-1}\left[\begin{smallmatrix}1\\\
-\\\ j\\\ -\\\ j+1\end{smallmatrix}\right]\right)$
$n$ | $(12\cdots n)$ |  | $1$ | $2$ | $\cdots$ | $n-2$
---|---|---|---
$n-1$ |  |  |
$n$ |  |  |
$E_{\sigma,t}=2\sum_{j=2}^{n-1}\left[\begin{smallmatrix}1\\\ -\\\ j\\\ -\\\
j+1\end{smallmatrix}\right]-\sum_{j=2}^{n-2}\left[\begin{smallmatrix}1\\\ -\\\
j\\\ -\\\ j+2\end{smallmatrix}\right]$
$n$ | $(1n)[23\cdots(n-1)]$ |  | $1$ | $2$ | $\cdots$ | $n-2$
---|---|---|---
$n-1$ |  |  |
$n$ |  |  |
$E_{\sigma,t}=\sum_{j=0}^{m-1}(-1)^{j}\left[\begin{smallmatrix}1\\\ -\\\
\sigma^{j}(n-1)\\\ -\\\ n\end{smallmatrix}\right]$
$n$ | $(abc)[d\cdots e]$ |  | $a$ | $d$ | $\cdots$ | $e$
---|---|---|---
$b$ |  |  |
$c$ |  |  |
$e_{t}$
$n$ | $(1\cdots k)[(k+1)\cdots n]$ |  | $1$ | $\cdots$ | $k-2$ | $k+1$ | $\cdots$ | $n$
---|---|---|---|---|---
$k-1$ |  |  |  |  |
$k$ |  |  |  |  |
$E_{\sigma,t}=\left(\frac{m}{k}\right)\left(\left[\begin{smallmatrix}1\\\ -\\\
k-1\\\ -\\\ k\end{smallmatrix}\right]+\left[\begin{smallmatrix}1\\\ -\\\ 2\\\
-\\\ 3\end{smallmatrix}\right]+\left[\begin{smallmatrix}1\\\ -\\\ 2\\\ -\\\
k\end{smallmatrix}\right]\right)$
|  |  | $+\frac{m}{k}\sum_{j=3}^{k-1}\left[\begin{smallmatrix}1\\\ -\\\ j\\\ -\\\ j+1\end{smallmatrix}\right]-\left[\begin{smallmatrix}1\\\ -\\\ j-1\\\ -\\\ j+1\end{smallmatrix}\right]+\left[\begin{smallmatrix}1\\\ -\\\ j-1\\\ -\\\ j\end{smallmatrix}\right]$

∎

Next we turn to the case $\lambda=(n-2,2)$. Rather than work out explicit
forms for the fixed polytabloids like in the proof of Part 1, we will only
sketch the proof since the details are nearly identical.

The shape of the tableaux themselves indicate how to find the fixed
polyabloids:

$\begin{tabular}[]{cccccccc}$\square\square$&$\underbrace{\square\square\square\cdots\square}_{n-4}$\\\
$\square\square$\end{tabular}.$

If $\sigma\in S_{n}$ has at least four fixed points, then place four of them
in the the $2\times 2$ block; the associated polytabloid is then fixed by
$\sigma$. If $\sigma$ has 1, 2, or 3 fixed points, then relabel the elements
so that the sets $\\{1\\}$, $\\{1,2\\}$, or $\\{1,2,3\\}$ are fixed,
respectively. In those cases, fill the $2\times 2$ block of a tableau as
follows, respectively:

$\begin{tabular}[]{cccccccc}${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1}$&$*$\\\
$*$&$*$\end{tabular},\qquad\begin{tabular}[]{cccccccc}${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1}$&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}
2}\\\
$*$&$*$\end{tabular},\qquad\begin{tabular}[]{cccccccc}${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}1}$&{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}
2}\\\
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}3}$&$*$\end{tabular}.$

Depending on the factorization of $\sigma$ into disjoint cycles, the remainder
of the entries in the tableau $t$ can be filled so that the element
$E_{\sigma_{,}t}$ (which is fixed by $\sigma$) can be expressed as a non-zero
sum of basis polytabloids.

If $\sigma$ fixes no element of $\\{1,\cdots,n\\}$, then we can again proceed
as in the proof of Part 1 and work on a case-by-case basis, depending on the
factorization of $\sigma$. In each of the following cases:

  * •

$\sigma$ is an $n$-cycle,

  * •

the smallest factor of $\sigma$ is a 2-cycle,

  * •

the smallest factor of $\sigma$ is a 3-cycle,

  * •

the smallest factor of $\sigma$ is a 4-cycle,

  * •

the smallest factor of $\sigma$ is a $k$-cycle, $5\leq
k\leq\lfloor\frac{n+1}{2}\rfloor$,

it follows by nearly identical arguments as above that we can produce a
tableau $t$ such that the element $E_{\sigma,t}$, which is fixed by $\sigma$,
is nonzero. We give the details of a sample case to conclude this sketch.

Let $n$ be even. Suppose $\sigma$ factors as a transposition and an
$(n-2)$-cycle. Then the order of $\sigma$ is $n-2$. Relabel the elements so
that $\sigma=(12)(3\cdots n)$ and let

$t=\begin{tabular}[]{cccccccc}1&2&3&$\cdots$&$n-2$\\\ $n-1$&$n$\end{tabular}.$

Then

$\sigma^{j}t=\begin{cases}\begin{tabular}[]{cccccccc}1&2&$j+3$&$\cdots$&$n$&3&$\cdots$&$j$\\\
$j+1$&$j+2$\end{tabular}&\text{if $2\leq j\leq n-4$ is even}\\\\[10.0pt]
\begin{tabular}[]{cccccccc}2&1&$j+3$&$\cdots$&$n$&3&$\cdots$&$j$\\\
$j+1$&$j+2$\end{tabular}&\text{if $3\leq j\leq n-3$ is odd.}\end{cases}$

