# Polynomiality of the faithful dimension for nilpotent groups
over finite truncated valuation rings

Mohammad Bardestani  John Abbott College, 21 275 Rue Lakeshore Road, Sainte-
Anne-de-Bellevue, QC, H9X 3L9, Canada<EMAIL_ADDRESS>,
Keivan Mallahi-Karai  Jacobs University, Bremen, Campus Ring I, Bremen 28759
Germany<EMAIL_ADDRESS>,  Dzmitry Rumintsau  Jacobs
University, Bremen, Campus Ring I, Bremen 28759 Germany  d.rumiantsau@jacobs-
university.de and  Hadi Salmasian  Department of Mathematics, University of
Ottawa, STEM Complex, 150 Louis-Pasteur Pvt, Ottawa, ON, Canada K1N 6N5
<EMAIL_ADDRESS>
###### Abstract.

Given a finite group $\mathrm{G}$, the faithful dimension of $\mathrm{G}$ over
$\mathbb{C}$, denoted by $m_{\mathrm{faithful}}(\mathrm{G})$, is the smallest
integer $n$ such that $\mathrm{G}$ can be embedded in
$\mathrm{GL}_{n}(\mathbb{C})$. Continuing the work initiated in [BMKS19], we
address the problem of determining the faithful dimension of a finite
$p$-group of the form $\mathscr{G}_{R}:=\exp(\mathfrak{g}_{R})$ associated to
$\mathfrak{g}_{R}:=\mathfrak{g}\otimes_{\mathbb{Z}}R$ in the Lazard
correspondence, where $\mathfrak{g}$ is a nilpotent $\mathbb{Z}$-Lie algebra
and $R$ ranges over finite truncated valuation rings

Our first main result is that if $R$ is a finite field with $p^{f}$ elements
and $p$ is sufficiently large, then
$m_{\mathrm{faithful}}(\mathscr{G}_{R})=fg(p^{f})$ where $g(T)$ belongs to a
finite list of polynomials $g_{1},\ldots,g_{k}$, with non-negative integer
coefficients. The latter list of polynomials is uniquely determined by the Lie
algebra $\mathfrak{g}$. Furthermore, for each $1\leq i\leq k$ the set of pairs
$(p,f)$ for which $g=g_{i}$ is a finite union of Cartesian products
$\mathscr{P}\times\mathscr{F}$, where $\mathscr{P}$ is a Frobenius set of
prime numbers and $\mathscr{F}$ is a subset of $\mathbb{N}$ that belongs to
the Boolean algebra generated by arithmetic progressions. This result
strengthens and unifies two theorems from [BMKS19] that establish
polynomiality-type properties of the faithful dimension with respect to $p$
and to $f$, separately.

Next we formulate a conjectural polynomiality property for the value of
$m_{\mathrm{faithful}}(\mathscr{G}_{R})$ in the more general setting where $R$
is a finite truncated valuation ring, and prove special cases of this
conjecture. In particular, we show that for a vast class of Lie algebras
$\mathfrak{g}$ that are defined by partial orders,
$m_{\mathrm{faithful}}(\mathscr{G}_{R})$ is given by a single polynomial-type
formula.

Finally, we compute $m_{\mathrm{faithful}}(\mathscr{G}_{R})$ precisely in the
case where $\mathfrak{g}$ is the free metabelian nilpotent Lie algebra of
class $c$ on $n$ generators and $R$ is a finite truncated valuation ring.

##  1\. Introduction

The _faithful dimension_ of a finite group $\mathrm{G}$ over $\mathbb{C}$ is
the smallest possible dimension of a faithful complex representation of
$\mathrm{G}$. Throughout this paper we denote the faithful dimension of
$\mathrm{G}$ by $m_{\mathrm{faithful}}(\mathrm{G})$. It is known that
$\mathrm{ed}_{\mathbb{C}}(\mathrm{G})\leq m_{\mathrm{faithful}}(\mathrm{G})$
for a finite group $\mathrm{G}$ (see [BF03, Proposition 4.15]), where
$\mathrm{ed}_{\mathbb{C}}(\mathrm{G})$ denotes the essential dimension of
$\mathrm{G}$ over $\mathbb{C}$ as defined by Buhler and Reichstein [BR97]. It
follows from a theorem of Karpenko and Merkurjev [KM08] that equality holds
for all finite $p$-groups (see also [Mer17]).

The current work is a continuation of [BMKS19] in which the question of
determining the faithful dimension was systematically studied for finite
$p$-groups associated to $\mathbb{Z}$-Lie algebras via the _Lazard
correspondence_. Let $\mathfrak{f}$ be a finite $\mathbb{Z}$-Lie algebra which
is nilpotent and has cardinality $p^{k}$ for a prime number $p$ that is
strictly larger than the nilpontency class of $\mathfrak{f}$. Then to
$\mathfrak{f}$ we can associate a finite $p$-group with underlying set
$\mathfrak{f}$ and with group multiplication defined by the Baker-Campbell-
Hausdorff formula, that is

$\displaystyle
x*y:=\sum_{n>0}\frac{(-1)^{n+1}}{n}\sum_{\begin{subarray}{c}(a_{1},b_{1}),\dots,(a_{n},b_{n})\\\
a_{j}+b_{j}\geq 1\end{subarray}}\frac{(\sum_{1\leq i\leq
n}a_{i}+b_{i})^{-1}}{a_{1}!b_{1}!\cdots
a_{n}!b_{n}!}(\mathrm{ad}_{x})^{a_{1}}(\mathrm{ad}_{y})^{b_{1}}\dots(\mathrm{ad}_{x})^{a_{n}}(\mathrm{ad}_{y})^{b_{n}-1}(y),$

for $x,y\in\mathfrak{f}$. By a result of Lazard [Khu98, Chapter 9] every
finite $p$-group of nilpotency class strictly smaller than $p$ is obtained in
this way from a unique Lie algebra $\mathfrak{f}$. It is natural to denote
this $p$-group by $\exp(\mathfrak{f})$.