If $j$ is even then $e_{\sigma^{j}t}$ is row-equivalent to the standard
polytabloid $\left[\begin{smallmatrix}1&2\\\ j+1&j+2\end{smallmatrix}\right]$,
while if $j$ is odd then $e_{\sigma^{j}t}$ is row-equivalent to the
polytabloid $\left[\begin{smallmatrix}2&1\\\ j+1&j+2\end{smallmatrix}\right]$.
Using Garnir calculus, we can rewrite this last polytabloid as a linear
combination of basis polytabloids:

$\left[\begin{smallmatrix}2&1\\\
j+1&j+2\end{smallmatrix}\right]=\left[\begin{smallmatrix}1&2\\\
j+1&j+2\end{smallmatrix}\right]-\left[\begin{smallmatrix}1&3\\\
2&j+2\end{smallmatrix}\right]+\left[\begin{smallmatrix}1&3\\\
2&j+1\end{smallmatrix}\right].$

Then $E_{\sigma,t}$ has the following expression as a sum of basis elements:

$E_{\sigma,t}=\left[\begin{smallmatrix}1&2\\\
3&n\end{smallmatrix}\right]+\sum_{j=3}^{n-1}\left[\begin{smallmatrix}1&2\\\
j&j+1\end{smallmatrix}\right]+\sum_{j=2}^{(n-2)/2}\left(\left[\begin{smallmatrix}1&2\\\
2&2j\end{smallmatrix}\right]-\left[\begin{smallmatrix}1&2\\\
2&2j+1\end{smallmatrix}\right]\right),$

which is nonzero by a dimension count.

Now that we have identified new classes of unisingular representations, we can
ask about the associated unirealization problem. If $\mathcal{S}^{\lambda}$ is
a Specht module for $S_{n}$ or $A_{n}$, then we can consider it as a
representation over $\mathbf{F}_{2}$ simply by reducing the matrix entries
modulo 2. And if $\mathcal{S}^{\lambda}$ is additionally unisingular, then the
unisingularity is preserved by reduction.

By [8, Main Theorem] the Specht module $\mathcal{S}^{(n-2,1,1)}$ is _never_
irreducible mod 2 since both $(n-2,1,1)$ and the conjugate partition
$(3,1,\dots,1)$ are 2-singular. In fact, the representation
$\mathcal{S}^{(n-2,1,1)}$ contains a copy of the trivial representation in its
decomposition modulo 2. On the other hand, the representation
$\mathcal{S}^{(n-2,2)}$ is irreducible for certain values of $n$; for $5\leq
n\leq 13$ it is irreducible precisely when $n=7,11$.

Since $S_{7}$ is not a transitive subgroup of $S_{15}$ or $S_{16}$, this
representation of $S_{7}$, which is 14-dimensional, cannot be unirealized over
$\mathbf{Q}$ by a hyperelliptic Jacobian; similarly for $S_{11}$. But all of
this raises two problems for further investigation:

  1. (1)

Determine which Specht modules are unisingular, either for $S_{n}$ or $A_{n}$.

  2. (2)

Given a unisingular $\mathcal{S}^{\lambda}$ for $S_{n}$ (or $A_{n}$) of
dimension $f^{\lambda}=2d$ which remains irreducible mod 2, determine whether
or not there are transitive groups $T\subseteq S_{2d+1}$ or $S_{2d+2}$ with
quotient $S_{n}$ (or $A_{n}$). This could give a bound on the minimal degree
needed for a hyperelliptic unirealization of the representation
$\mathcal{S}^{\lambda}$.

##  5\. A Conjecture on Non-Unisingularity

If $n$ is odd then we conjecture that a certain Specht module is never
unisingular for $S_{n}$, but is unisingular for $A_{n}$. We first state the
conjecture and then provide evidence.

######  Conjecture 5.0.1.

Let $n\geq 5$ be odd. Then the Specht module $\mathcal{S}^{(n-2,2)^{\prime}}$
does not afford 1 on the class of elements of that factor as a transposition
and an $(n-2)$-cycle. Otherwise, every element has 1 as an eigenvalue.

When $n$ is odd the class of permutations of Conjecture 5.0.1 is odd, hence
does not meet $A_{n}$. A consequence of our conjecture is that the restriction
$\mathcal{S}^{(n-2,2)^{\prime}}\downarrow_{A_{n}}$ is unisingular. However,
because $(n-2,2)^{\prime}\neq(n-2,2)$,

$\mathcal{S}^{(n-2,2)^{\prime}}\downarrow_{A_{n}}=\mathcal{S}^{(n-2,2)}\downarrow_{A_{n}}$

which we already knew to be unisingular by Theorem 1.2.2.

Continuing, let $C_{(n-2,2)}$ be the class of elements of $S_{n}$ that
decompose as a product of an $(n-2)$-cycle and a transposition. To show that
this class does not afford 1 as an eigenvalue in the representation
$\mathcal{S}^{(n-2,2)^{\prime}}$ we will

  1. (1)

compute the multiplicity of $(x-1)$ in the characteristic polynomial of the
permutation module $\mathcal{M}^{(n-2,2)^{\prime}}$ on $C_{(n-2,2)}$, and

  2. (2)

compare this to the multiplicities of $(x-1)$ in the simple factors of
$\mathcal{M}^{(n-2,2)^{\prime}}$ on $C_{(n-2,2)}$; since
$\mathcal{S}^{(n-2,2)^{\prime}}$ appears once as a factor of
$\mathcal{M}^{(n-2,2)^{\prime}}$, we can just compare powers of $(x-1)$ and
conclude that $\mathcal{S}^{(n-2,2)^{\prime}}$ is not unisingular.

An advantage to getting the eigenvalue-1 multiplicity in this way is that the
permutation modules are easy to generate in Magma.