A general class of examples of the above construction can be given as follows.
Let $\mathfrak{g}$ be a nilpotent $\mathbb{Z}$-Lie algebra that is finitely
generated as an abelian group, and let $R$ be a finite commutative unital ring
of cardinality $p^{k}$ where $p$ is a prime number strictly larger than the
nilpotency class of $\mathfrak{g}$. Then the above construction yields a
finite $p$-group $\mathscr{G}_{R}:=\exp(\mathfrak{g}_{R})$ where
$\mathfrak{g}_{R}:=\mathfrak{g}\otimes_{\mathbb{Z}}R$. For example, the group
of unitriangular matrices over $R$ is obtained in this fashion from the Lie
algebra of strictly upper triangular matrices over $R$.

Let $\mathbb{F}_{q}$ denote a finite field of order $q:=p^{f}$. A central
result of our previous work [BMKS19] is about the dependence of
$m_{\mathrm{faithful}}(\mathscr{G}_{\mathbb{F}_{q}})$ on $p$ and $f$. More
precisely, we proved the following two assertions (see Theorems 2.5 and 2.7 of
[BMKS19]):

  * (A)

There exist finitely many polynomials $g_{1},\ldots,g_{k}$ with non-negative
integer coefficients such that for every prime $p$ we have
$m_{\mathrm{faithful}}(\mathscr{G}_{\mathbb{F}_{p}})=g(p)$ where $g=g_{i}$ for
some $1\leq i\leq k$. The set of $p$ for which $g=g_{i}$ is definable in terms
of existence of solutions of polynomial systems modulo $p$.

  * (B)

Given any sufficiently large prime $p$, there exist finitely many polynomials
$g_{1},\ldots,g_{k}$, depending only on $\mathfrak{g}$ and on $p$, such that
for every $q:=p^{f}$ with $f\in\mathbb{N}$ we have
$m_{\mathrm{faithful}}(\mathscr{G}_{\mathbb{F}_{q}})=fg(q)$ where $g=g_{i}$
for some $1\leq i\leq k$. The set of $f\in\mathbb{N}$ for which $g=g_{i}$ is a
finite union of arithmetic progressions.

In statement A only $p$ varies, while in statement B only $f$ varies and it is
not clear how the $g_{i}$ depend on $p$. Statements A and B suggest that
$m_{\mathrm{faithful}}(\mathscr{G}_{\mathbb{F}_{q}})$ should possess a
stronger polynomiality-type property, where both $p$ and $f$ are allowed to
vary simultaneously. We did not succeed in proving a unifying assertion in
[BMKS19] (see Remark 2.8 of the latter reference).

The first major result of the present paper (Theorem 1.1) proves such a
unifying assertion. For a polynomial $g(T)\in\mathbb{Z}[T]$, we denote the set
of primes $p$ for which the congruence $g(T)\equiv 0\ (\mathrm{mod}\ p)$ has a
solution by $V_{g}$. We call a set of prime numbers a _Frobenius set_ if it is
in the Boolean algebra generated by the sets $V_{g}$. The Frobenius sets are
closely related to Serre’s Frobenian sets (see [BMKS19] or [Lag83] for a
precise statement).

######  Theorem 1.1.

Let $\mathfrak{g}$ be a nilpotent $\mathbb{Z}$-Lie algebra which is finitely
generated as an abelian group. Then there exist

  * (a)

a partition $\left\\{\mathscr{P}_{1},\dots,\mathscr{P}_{r}\right\\}$ of the
set of primes larger than $M=M(\mathfrak{g})$ into Frobenius sets,

  * (b)

a partition $\\{\mathscr{F}_{1},\dots,\mathscr{F}_{s}\\}$ of natural numbers
into arithmetic progressions, and

  * (c)

polynomials $g_{ij}(T)$, for $1\leq i\leq r$ and $1\leq j\leq s$, with non-
negative integer coefficients,

such that for all $q:=p^{f}$ where
$(p,f)\in\mathscr{P}_{i}\times\mathscr{F}_{j}$ for some $1\leq i\leq r$ and
$1\leq j\leq s$, we have

$m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{\mathbb{F}_{q}})=fg_{ij}(q).$

A natural extension of the class of $p$-groups $\mathscr{G}_{\mathbb{F}_{q}}$
is the groups $\mathscr{G}_{R}$ with $R\cong\mathcal{O}/\mathfrak{p}^{d}$,
where $\mathcal{O}$ is the valuation ring of a local field and $\mathfrak{p}$
is the prime ideal of $\mathcal{O}$. Up to isomorphism, every finite local
ring with a nilpotent and principal maximal ideal is of the form
$\mathcal{O}/\mathfrak{p}^{d}$ (see [Del84, McL73]). Such rings are called
_finite truncated valuation rings_.

Let $R$ be a finite truncated valuation ring with maximal ideal
$\mathfrak{m}$. We use $d=d_{R}$ to denote the smallest positive integer
satisfying $\mathfrak{m}^{d}=0$. If $p=p_{R}:=\mathrm{char}(R/\mathfrak{m})$,
then $pR=\mathfrak{m}^{e}$ for some $e=e_{R}\geq 1$ and
$\mathcal{O}/\mathfrak{m}\cong\mathbb{F}_{p^{f}}$ for some $f=f_{R}\geq 1$.
Henceforth we refer to the quadruple $(d,p,e,f)$ as the _associated
parameters_ of $R$. It turns out that we can formulate the problem (and our
results) only in terms of $(d,p,e,f)$, even though these parameters do not
identify $R$ uniquely up to isomorphism. We conjecture the following extension
of Theorem 1.1 for $m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{R})$.