Recall that the permutation module $\mathcal{M}^{(n)}$ and the Specht module
$\mathcal{S}^{(n)}$ are isomorphic to the trivial module. Then, for example,

$\displaystyle\mathcal{M}^{(n)}$ $\displaystyle=\mathcal{S}^{(n)}$
$\displaystyle\mathcal{M}^{(n-1,1)}$
$\displaystyle=\mathcal{S}^{(n-1,1)}\oplus\mathcal{S}^{(n)}$
$\displaystyle\mathcal{M}^{(n-2,2)}$
$\displaystyle=\mathcal{S}^{(n-2,2)}\oplus\mathcal{S}^{(n-1,1)}\oplus\mathcal{S}^{(n)}$
$\displaystyle\mathcal{M}^{(n-2,1,1)}$
$\displaystyle=\mathcal{S}^{(n-2,1,1)}\oplus\mathcal{S}^{(n-2,2)}\oplus
2\mathcal{S}^{(n-1,1)}\oplus\mathcal{S}^{(n)}.$

Now we work through the simplest case where $n=5$ to give a sense of what is
involved, though this process generalizes naturally to larger $n$. Our goal is
in this case is to show that the characteristic polynomial of the class
$C_{(n-2,2)}$ in the representation $\mathcal{S}^{(2,2,1)}$ is not divisible
by $(x-1)$.

The trivial module $\mathcal{M}^{(5)}=\mathcal{S}^{(5)}$ has one power of
$(x-1)$ on the class $C_{(n-2,2)}$. Next, there is a unique transitive action
of $S_{5}$ on 5 points, decomposing into
$\mathcal{S}^{(4,1)}\oplus\mathcal{S}^{(5)}$ as above. By computing with the
character of this representation, we conclude that $\mathcal{S}^{(4,1)}$ has
one power of $(x-1)$ on the class $C_{(n-2,2)}$; this is basic representation
theory, and also easily implementable in Magma.

To construct the module $\mathcal{M}^{(3,2)}$, we search the TransitiveGroups
database for transitive actions of $S_{5}$ on 10 points. There are exactly two
such actions. Setting T:=TransitiveGroups(10), the two transitive groups
isomorphic to $S_{5}$ are T[12] and T[13].

The group T[13] is the one we want, which we verify by decomposing the
representation in Magma into irreducibles and comparing dimensions of the
simple factors agains those of $\mathcal{S}^{(3,2)}$, $\mathcal{S}^{(4,1)}$,
and $\mathcal{S}^{(5)}$. The following code produces matrix generators of the
permutation representation, say $\mathtt{a}$ and $\mathtt{b}$ (this is the
result of the command M:=Maximal;), and then decomposes the permutation
representation into irreducibles.

T:=TransitiveGroups(10);
---
M := PermutationModule(T[13], Rationals());
M:Maximal;
A:=MatrixAlgebra<Rationals(), 10 | a,b>;
MM:=RModule(A);
B:=CompositionFactors(MM);
B;

The characteristic polynomial of the class $C_{(n-2,2)}$ on
$\mathcal{M}^{(3,2)}$ is exactly divisible by $(x-1)^{3}$, thus we conclude
that $\mathcal{S}^{(3,2)}$ has exactly one power of $(x-1)$ on the same class.

Iterating this procedure, we find that there are exactly three transitive
permutation representations of $S_{5}$ on 20 points; if
TT:=TransitiveGroups(20), then they are TT[30], TT[32], and TT[35]. By
decomposing each module, comparing to the known decomposition of
$\mathcal{M}^{(3,1,1)}$ above, and using the fact that both 4-dimensional
components of $\mathcal{M}^{(3,1,1)}$ are isomorphic as $S_{5}$-modules, we
conclude that TT[30] is the permutation group that we seek for our
decomposition. Computing characteristic polynomials of each side, we conclude
that the characteristic polynomial of $\mathcal{S}^{(3,1,1)}$ on $C_{(n-2,2)}$
is divisible by exactly one power of $(x-1)$.

The permutation module $\mathcal{M}^{(2,2,1)}$ has dimension 30 and decomposes
as

(5.0.2)
$\displaystyle\mathcal{M}^{(2,2,1)}=\mathcal{S}^{(2,2,1)}\oplus\mathcal{S}^{(3,1,1)}\oplus
2\mathcal{S}^{(3,2)}\oplus 2\mathcal{S}^{(4,1)}\oplus\mathcal{S}^{(5)}.$

There are three transitive permutation representations of $S_{5}$ on 30
points; if

TTT:=TransitiveGroups(30), then they are TTT[22], TTT[25], and TTT[27]. A
similar analysis to the ones above shows that TTT[27] is the representation we
seek. We find that the characteristic polynomial of $\mathcal{M}^{(2,2,1)}$ on
$C_{(n-2,2)}$ has eigenvalue 1 of multiplicity 6. The decomposition (5.0.2)
has 7 summands, 6 of which have been shown to have eigenvalue 1 of
multiplicity 1. We conclude that $\mathcal{S}^{(2,2,1)}$ is not unisingular.

As $n$ gets larger, the decompositions of $\mathcal{M}^{(2,2,1,\dots,1)}$ as
in (5.0.2) get more complicated, but this same strategy is, in principle,
implementable. For $n=15,\dots,23$, the online Atlas at
http://brauer.maths.qmul.ac.uk/Atlas/v3/, gives explicit matrix generators for
the representations $\mathcal{S}^{(n-2,2)^{\prime}}$ for $n=15,\dots,23$.

######  Remark 5.0.3.

In http://brauer.maths.qmul.ac.uk/Atlas/v3/ it is not stated explicitly
whether the precomputed representation of $S_{n}$ of degree $n(n-3)/2$ is
$\mathcal{S}^{(n-2,2)}$ or $\mathcal{S}^{(n-2,2)^{\prime}}$. By comparing
traces of elements against the known character tables of these groups, we
verify that they are indeed $\mathcal{S}^{(n-2,2)^{\prime}}$.