######  Conjecture 1.2.

Let $\mathfrak{g}$ be as in Theorem 1.1. Then the constant $M(\mathfrak{g})$,
the partitions $\\{\mathscr{P}_{1},\ldots,\mathscr{P}_{r}\\}$ and
$\\{\mathscr{F}_{1},\ldots,\mathscr{F}_{s}\\}$, and the polynomials
$g_{ij}(T)$ can be chosen such that for every finite truncated valuation ring
$R$ with associated parameters $(d,p,e,f)$ we have

$m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{R})=f\sum_{\ell=0}^{e-1}g_{ij}(p^{f(d-\ell)})\quad\text{
whenever }(p,f)\in\mathscr{P}_{i}\times\mathscr{F}_{j}.$

We are unable to prove Conjecture 1.2 in full generality, but we establish two
results that support this conjecture. The first result, Theorem 1.3, shows
that $m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{R})$ is not greater than the
conjectural value, but grows with the same rate as $d$ tends to infinity.

######  Theorem 1.3.

Let $\mathfrak{g}$, $\\{\mathscr{P}_{i}\\}_{i=1}^{r}$,
$\\{\mathscr{F}_{i}\\}_{i=1}^{s}$, and the $g_{ij}(T)$ be as in Theorem 1.1.
Set $\mathscr{P}:=\mathscr{P}_{i_{\circ}}$,
$\mathscr{F}:=\mathscr{F}_{j_{\circ}}$, and $g(T):=g_{i_{\circ}j_{\circ}}(T)$,
where $1\leq i_{\circ}\leq r$ and $1\leq j_{\circ}\leq s$. Then there exists a
constant $M>0$ such that for every finite truncated valuation ring $R$ with
associated parameters $(d,p,e,f)$ satisfying
$(p,f)\in\mathscr{P}\times\mathscr{F}$ and $p>M$, we have

$fg(p^{fd})\leq m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{R})\leq
f\sum_{\ell=0}^{e-1}g(p^{f(d-\ell)}).$

######  Remark 1.4.

Note that the above theorem establishes Conjecture 1.2 in the case $e=1$. In
this case, the rings $R$ are of the form $\mathcal{O}/\mathfrak{p}^{d}$ where
$\mathcal{O}$ is the ring of integers of a finite unramified extension of
$\mathbb{Q}_{p}$.

The second result, Theorem 1.5, establishes the conjecture for groups
associated to _pattern Lie algebras_. Before stating Theorem 1.5 we recall
some definitions. Let $\prec$ be a partial order on the set
$[n]:=\\{1,2,\dots,n\\}$. To this partial order, we assign the
$\mathbb{Z}$-Lie subalgebra $\mathfrak{g}_{\prec}$ of
$\mathfrak{gl}(n,\mathbb{Z})$ that is spanned by the $e_{ij}$ satisfying
$i\prec j$, where the $e_{ij}$ denote the standard basis vectors of
$\mathfrak{gl}(n,\mathbb{Z})$. Many nilpotent Lie algebras of interest (e.g.,
nilradicals of parabolic subalgebras of semisimple Lie algebras) are pattern
Lie algebras.

Without loss of generality we assume that $\prec$ is compatible with the usual
ordering of $[n]$, so that elements of $\mathfrak{g}_{\prec}$ are strictly
upper triangular matrices. For any $i\prec j$ define

(1.1) $\alpha(i,j):=\\#\\{k\in[n]:i\prec k\prec j\\}.$

A pair $(i,j)$ with $i\prec j$ is called _extreme_ if $i$ is minimal and $j$
is maximal with respect to $\prec$. The set of extreme pairs will be denoted
by $I_{\mathrm{ex}}$.

######  Theorem 1.5.

Let $\mathfrak{g}:=\mathfrak{g}_{\prec}$ with $\mathfrak{g}_{\prec}$ as above,
and let $R$ be a finite truncated valuation ring with associated parameters
$(d,p,e,f)$ such that

$p>\max\\{\alpha(i,j):i\prec j\\}+1.$

Then

(1.2)
$m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{R})=\sum_{\ell=0}^{e-1}\sum_{(i,j)\in
I_{\mathrm{ex}}}fp^{f(d-\ell)\alpha(i,j)}.$

Special cases of Theorem 1.5 were proved in [BMKS19] (for $R=\mathbb{F}_{q}$)
and [BMKS16] (for Heisenberg and unitriangular groups over general $R$).

We conclude this paper by computing
$m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{R})$ when
$\mathfrak{g}\cong\mathfrak{m}_{n,c}$, where $\mathfrak{m}_{n,c}$ is the free
metabelian nilpotent Lie algebra of class $c$ on $n$ generators. Recall that

$\mathfrak{m}_{n,c}:=\mathfrak{f}_{n,c}/[[\mathfrak{f}_{n,c},\mathfrak{f}_{n,c}],[\mathfrak{f}_{n,c},\mathfrak{f}_{n,c}]],$

where $\mathfrak{f}_{n,c}$ denotes the free nilpotent $\mathbb{Z}$-Lie algebra
of class $c$ on $n$ generators. Indeed $\mathfrak{m}_{n,c}$ is the largest
metabelian quotient of $\mathfrak{f}_{n,c}$. The following theorem is a
generalization of [BMKS16, Theorem 2.15], which corresponds to the special
case $\mathfrak{m}_{2,c}$. Special cases of
$\mathfrak{m}_{3,5},\mathfrak{m}_{3,6}$ and $\mathfrak{m}_{n,4}$ for all
$n\geq 2$ over finite fields were computed by Tielker [Tie18].