For $n=5,7,9$ this procedure works since the TransitiveGroups database has all
permutation groups of degree $\leq 48$. For $n=11,13$ we construct the modules
individually since they are out of range of the TransitiveGroups database. For
$n=15,\dots,23$ we appeal to the representations precomputed in
http://brauer.maths.qmul.ac.uk/Atlas/v3/. In all instances, the
representations afford 1 as an eigenvalue on all classes except $C_{(n-2,2)}$,
and on that class, evaluating the characteristic polynomial at 1 gives

$n$ | 5 | 7 | 9 | 11 | 13 | 15 | 17 | 19 | 21 | 23
---|---|---|---|---|---|---|---|---|---|---
$\chi_{(n-2,2)^{\prime}}(1)$ | 6 | 20 | 56 | 144 | 352 | 832 | 1920 | 4352 | 9728 | 21504

This pattern suggests that the character value on this class, if $n=2k+1$, is
$2^{k-1}(2k-1)$.

##  6\. Concluding Remarks

The paper [6] was concerned with specific families (certain groups of Lie
type) and specific representations (Steinberg) that are unisingular. We
conclude by observing that many finite simple groups that are not covered by
the results of [6] have unisingular representations over $\mathbf{F}_{2}$ as
well. We list just a few of them here, all of which have Schur indicator ‘+’
or ‘$-$’, and are therefore symplectic. However, none of these representations
are candidates for a hyperelliptic unirealization over $\mathbf{Q}$ because
the degree of the polynomial required does not divide the order of the group.
So if a hyperelliptic realization of any of these groups is possible, it would
have to be over a nontrivial field extension of $\mathbf{Q}$ via a quotient of
a larger group.

Group | ${\rm L}_{2}(13)$ | ${\rm L}_{3}(3)$ | ${\rm U}_{3}(3)$ | $M_{11}$ | $M_{12}$ | $M_{22}$ | $\operatorname{Aut}(M_{22})$ | $M_{23}$ | $J_{1}$ | $J_{2}$ | ${\rm Sz}(8)$
---|---|---|---|---|---|---|---|---|---|---|---
Dimension | 14 | 26 | 14 | 44 | 44, 144 | 34 | 34 | 120 | 76 | 36, 84 | 64

######  Remark 6.0.1.

  1. (1)

The groups $\operatorname{Aut}(M_{22})$ and ${\rm L}_{3}(4)\cdot 2_{2}$ have
been shown to be Galois over $\mathbf{Q}$ in [13], and the Steinberg
representation of ${\rm L}_{3}(4)$ of dimension 64 is unisingular by [6]).
However, the degrees of the polynomials in [13] are not the ones required for
hyperelliptic unirealization of these groups.

More recently, in [19], Zywina showed that ${\rm L}_{3}(4)$ is Galois over
$\mathbf{Q}$ using a degree-21 specialization $f(x)$ of Malle’s parametric
polynomials realizing ${\rm L}_{3}(4)\cdot 2_{2}$. Since $\deg f=21$, we get a
20 dimensional permutation representation

${\rm L}_{3}(4)\hookrightarrow
S_{21}\hookrightarrow\operatorname{GL}_{20}(\mathbf{Z}).$

This is precisely the 20-dimensional representation defined over $\mathbf{Z}$
in the online Atlas. One easily checks that this representation is unisingular
and irreducible over $\mathbf{Q}$. However, this representation decomposes
over $\mathbf{F}_{2}$ as two 9-dimensional representations, and two
1-dimensionals.

  2. (2)

The representation of ${\rm Sz}(8)$ listed in the table above is the Steinberg
but, as recently as the preprint [19], it is is not known if ${\rm Sz}(8)$ is
Galois over $\mathbf{Q}$, let alone unirealizable.

## References

  * [1] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, An ATLAS of Finite Groups, Clarendon Press, Oxford, $1985$.
  * [2] J. Cullinan, Local-global properties of torsion points on three-dimensional abelian varieties. J. Algebra. 311 (2007), 736-774.
  * [3] J. Cullinan, A computational approach to the 2-torsion structure of abelian threefolds. Math. of Comp. 78 (2009), 1825-1836.
  * [4] J. Cullinan, Fixed-point subgroups of $\mathrm{GL}_{3}(q)$. _J. Group Theory_ 22 no. 5 (2019) 893-914
  * [5] J. Cullinan, J. Yelton. Divisibility of torsion subgroups of abelian surfaces over number fields. To appear in _Canadian Journal of Mathematics_. https://arxiv.org/pdf/1912.02356.pdf
  * [6] J. Cullinan, A.E. Zalesski. Unisingular representations in arithmetic and Lie theory. To appear in _European Journal of Mathematics_. https://arxiv.org/pdf/2011.04390.pdf
  * [7] R. Guralnick, P.H. Tiep. Finite simple unisingular groups of Lie type. _J. Group Theory_ 6 (2003), no. 3, 271-310.
  * [8] G. James, A. Mathas. The irreducible Specht modules in characteristic 2. _Bull. London Math. Soc._ 31 (1999), no. 4, 457-462.
  * [9] N.M. Katz, Galois properties of torsion points on abelian varieties, Inv. Math. 62 (1981), 481-502
  * [10] The LMFDB Collaboration, The L-functions and Modular Forms Database, http://www.lmfdb.org
  * [11] G. Malle. Multi-parameter polynomials with given Galois group. Algorithmic methods in Galois theory. _J. Symbolic Comput._ 30 (2000), no. 6, 717-731.
  * [12] G. Malle. Explicit realization of the Dickson groups $G_{2}(q)$ as Galois groups. _Pacific J. Math._ 212 (2003), no. 1, 157-167.
  * [13] G. Malle. Polynomials with Galois Groups $\operatorname{Aut}(M_{22})$, $M_{22}$, and $\operatorname{PSL}_{3}(\mathbf{F}_{4})\cdot 2_{2}$ over $\mathbf{Q}$. _Math. of Comp._ 51 (1988), no. 184, 761-768.
  * [14] B. Sagan. The symmetric group. Representations, combinatorial algorithms, and symmetric functions. Second edition. Graduate Texts in Mathematics, 203. Springer-Verlag, New York, 2001.
  * [15] J.-P. Serre. Letter to J. Cullinan.
  * [16] J. Yelton. Lifting images of standard representations of symplectic groups. To appear in _Manuscripta Mathematica_. https://arxiv.org/pdf/1903.06148.pdf
  * [17] A.E. Zalesski, On eigenvalues of group elements in representations of simple algebraic groups and finite Chevalley groups, Acta Appl. Math. 108 (2009), 175-195
  * [18] Y. Zarhin. Non-isogenous elliptic curves and hyperelliptic jacobians. Preprint https://arxiv.org/pdf/2105.03783.pdf
  * [19] D. Zywina. Inverse Galois problem for small simple groups.
http://pi.math.cornell.edu/$\sim$zywina/papers/smallGalois.pdf