######  Theorem 1.6.

Set $\mathfrak{g}:=\mathfrak{m}_{n,c}$ where $n,c\geq 2$. Then for $p$
sufficiently large we have

$m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{R})=(c-1){n+c-2\choose
c}\sum_{\ell=0}^{e-1}fp^{f(d-\ell)}.$

Let us now explain the key ideas that are used in the present paper. Similar
to our previous work [BMKS19], we heavily use the Kirillov orbit method for
finite $p$-groups, as well as the notion of the commutator matrix associated
to a nilpotent Lie algebra, defined originally by Grunewald and Segal [GS84].
Another important tool is a theorem from [BMKS19] which relates the faithful
dimension to the question of existence of rational points in general position
on certain rank varieties. However, the proof of Theorem 1.1 requires two new
ideas: first, a theorem of Ax from his seminal work [Ax68] (see also [vdD91])
which roughly states that any elementary statement in the language of finite
fields is equivalent to one about single-variable polynomials; second, a
theorem of Dedekind about factorization of the reduction modulo $p$ of a monic
polynomial with integer coefficients. In addition, in the proof of Theorem 1.3
we need a (not so well known) variant of the multivariate Hensel’s lemma that
does not require any smoothness or genericity assumption (see Proposition
3.3). This variant was first established by Ax and Kochen [AK65] using
ultrafilter methods (for further references, see Section 3).

### Acknowledgement

The authors thank Emmanuel Breuillard, Jamshid Derakhshan, Julia Gordon, and
Christopher Voll for stimulating conversations. The research of Hadi Salmasian
was partially supported by an NSERC Discovery Grant (RGPIN-2018-04044).

##  2\. Polynomiality along all finite fields

In this section we will prove Theorem 1.1. We begin with recalling a result of
Ax from model theory of finite fields. Let
${\mathcal{Q}}:=\\{p^{f}\,:\,p\text{ is prime and }f\in\mathbb{N}\\}$ be the
set of all prime powers. For $n\geq 1$, denote by ${\mathcal{B}_{n}}$ the
Boolean algebra on ${\mathcal{Q}}$ generated by the subsets

$Z_{h_{1},\ldots,h_{k}}:=\\{q\in\mathcal{Q}:\text{ the equations
}h_{i}(T_{1},\ldots,T_{n})=0,1\leq i\leq k,\text{ have a solution in
}\mathbb{F}_{q}\\},$

where $h_{1},\ldots,h_{k}$ range over all polynomials in
$\mathbb{Z}[T_{1},\dots,T_{n}]$. Also, let $\mathcal{B}$ be the Boolean
algebra generated by finite subsets of $\mathcal{Q}$ and the sets $Z_{h}$
where $h\in\mathbb{Z}[T_{1}]$. In [Ax68, Theorem 11], Ax proved that for all
$n\geq 1$, we have ${\mathcal{B}_{n}}\subseteq\mathcal{B}$. This statement
generalizes [Ax67, Theorem 1], which proves a similar assertion for primes. It
also follows from a result of van den Dries in [vdD91, Subsec. (2.3)].

In order to prove Theorem 1.1 we need a refinement of the aforementioned
result of Ax (see Proposition 2.4). This refinement may be well known to the
experts in the field (see e.g. [Ax68, Theorem 11’] for a somewhat similar
statement), but we were unable to locate the required formulation in the
literature. For this reason and for convenience of the reader we will supply a
proof. We would like to thank J. Derakhshan for communicating the main steps
of this proof to us.

Let us recall some basic facts from algebraic number theory. Let
$h(T)\in\mathbb{Z}[T]$ be a monic polynomial. The discriminant of $h(T)$ is
defined to be

$\mathrm{Disc}_{h}:=\prod_{1\leq i<j\leq n}(\alpha_{i}-\alpha_{j})^{2},$

where $\alpha_{1},\dots,\alpha_{n}$ are the roots of $h(T)$. Note that since
$h(T)$ is monic, $\mathrm{Disc}_{h}\in\mathbb{Z}$. We denote the splitting
field of $h(T)$ by $E$, that is, $E:=\mathbb{Q}(\alpha_{1},\dots,\alpha_{n})$,
and the ring of algebraic integers of $E$ by $\mathcal{O}_{E}$.

Let $p$ be a prime number that does not divide $\mathrm{Disc}_{h}$. Then $p$
is unramified in every simple extension $\mathbb{Q}(\alpha_{i})$ (see [Mar18,
Theorem 3.24]), hence by [Mar18, Theorem 4.31] it is also unramified in the
compositum of the $\mathbb{Q}(\alpha_{i})$, that is, in $E$. Therefore to
every prime ideal $\mathfrak{q}$ of $\mathcal{O}_{E}$ that lies over
$p\mathbb{Z}$ (i.e., $\mathfrak{q}\cap\mathbb{Z}=p\mathbb{Z}$) one can
associate a Frobenius automorphism [Mar18, Theorem 4.32], which is the unique
element $\sigma_{\mathfrak{q}}\in\mathrm{Gal}(E/\mathbb{Q})$ that satisfies

$\sigma_{\mathfrak{q}}(\alpha)\equiv\alpha^{p}\mod\mathfrak{q}\quad\text{ for
all }\alpha\in\mathcal{O}_{E}.$

The $\sigma_{\mathfrak{q}}$ form a conjugacy class of
$\mathrm{Gal}(E/\mathbb{Q})$, denoted by the Artin symbol
$\left(\frac{E/\mathbb{Q}}{p}\right)$.

Denote the reduction of $h(T)$ modulo $p$ by $\bar{h}$. Since
$p\nmid\mathrm{Disc}_{h}$, the roots of $\bar{h}(T)$ are also simple. For the
proof of the next theorem, see [Jac85, Theorems 4.37 and 4.38].

######  Theorem 2.1 (Dedekind).

Let $h(T)$ and $p$ be as above. Suppose

(2.1) $\bar{h}(T)=h_{1}(T)\cdots h_{r}(T)$

is the factorization of $\bar{h}(T)$ into irreducible polynomials over
$\mathbb{F}_{p}$. Set $n_{i}:=\deg h_{i}$ for $1\leq i\leq r$. Let
$\mathfrak{q}$ be a prime ideal of $\mathcal{O}_{E}$ that lies over
$p\mathbb{Z}$. Then the cycle decomposition of $\sigma_{\mathfrak{q}}$, when
viewed as a permutation of the roots of $h(T)$, is of the form
$\varsigma_{1}\cdots\varsigma_{r}$ with each $\varsigma_{i}$ a cycle of length
$n_{i}$.

######  Remark 2.2.

Theorem 2.1 is not stated in [Jac85] exactly in the given form. Jacobson
proves (by an argument originally due to Tate) that for every prime ideal
$\mathfrak{p}$ of $\mathcal{O}_{h}:=\mathbb{Z}[\alpha_{1},\ldots,\alpha_{n}]$
that lies over $p\mathbb{Z}$, there exists a unique
$\sigma_{\mathfrak{p}}\in\mathrm{Gal}(E/\mathbb{Q})$ such that for all
$\alpha\in\mathcal{O}_{h}$ one has
$\sigma_{\mathfrak{p}}(\alpha)\equiv\alpha^{p}\mod\mathfrak{p}$, and
furthermore the restriction of $\sigma_{\mathfrak{p}}$ to the roots of $h(T)$
has the desired cycle decomposition. Now if
$\mathfrak{q}\subset\mathcal{O}_{E}$ is a prime ideal that lies over
$p\mathbb{Z}$, then $\sigma_{\mathfrak{q}}$ also satisfies
$\sigma_{\mathfrak{q}}(\alpha)\equiv\alpha^{p}\mod\mathfrak{p}$ for
$\alpha\in\mathcal{O}_{h}$, where
$\mathfrak{p}:=\mathfrak{q}\cap\mathcal{O}_{h}$. Thus by the uniqueness of
$\sigma_{\mathfrak{p}}$ we obtain
$\sigma_{\mathfrak{p}}=\sigma_{\mathfrak{q}}$.

######  Lemma 2.3.

Let $h(T)$, $p$, $\sigma_{\mathfrak{q}}$, and $n_{i}$ for $1\leq i\leq r$ be
as in Theorem 2.1. Denote by $S_{p}(h)$ the set of positive integers $n\geq 1$
such that $\bar{h}(T)$ has a root in $\mathbb{F}_{p^{n}}$. Then

$S_{p}(h)=\bigcup_{n=1}^{r}n_{i}\mathbb{Z}=\\{n\geq
1:\mathrm{Fix}(\sigma_{\mathfrak{q}}^{n})\neq\varnothing\\},$

where $\mathrm{Fix}(\sigma_{\mathfrak{q}}^{n})$ denotes the set of fixed
points of $\mathrm{Fix}(\sigma_{\mathfrak{q}}^{n})$, viewed as a permutation
of the roots of $h(T)$.

###### Proof.

From the cycle decomposition of $\sigma_{\mathfrak{q}}$ in Theorem 2.1 it
follows immediately that $\sigma_{\mathfrak{q}}^{n}$ has a fixed point if and
only if $n_{i}|n$ for some $1\leq i\leq r$. This proves the second equality.
Thus it remains to verify that $S_{p}(h)=\bigcup_{i=1}^{r}n_{i}\mathbb{Z}$.

First observe that for each $1\leq i\leq r$, the quotient
$\mathbb{F}[x]/(h_{i}(x))$ is a field of order $p^{n_{i}}$ in which
$\bar{h}(x)$ has a root. Recall that $k|l$ if and only if $\mathbb{F}_{p^{l}}$
has a subfield isomorphic to $\mathbb{F}_{p^{k}}$. Thus if $n\in
n_{i}\mathbb{Z}$ for some $1\leq i\leq r$, then $\bar{h}(x)$ has a root in
$\mathbb{F}_{p^{n}}$. This proves
$S_{p}(h)\supseteq\bigcup_{i=1}^{r}n_{i}\mathbb{Z}$. For the reverse
inclusion, assume that $\bar{h}(x)$ has a root in $\mathbb{F}_{p^{n}}$. It
follows that there exists $1\leq i\leq r$ such that $h_{i}(x)$ has a root in
$\mathbb{F}_{p^{n}}$. Call this root $\alpha$. Since $h_{i}(x)$ is irreducible
over $\mathbb{F}_{p}$, it follows that the subfield $\mathbb{F}_{p}(\alpha)$
has $p^{n_{i}}$ elements. Thus $\mathbb{F}_{p^{n_{i}}}$ is isomorphic to a
subfield of $\mathbb{F}_{p^{n}}$, and consequently $n_{i}$ divides $n$. ∎

Define an equivalence relation $\sim$ on $\mathrm{Gal}(E/\mathbb{Q})$ by

$\tau_{1}\ \sim\ \tau_{2}\text{ iff
}\tau_{1}=\sigma\tau_{2}^{j}\sigma^{-1}\text{ for some
$\sigma\in\mathrm{Gal}(E/\mathbb{Q})$,}$

where $\mathrm{gcd}(j,\mathrm{ord}(\tau_{2}))=1$. An equivalence class of
$\sim$ is called a division, or an Abteilung. Following [Lag83], for any
division $C\subseteq\mathrm{Gal}(E/\mathbb{Q})$ we consider the set

$\mathscr{E}(C):=\left\\{p\ :\ \left(\frac{E/\mathbb{Q}}{p}\right)\subseteq
C\right\\}.$

In the following proposition, let $S(h;M)$ for $M>0$ and a monic polynomial
$h\in\mathbb{Z}[T]$ denote the set of pairs $(p,f)$ where $p>M$ is prime,
$f\geq 1$, and the reduction modulo $p$ of $h(T)$ has a root in
$\mathbb{F}_{p^{f}}$.