# ViralBERT: A User Focused BERT-Based Approach to Virality Prediction

Rikaz Rameez  University College LondonGower StreetLondonUKWC1E 6BT
<EMAIL_ADDRESS>,  Hossein A. Rahmani  University College
LondonGower StreetLondonUKWC1E 6BT<EMAIL_ADDRESS>and  Emine Yilmaz
University College LondonGower StreetLondonUKWC1E 6BT<EMAIL_ADDRESS>
(2022)

###### Abstract.

Recently, Twitter has become the social network of choice for sharing and
spreading information to a multitude of users through posts called ‘tweets’.
Users can easily re-share these posts to other users through ‘retweets’, which
allow information to cascade to many more users, increasing its outreach.
Clearly, being able to know the extent to which a post can be retweeted has
great value in advertising, influencing and other such campaigns. In this
paper we propose ViralBERT, which can be used to predict the virality of
tweets using content- and user-based features. We employ a method of
concatenating numerical features such as hashtags and follower numbers to
tweet text, and utilise two BERT modules: one for semantic representation of
the combined text and numerical features, and another module purely for
sentiment analysis of text, as both the information within text and it’s
ability to elicit an emotional response play a part in retweet proneness. We
collect a dataset of 330k tweets to train ViralBERT and validate the efficacy
of our model using baselines from current studies in this field. Our
experiments show that our approach outperforms these baselines, with a $13\%$
increase in both F1 Score and Accuracy compared to the best performing
baseline method. We then undergo an ablation study to investigate the
importance of chosen features, finding that text sentiment and follower
counts, and to a lesser extent mentions and following counts, are the
strongest features for the model, and that hashtag counts are detrimental to
the model.

††journalyear: 2022††copyright: rightsretained††conference: Adjunct
Proceedings of the 30th ACM Conference on User Modeling, Adaptation and
Personalization; July 4–7, 2022; Barcelona, Spain††booktitle: Adjunct
Proceedings of the 30th ACM Conference on User Modeling, Adaptation and
Personalization (UMAP ’22 Adjunct), July 4–7, 2022, Barcelona, Spain††doi:
10.1145/3511047.3536415††isbn: 978-1-4503-9232-7/22/07

##  1\. Introduction

The zeitgeist of recent times has been characterised by the rise of the
internet as a central component in our lives, and a huge factor in this is the
communication between users on social media allowing borderless conversations
between groups of people. Twitter is one of the biggest social media platforms
in the world with over 300 million monthly active users. These users can share
‘tweets’ - a piece of text up to 280 characters in length with the option to
be accompanied with photos, GIFs or videos. Around 500 million tweets are
posted every day on the platform, so there is a huge amount of content and
interaction between users. Users can interact with tweets by sharing the post
to their follower network through retweets, liking them, or replying to the
post with their own tweet. These interactions show that a tweet contains
information that users have interest in and by retweeting, this information
gets spread to other users. A small number of tweets get more interactions
than others, so gain attention from a relatively large number of users on
Twitter - this indicates that the tweet has gone viral. Virality can be used
to determine popularity and engagement of trends and topics not only on
Twitter but throughout society as a whole. Indeed, there has been an
increasing number of studies on the phenomenon of viral posts affecting
social, economic and political outcomes so there is importance in having the
capability to forecast the popularity of a tweet (Borges-Tiago et al., 2019).
Kouzy et al. (2020) found that false or unverifiable content about the
COVID-19 pandemic could just as easily gain a high amount of user interactions
as verified information, showing the extent to which misinformation can
rampantly spread on social media (as it did in the pandemic (Thompson and
Volkmer, 2021)). In addition, being able to model Twitter popularity would
allow tweets to be engineered towards user engagement, which has uses in viral
marketing or public information campaigns for example. There are analogous
popularity metrics on other social media (Likes/Shares on Facebook, Instagram,
YouTube etc.), so this study could also ignite research into these platforms.

A large amount of current studies on tweet engagement focus on modelling the
network structure of Twitter and how tweets propagate between users and
demographics (Weng et al., 2013; Lymperopoulos, 2021). This allows some
understanding of Twitter’s userbase and post structure and why users share
posts between each other. However, there is limited recent research on
predicting virality or popularity, and these studies focus on a specific
subset of tweets (Jiménez-Zafra et al., 2021) or certain Twitter users (Pancer
and Poole, 2016) instead of tweets from the entire userbase. Focusing on a
specific subset of tweets or users allows adequate performance on a minority
of tweets, but no generalisation to Twitter users and posts as a whole.
Predicting virality poses a challenge as it can be affected by a multitude of
factors, many of which are not easily quantifiable such as relatability to
users, creativity of content and relevancy to current social climates -
modelling this widespread user appeal is a difficult problem. Furthermore,
most tweets are never retweeted, so it is also difficult to build a large,
balanced and representative dataset on all Twitter users for ML use due to
zero inflation. In this paper, we build our own dataset of a sample of 330k
tweets to train our model. We then explore whether we can predict the degree
of virality of a tweet using novel deep leaning methods on this problem,
namely using the BERT architecture (Devlin et al., 2018). We create a well
performing virality prediction model using textual, content-based, and user-
based features from a tweet, called ViralBERT. Our model incorporates a
specific BERT module for sentiment of tweet text, as emotional arousal has
been highly linked with virality in past works (Berger and Milkman, 2012), and
this correlation is further investigated in this paper. ViralBERT is compared
with models in the most recent work on predicting Twitter virality, and the
importance of features is investigated as well.