######  Proposition 2.4.

Let $h(T)\in\mathbb{Z}[T]$ be monic. Then there exists $M_{\circ}>0$, only
depending on $h(T)$, such that
$S(h;M_{\circ})=\bigcup_{i=1}^{r}\mathcal{P}_{i}\times\mathcal{N}_{i}$ for
some $r\in\mathbb{N}$, where the $\mathcal{P}_{i}$ and the $\mathcal{N}_{i}$
satisfy the following properties:

  * (a)

The $\mathcal{P}_{i}$ form a partition of the set of primes bigger than
$M_{\circ}$ into Frobenius sets, and

  * (b)

Each $\mathcal{N}_{i}$ is a finite union of sets of the form $k\mathbb{N}$ for
$k\in\mathbb{N}$.

###### Proof.

Without loss of generality we can assume that $h(T)$ is irreducible over
$\mathbb{Q}$. As before let $E$ denote the splitting field of $h(T)$. We
denote the divisions of $\mathrm{Gal}(E/\mathbb{Q})$ by $C_{1},\dots,C_{r}$.
We set $M_{\circ}:=1+|\mathrm{Disc}_{h}|$. First we show that if
$p,p^{\prime}>M_{\circ}$ are primes such that

$\left(\frac{E/\mathbb{Q}}{p}\right)\cup\left(\frac{E/\mathbb{Q}}{p^{\prime}}\right)\subseteq
C_{i}\text{ for some $1\leq i\leq r$,}$

then $S_{p}(h)=S_{p^{\prime}}(h)$. Choose
$\sigma_{\mathfrak{q}}\in\left(\frac{E/\mathbb{Q}}{p}\right)$ and
$\sigma_{\mathfrak{q}^{\prime}}\in\left(\frac{E/\mathbb{Q}}{p^{\prime}}\right)$.
Then $\sigma_{\mathfrak{q}^{\prime}}=\tau\sigma_{\mathfrak{q}}^{j}\tau^{-1}$,
where $\mathrm{gcd}(j,\mathrm{ord}(\sigma_{\mathfrak{q}}))=1$. By Lemma 2.3,
if $n\in S_{p}(h)$ then
$\mathrm{Fix}(\sigma_{{\mathfrak{q}}}^{n})\neq\varnothing$, hence also
$\mathrm{Fix}(\sigma_{{\mathfrak{q}}}^{nj})\neq\varnothing$. Since
$\sigma_{\mathfrak{q}^{\prime}}^{n}$ is conjugate to
$\sigma_{{\mathfrak{q}}}^{nj}$, it follows that
$\mathrm{Fix}(\sigma_{\mathfrak{q}^{\prime}}^{n})\neq\varnothing$, so that
$n\in S_{p^{\prime}}(h)$. This proves $S_{p}(h)\subseteq S_{p^{\prime}}(h)$,
and by symmetry we have the reverse inclusion, hence
$S_{p}(h)=S_{p^{\prime}}(h)$.

From what we just proved it follows that $S_{p}(h)$ depends only on the index
$1\leq i\leq r$ for which $\left(\frac{E/\mathbb{Q}}{p}\right)\subseteq
C_{i}$. It is shown in [Lag83, Theorem 1.1] that every set of primes that
differs by only a finite set from $\mathscr{E}(C)$ for some division $C$ is a
Frobenius set.111We remark that Lagarias uses different terminology. In
particular, in [Lag83] our Frobenius sets are called SPC sets, the sets
$\mathscr{E}(C)$ are called elementary Frobenius sets, and the sets in the
Boolean algebra generated by the $\mathscr{E}(C)$ for divisions $C$ are called
Frobenius sets. Thus we can set $\mathcal{P}_{i}:=\mathscr{E}(C_{i})$ for
$1\leq i\leq r$. The existence of the $\mathcal{N}_{i}$ now follows from Lemma
2.3. ∎

For the proof of Theorem 1.1 we need one more ingredient: a result from our
previous work [BMKS19] which relies on the Kirillov orbit method for finite
$p$-groups. To state this result, we need to recall the notion of the
_commutator matrix_ of a nilpotent $\mathbb{Z}$-Lie algebra.

Given a finitely generated abelian group $\Gamma$, we call a subset
$S\subseteq\Gamma$ a _semibasis_ if it represents a basis over $\mathbb{Z}$ of
the free abelian group $\Gamma/\Gamma_{\mathrm{tor}}$, where
$\Gamma_{\mathrm{tor}}$ denotes the subgroup of torsion elements of $\Gamma$.
Clearly $\\#S=\mathrm{rk}_{\mathbb{Z}}\Gamma$.

Now let $\mathfrak{g}$ be a nilpotent $\mathbb{Z}$-Lie algebra which is
finitely generated as an abelian group. Let ${\bf w}_{1},\ldots,{\bf
w}_{l_{1}}$ be a semibasis of
$\mathrm{Z}(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}]$. Let ${\bf
w}_{l_{1}+1},\ldots,{\bf w}_{m}$ be elements of $\mathfrak{g}$ that represent
a semibasis of
$[\mathfrak{g},\mathfrak{g}]/(\mathrm{Z}(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}])$.
Finally, let ${\bf z}_{1},\ldots,{\bf z}_{l_{2}}$ be elements of
$\mathfrak{g}$ that represent a semibasis of
$\mathrm{Z}(\mathfrak{g})/(\mathrm{Z}(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}])$.
Then the vectors $\\{{\bf w}_{1},\ldots,{\bf w}_{m},{\bf z}_{1},\ldots,{\bf
z}_{l_{2}}\\}$ are $\mathbb{Z}$-linearly independent and
$\mathrm{rk}_{\mathbb{Z}}(\mathrm{Z}(\mathfrak{g})+[\mathfrak{g},\mathfrak{g}])=l_{2}+m$.

Let ${\bf v}^{\prime}_{1},\ldots,{\bf v}^{\prime}_{n}$ be elements of
$\mathfrak{g}$ that represent a semibasis of
$\mathfrak{g}/\mathrm{Z}(\mathfrak{g})$. For $1\leq i<j\leq n$, there exist
integers $\eta_{ij}^{k}$, $l_{1}+1\leq k\leq m$, such that if we set

${\bf v}^{\prime}_{i,j}:=\left([{\bf v}^{\prime}_{i},{\bf
v}^{\prime}_{j}]-\sum_{k=l_{1}+1}^{m}\eta^{k}_{ij}{\bf w}_{k}\right),$

then ${\bf
v}^{\prime}_{i,j}+([\mathfrak{g},\mathfrak{g}]\cap\mathrm{Z}(\mathfrak{g}))\in([\mathfrak{g},\mathfrak{g}]/[\mathfrak{g},\mathfrak{g}]\cap\mathrm{Z}(\mathfrak{g}))_{\mathrm{tor}}$.
Set $K$ equal to the exponent of the group
$\left([\mathfrak{g},\mathfrak{g}]/[\mathfrak{g},\mathfrak{g}]\cap\mathrm{Z}(\mathfrak{g})\right)_{\mathrm{tor}}$,
so that $K{\bf
v}^{\prime}_{i,j}\in[\mathfrak{g},\mathfrak{g}]\cap\mathrm{Z}(\mathfrak{g})$
for $1\leq i<j\leq n$. Now set ${\bf v}_{i}:=K{\bf v}^{\prime}_{i}$ for $1\leq
i\leq n$. Then there exist integers $\lambda_{ij}^{k}$ such that

$[{\bf v}_{i},{\bf v}_{j}]-\sum_{k=1}^{m}\lambda^{k}_{ij}{\bf
w}_{k}\in\big{(}[\mathfrak{g},\mathfrak{g}]\cap\mathrm{Z}(\mathfrak{g})\big{)}_{\mathrm{tor}}\text{
for $1\leq i<j\leq n$}.$

We remark that $\lambda_{ij}^{k}=K^{2}\eta_{ij}^{k}$ for $l_{1}+1\leq k\leq
m$. We set $\lambda_{ii}^{k}:=0$, and for $i>j$ we define
$\lambda_{ij}^{k}:=-\lambda_{ji}^{k}$. Finally, for $1\leq i,j\leq n$ we
define the linear forms

$\Lambda_{ij}(T_{1},\dots,T_{m}):=\sum_{k=1}^{m}\lambda_{ij}^{k}T_{k}\in\mathbb{Z}[T_{1},\dots,T_{m}].$

It is clear that $\Lambda_{ii}=0$ and $\Lambda_{ij}=-\Lambda_{ji}$ for $1\leq
i,j\leq n$. The commutator matrix of $\mathfrak{g}$ (relative to the chosen
ordered basis) is the $n\times n$ skew-symmetric matrix of linear forms
defined by

(2.2)
$F_{\mathfrak{g}}(T_{1},\dots,T_{m}):=[\Lambda_{ij}(T_{1},\dots,T_{m})]_{1\leq
i,j\leq n}.$

See [GS84, Vol05, Vol04, OV15, AKOV13, SV14] as other references for
$F_{\mathfrak{g}}$. In the following theorem,
$\mathrm{rk}_{\mathbb{F}_{q}}(\cdot)$ denotes the rank of a matrix with
entries in the field $\mathbb{F}_{q}$.

######  Theorem 2.5.

Let $\mathfrak{g}$ be a nilpotent $\mathbb{Z}$-Lie algebra which is finitely
generated as an abelian group. Then there exists a constant $M>0$, depending
only on $\mathfrak{g}$, such that for every prime power $q:=p^{f}$ with $p>M$
and $f\geq 1$ we have

$m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{q})=\min\left\\{\sum_{\ell=1}^{l_{1}}fq^{\frac{\mathrm{rk}_{\mathbb{F}_{q}}(F_{\mathfrak{g}}(x_{\ell
1},\dots,x_{\ell m}))}{2}}:\begin{pmatrix}x_{11}&\dots&x_{1l_{1}}\\\
\vdots&\ddots&\vdots\\\
x_{l_{1}1}&\dots&x_{l_{1}l_{1}}\end{pmatrix}\in\mathrm{GL}_{l_{1}}(\mathbb{F}_{q})\right\\}+fl_{2},$

where $m:=\mathrm{rk}_{\mathbb{Z}}([\mathfrak{g},\mathfrak{g}])$,
$l_{1}:=\mathrm{rk}_{\mathbb{Z}}([\mathfrak{g},\mathfrak{g}]\cap\mathrm{Z}(\mathfrak{g}))$
and
$l_{2}:=\mathrm{rk}_{\mathbb{Z}}(\mathrm{Z}(\mathfrak{g})/\mathrm{Z}(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}])$.