##  2\. Background

There are several studies on analyzing the trends of and modelling user
interactions on Twitter. Lymperopoulos (2021) investigates the temporal growth
of retweet numbers and identifies a pattern of virality evolution for all
tweets - they start with rapid growth and then asymptotically reach a virality
limit after a certain amount of time. Kwak et al. (2010) explores several
different ways of modelling Twitter usage. The Pagerank score of a user is
compared to the amount of followers that a user has and the amount of retweets
a user gets. It is found that Pagerank is similar to the follower amounts but
having a large amount of followers doesn’t necessarily mean that a tweet from
this user will have a high amount of retweets. Furthermore, the paper
emphasises that retweets are important for user exposure - even if a user
doesn’t have a lot of followers, being retweeted will give them exposure to a
large amount of users.

There have also been some studies into modelling tweets with the aim of
predicting if a post will be popular. Jiménez-Zafra et al. (2021) focuses on
virality depending on sentiments (and other features) during the Catalan
referendum. They find that likes, hashtags, URLs, mentions, followers, and
following numbers have significant effects on retweet numbers, and reply
numbers and verification status have negative effects on retweet numbers.
Notably, they also find that positive terms contribute negatively to retweets
count and negative terms contribute positively - similar conclusions have been
made by Pancer and Poole (2016). Nesi et al. (2018) assess the degree of
retweeting, and treat this problem as a multi-class classification problem
using retweet class boundaries (0, 1-10, 11-100, 101-1000, 1000+). The
features used are the hashtags, mentions and URLs in a tweet, the number of
favourites, and tweet author’s followers and following numbers. The models
used on these features are Random Forests and their own model, RPART, which is
form of decision tree. They find that mentions count is the most important
metric in predicting virality degree. An observation from this study is that
favourites count is used, but as it is an indicator of user engagement it
should not have been used.

These studies provide insight into methodologies that could be used in this
study. The key observations for this study are the features that could be used
in addition to tweet text, such as content and author based features. In
addition, retweets are used as the principal indicator of virality as retweets
increase user visibility of a tweet and are driven mainly by tweet content, so
this should inform our labels. Virality prediction could also be turned into a
classification problem, which can allow for multiple tasks to be used to
inform a prediction, and allow for different classification metrics to be
used. Although ML methods have been used in previous works, there is a clear
lack of studies in using deep learning algorithms and NLP techniques to
predict tweet virality - this will be the focus and novelty of this work.

##  3\. Proposed Methodology

Architecture using tweet text and numerical features for BERTweet, then
concatenating the output with RoBERTa sentiment analysis for classification
layers

Figure 1. ViralBERT architecture

In this section, we describe our proposed approach for virality prediction,
called ViralBERT, which captures both numerical and textual information from
tweets. The architecture is shown in Figure 1. Our model generates a high
dimensional representation using a combination of tweet text and numerical
features, as well as a probability distribution of the sentiment of tweet
text, then feeds this into a classification model to produce a virality class
prediction. Sentiment is used here as several studies (Jiménez-Zafra et al.,
2021; Pancer and Poole, 2016) find it directly affects the propagation of a
tweet. Therefore, we use BERT (Devlin et al., 2018) on the tweet text for both
this sentiment task and tweet text semantics task - BERT is a bidirectional
transformer based language model that is the current state-of-the-art for NLP
models. The BERT modules we use are detailed below.

###  3.1. BERT

Tweets, due to their short length, tend to use informal grammar and shortened
vocabulary (e.g hashtags, abbreviations, etc.) so there is a challenge in
choosing a pre-trained model adapted to this domain as opposed to the general
domain such as Wikipedia. The pre-trained model we use is BERTweet (Nguyen et
al., 2020) which is already fine-tuned on a corpus of 850M tweets, meaning
that it has a deep understanding of language used in tweets from a wide range
of topics and users. Indeed, BERTweet’s vocabulary contains many token
mappings that wouldn’t be in a conventional BERT vocabulary such as emojis and
colloquialisms, so BERTweet’s specific tokeniser was used. BERTweet is based
on BERT base, with 12 transformer blocks, 768 hidden units and 12 self-
attention heads and produces a 768 dimension $[CLS]$ embedding. The input into
this part of the model is the tweet text, $T$, concatenated with the non-
normalised numerical features ($N_{0},N_{1}...,N_{i}$), which are specified in
Section 4.3. This is separated using BERT’s $[SEP]$ tokens to indicate
multiple sentences for each feature, so during training it is being fine-tuned
to learn virality from these features concatenated as sentences. Formally, the
input to BERTweet is as follows, and the output produced afterwards is a
$[CLS]$ embedding, $h_{B}$.

(1) $X_{BERT}=[CLS]\oplus T\oplus[SEP]\oplus N_{0}\oplus[SEP]\oplus
N_{1}\oplus[SEP]\oplus...\oplus N_{i}\oplus[SEP]$

###  3.2. RoBERTa

For sentiment analysis we use a RoBERTa (Liu et al., 2019) based pre-trained
model. This is an optimised method for pre-training BERT where key
hyperparameters are modified such as removing the Next Sentence Prediction
objective and training with larger mini-batches. This model is from the
TweetEval benchmark (Barbieri et al., 2020) so is also specific to tweet
texts. The output of this model is a likelihood for a tweet having each of
negative, neutral or positive sentiment so is a 3 dimensional softmaxed
probability distribution, which we call $S$. Sentiment analysis is used as a
task alongside BERTweet so it is an additional numerical input for the final
virality classification layers and it is fine-tuned for this task during
training.

###  3.3. Classification Layers

The output of BERTweet, $h_{B}$, and sentiment probabilities, $S$, is
concatenated as input into the classification layer:

(2) $X_{CLS}=BERTweet(X_{BERT})\oplus S=h_{B}\oplus S$

BERT is fine-tuned for virality prediction using this layer - using fine-
tuning allows ease of implementation, faster training and performance gains as
we build on pre-trained models (Huang et al., 2019). The classification layers
we use has a fully connected flat MLP structure with tanh activation - the
dimensions of this layer are based on $X_{CLS}$, so there are 771 hidden
units. Next, a dropout of 0.1 is applied before a final linear prediction
layer, outputting probabilities of belonging to a virality class after being
fed into a softmax function. Experimentation was done on increasing the amount
of layers for the classifier and using CNNs instead of ANNs, but performance
was not improved by either.

##  4\. Experimental Setup

###  4.1. Data Collection

The datasets for this study are collected through the Twitter API
v2111https://developer.twitter.com/en/docs/twitter-api using Python. The
Twitter API allows context
annotations222https://developer.twitter.com/en/docs/twitter-
api/annotations/overview to be used to extract tweets that fall under a
certain context, categorised by domains and entities. For this study, we use 8
domain entity pairings - Cryptocurrencies, TV & Movies, Pets, Video Games,
Cell Phones, COVID-19, Football and K-pop. These topics are chosen as they are
generally popular over time so would have consistent virality criteria during
data collection, and are a good representation of a large amount of Twitter
posts. Using these topics allow us to filter a large amount of the spam posts
that are present on Twitter, and any that remain would be in the lower
virality classes so do not affect the model as much. For this study, only
original (not retweeted) English tweets are collected. These tweet fields we
collect are text, creation time, number of hashtags, number of mentions, and
tweet source client. Fields collected from the user are followers and
following count, and verified status. Retweet, like, reply and quote counts
are retrieved after 24 hours of creation as by this time the virality limit of
a tweet would have been reached (Chen and Tan, 2018; Lymperopoulos, 2021), and
these metrics are collected to indicate virality. This information is all
connected using the tweet’s ID and duplicate IDs are removed. The original
dataset consists of 2.1 million tweets collected over 26 days, although this
was reduced as detailed in Section 4.2.

###  4.2. Data Exploration and Labelling

Data analysis is conducted on the data to ensure data was appropriate for
modelling and to explore trends within the data. Within the original dataset
over ninety percent of the tweets are tweets that have not been retweeted,
which is indicative of how actual users interact with tweets. However, this
skewed class distribution will lead to poor performance for an ML model.
Therefore, the dataset is made more balanced (although not completely
balanced) by reducing the number of tweets with zero retweets to be equal the
amount with at least one retweet. The total amount of tweets remaining after
this reduction is 330k with half of the tweets having zero retweets.

We decide that retweets are best used as the label as it is more indicative of
tweet propagation to other users through social cascades (Cheung et al., 2017;
Jiménez-Zafra et al., 2021), so provide more exposure and engagement than
likes (Pancer and Poole, 2016). We use retweet class bands to turn the problem
into a multi-class classification problem similar to Nesi et al. (2018). Using
classes instead of raw retweet numbers allows the problem to be translated to
other platforms and allows the exploration of additional virality metrics for
labelling in the future. We select the following four class boundaries for
virality degree: 0 retweets (164k tweets); 1 retweet (98k tweets); 2-20
retweets (60k tweets); 21+ retweets (5k tweets). From the tweet numbers in
each class it is evident that there is still significant class imbalance
present.

###  4.3. Training Parameters

We use Class-Balanced Focal Loss (Cui et al., 2019) as the loss function to
aid in handling the class imbalance (as half of the dataset has zero
retweets). This calculates loss with an additional class balancing factor,
which is especially useful for long-tailed datasets such as ours (Huang et
al., 2021). We apply AdamW (Loshchilov and Hutter, 2017) as the optimization
algorithm to update model parameters and to allow for better generalisation.
The numerical features chosen for all models are hashtags, mentions,
followers, following, verified status, and text length. ‘Hashtags’ and
‘mentions’ are the amount of these elements in the tweet text, so is a feature
derived from tweet text. In addition, the sentiment of tweet text is added as
a feature for baselines. Tweet text embeddings are not used as none of the
literature we review utilises them. Note that all of these features are
collected when the tweet is collected and would stay consistent. With the
exception of ViralBERT, a min-max scalar is used for the numerical features as
there is a large variation in the values (e.g. hashtags being much lower than
follower counts). The ratio for training, validation and test sets is
80:10:10, with data being split randomly into these subsets. A data loader
class with a batch size of 32 is used, and models are trained until
convergence on the validation set.

###  4.4. Baselines

We compare the performance of ViralBERT with the following baseline methods
that were developed for similar tasks in the literature. Their inputs are
numerical features as detailed in Section 4.3 and are built using Pedregosa et
al. (2011):

  * •

Logistic Regression (Hong et al., 2011): Utilises Newton’s method for gradient
optimisation. This method has been used for predicting popular messages in
Twitter (Hong et al., 2011).

  * •

Support Vector Machine (SVM) (Ma et al., 2013; Nesi et al., 2018): Uses hinge
loss and SGD optimization. This method used for predicting the popularity of
newly emerging hashtags in Twitter (Ma et al., 2013), as well as for assessing
the retweet proneness (Nesi et al., 2018).

  * •

Decision Tree (Nesi et al., 2018): Uses Gini impurity score to measure quality
of a split with no max depth. This method has also been used for assessing
retweet proneness (Nesi et al., 2018).

  * •

Random Forest (Palovics et al., 2013; Nesi et al., 2018): Uses 100 trees with
no max depths. This baseline is based on previous work, which focused on the
problem of temporal prediction of retweet count (Palovics et al., 2013) and
likelihood of a retweet (Nesi et al., 2018).

Two more baselines are added using either numerical or text features only to
evaluate how these features perform.

  * •

MLPNum: Fully connected Multi Layer Perceptron for numerical features. Has one
hidden layer with 32 units.

  * •

ViralBERT Text: Using BERTweet on tweet text only (no numerical features)

To foster the reproducibility of our study, we have made the code for all
models including ViralBERT open source. 333https://github.com/RikazR/ViralBERT

###  4.5. Evaluation Metrics

To evaluate the effectiveness of our classifiers we employ the following
metrics on a holdout test set after training: accuracy, which is the ratio of
correct predictions over total number of instances evaluated; recall, which is
the fraction of actual classifications for a class that were correctly
identified; precision, which is the proportion of model predictions for a
class that were actually correct; F1 Score, which is the harmonic mean of
precision and recall. As accuracy and F1 scores consider both the positive and
negative classifications for evaluation, these are ideal for discriminating
our optimal solution. These metrics are balanced using the macro weighting, so
there is an equal weighting and importance for each class allowing metrics to
be more indicative of predictions of all classes (Grandini et al., 2020).

##  5\. Results

Table 1. Evaluation metrics on models. The best performing results are shown in bold. Method | F1 Score | Precision | Recall | Accuracy
---|---|---|---|---
Logistic Regression | 0.235 | 0.503 | 0.277 | 0.277
SVM | 0.221 | 0.320 | 0.271 | 0.271
Decision Tree Classifier | 0.405 | 0.402 | 0.408 | 0.408
Random Forest Classifier | 0.458 | 0.562 | 0.435 | 0.435
MLPNum | 0.213 | 0.235 | 0.268 | 0.268
ViralBERT Text | 0.410 | 0.415 | 0.409 | 0.409
ViralBERT | 0.523 | 0.609 | 0.494 | 0.494

Table 1 shows the results for our different model architectures. Solely
utilizing ViralBERT with text does not give optimal performance compared to
the baseline models. There is a significant performance increase when adding
numerical features to ViralBERT, in comparison to only using text so numerical
features are also important for virality prediction. By fine-tuning ViralBERT
to the concatenated input of numerical features, along with training
classification layers, we are able to achieve evaluation metrics that are
higher than the baselines. This could mean that adding numerical features to
text allows ViralBERT to gain context into features that could help with
prediction in addition to using the semantics of the text input.

Table 2. Comparison of removing features on performance of ViralBERT. The most significant differences are shown in bold. Feature removed | F1 Score | Accuracy
---|---|---
ViralBERT | 0.523 | 0.494
Sentiment | 0.438 | 0.432
Hashtags | 0.531 | 0.502
Mentions | 0.490 | 0.466
Followers | 0.429 | 0.435
Following | 0.502 | 0.474
Verified | 0.518 | 0.491
Text Length | 0.523 | 0.493

We conduct a study to measure the importance of each feature to ViralBERT by
removing a feature from the input and measuring the performance of the model.
For sentiment, the RoBERTa sentiment module is removed. Table 2 shows the
results of removing features from the input in comparison to ViralBERT.
Firstly, removing either sentiment or follower counts completely from the
network reduces the performance of the model significantly compared to other
features. This is intuitive, as a user with a greater amount of followers is
likely to gain more traction on their tweets (Hong et al., 2011), and tweets
that induce a greater emotional response in the users reading them may get
retweeted more (Hemsley et al., 2018). Tweet sentiments allow the model to
detect viral tweets from less popular users, and using both of these user-
focused features will give the best model performance. Furthermore, removing
mentions count or following counts have an effect on performance, albeit
lesser than the former two features. This could be due to more popular users
tending to follow others less (to have a high follower-following ratio). Also,
tweets having lots of mentions lack readability and utilise valuable space
that could be used for information that is useful to users, meaning it is less
likely to be retweeted. The most surprising result is that removing hashtag
counts slightly improves the performance of the model. This could mean that
this count is unrepresentative of virality, so adding this feature makes the
BERT learn worse representations from the inputs. These results should be
explored further in future work with a larger dataset and a more comprehensive
ablation study, including testing the interaction between each feature, to
allow for a deeper understanding of the reasons these have an adverse effect
on performance.

##  6\. Conclusion and Future Work

In this paper, we propose a BERT-based approach for predicting tweet virality
using both tweet text and numerical features, called ViralBERT. These
numerical features contain both content based features such as hashtag counts
and user based features such as author follower count. We collect a dataset of
tweets from the general userbase of Twitter, and train our proposed models on
this dataset, showing that our BERT-based approach outperforms current
approaches to virality prediction with an F1 score of 0.523 and Accuracy of
$0.494$. We find that text sentiment and follower counts, and to a lesser
extent mentions and following counts, are the most important features in
determining tweet virality, and that hashtag counts negatively affect the
model performance.

Future work can be done on increasing the complexity of the proposed
approaches and collecting a more balanced dataset, which was one of the
challenges of this study. Methods of balancing the class imbalance present in
the dataset could be used, for example using a hybrid multi task architecture
(Yang et al., 2020) on top of BERT and other such methods outlined in Huang et
al. (2021). As follower counts were found to affect virality, the dataset
could be improved further by specifically collecting viral tweets from less
popular users. Additionally, a larger dataset could have been collected to
improve the model further (as the dataset was reduced after collection), and
data from other social media could be investigated with this model to test
adaptability. Lastly, work should be done into incorporating more languages to
the dataset to measure how well the models adapt to completely different tweet
texts, and a module for media (images, GIFs, videos) could also be introduced
through Equation 2 to determine virality through this new feature type.

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