###### Proof.

This is proved in [BMKS19, Theorem 5.3]. ∎

###### Proof of Theorem 1.1.

Let $p$ be a prime number and set $q:=p^{f}$ for some $f\geq 1$. Since
$F_{\mathfrak{g}}(T_{1},\dots,T_{m})$ is a skew-symmetric $n\times n$ matrix
where $n:=\mathrm{rk}_{\mathbb{Z}}(\mathfrak{g}/\mathrm{Z}(\mathfrak{g}))$,
the rank of $F_{\mathfrak{g}}(x_{1},\dots,x_{m})$ for
$x_{1},\ldots,x_{m}\in\mathbb{F}_{q}$ is an even number no larger than $n$.
Let $\mathcal{M}$ be the set of all integer vectors
$\mu=(a_{1},\dots,a_{l_{1}})\in\mathbb{Z}^{l_{1}}$ satisfying $0\leq
a_{1}\leq\cdots\leq a_{l_{1}}\leq n/2$. To each $\mu\in\mathcal{M}$ we assign
a polynomial $g_{\mu}(T)$ defined by

$g_{\mu}(T):=T^{a_{1}}+\dots+T^{a_{l_{1}}}+l_{2}.$

We order the vectors $\mu$ according to the reverse lexicographical order,
i.e. $\mu\lhd\mu^{\prime}$ if the rightmost non-zero component of the vector
$\mu^{\prime}-\mu$ is positive. If $\mu\lhd\mu^{\prime}$ and $q>l_{1}$, then
we can easily see that

(2.3) $g_{\mu}(q)<g_{\mu^{\prime}}(q).$

We sort elements of $\mathcal{M}$ as $\mu_{1}\lhd\dots\lhd\mu_{N}$, where
$N:=\\#\mathcal{M}$. For $\mu=(a_{1},\ldots,a_{l_{1}})\in\mathcal{M}$, we
define an affine variety

${\bf X}_{\mu}:=\left\\{(x_{ij})\in\mathrm{Mat}_{l_{1}\times
m}(\mathbb{C}):\mathrm{rk}_{\mathbb{C}}F_{\mathfrak{g}}(x_{i1},\dots,x_{im})=2a_{i}\text{
for }1\leq i\leq l_{1},\ \det\left(x_{ij}\right)_{1\leq i,j\leq l_{1}}\neq
0\right\\}.$

Note that the non-vanishing condition on the determinant can be turned into an
equation by introducing a new variable, standing for the inverse of the
determinant. We also remark that ${\bf X}_{\mu}$ is defined over $\mathbb{Z}$
because $F_{\mathfrak{g}}(T_{1},\dots,T_{m})$ is an integer matrix. For
$\mu\in\mathcal{M}$ set

$\Sigma_{\mu}:=\left\\{p^{f}\,:\,p\text{ is prime},\ p>\max\\{l_{1},M_{1}\\},\
f\geq 1,\ \text{ and }{\bf
X}_{\mu}(\mathbb{F}_{p^{f}})\neq\varnothing\right\\},$

where $M_{1}$ is the lower bound constant of Theorem 2.5. Theorem 2.5 and
(2.3) imply that
$m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{p^{f}})=fg_{\mu_{k}}(p^{f})$ whenever

$p^{f}\in\mathscr{Q}_{k}:=\Sigma_{\mu_{k}}\setminus\bigcup_{1\leq
i<k}\Sigma_{\mu_{i}}.$

Recall that $\mathcal{Q}$ denotes the set of prime powers. By Ax’s theorem
[Ax68, Theorem 11] stated in the beginning of this section, the sets
$\mathscr{Q}_{k}$ can be expressed in terms of unions, intersections, and
complements of finitely many of the generators of the Boolean algebra
$\mathcal{B}$, that is, finite subsets of $\mathcal{Q}$ and the sets $Z_{h}$
for $h(T)\in\mathbb{Z}[T]$. For $M>0$ set

$\mathcal{Q}^{M}:=\\{p^{f}\in\mathcal{Q}\,:\,p>M\\}.$

For a non-constant polynomial $h(T)\in\mathbb{Z}[T]$, there exists a monic
polynomial $h_{1}(T)\in\mathbb{Z}[x]$ and a constant $M_{h}>0$ such that
$Z_{h}\cap\mathcal{Q}^{M_{h}}=Z_{h_{1}}\cap\mathcal{Q}^{M_{h}}$ (this is
proved easily using a change of variable $T\mapsto T/T_{0}$ for a suitable
$T_{0}\in\mathbb{Z}$). It follows that for $M^{\prime}$ sufficiently large,
the sets $\mathscr{Q}_{k}\cap\mathcal{Q}^{M^{\prime}}$ are in the Boolean
algebra of subsets of $\mathcal{Q}^{M^{\prime}}$ that is generated by the sets
$Z_{h}\cap\mathcal{Q}^{M^{\prime}}$ for monic $h(T)\in\mathbb{Z}[T]$.
Proposition 2.4 implies that there exists a constant $M(\mathfrak{g})$ such
that each of the sets
$\tilde{\mathscr{Q}}_{k}:=\left\\{(p,f)\,:\,p^{f}\in\mathscr{Q}_{k}\text{ and
}p>M(\mathfrak{g})\right\\}$ for $1\leq k\leq N$ is a finite union of
Cartesian products $\mathcal{P}\times\mathcal{N}$ where $\mathcal{P}$ is a
Frobenius set and $\mathcal{N}$ is in the Boolean algebra generated by
arithmetic progressions. The existence of the desired partitions
$\mathscr{P}_{1},\ldots,\mathscr{P}_{r}$ and
$\mathscr{F}_{1},\ldots,\mathscr{F}_{s}$ now follows from the following
general remark: let $\mathscr{X}$ and $\mathscr{Y}$ be two arbitrary sets, and
let $\\{\mathscr{Z}_{i}\\}_{i\in\mathcal{I}}$ be a partition of
$\mathscr{X}\times\mathscr{Y}$ into finitely many sets such that
$\mathscr{Z}_{i}=\bigcup_{j=1}^{N_{i}}X_{i,j}\times Y_{i,j}$ for
$i\in\mathcal{I}$, where $N_{i}\in\mathbb{N}$. Then there exist partitions
$\\{X_{k}\\}_{k\in\mathcal{I}_{X}}$ of $\mathscr{X}$ and
$\\{Y_{l}\\}_{l\in\mathcal{I}_{Y}}$ of $\mathscr{Y}$, where
$|\mathcal{I}_{X}|,|\mathcal{I}_{Y}|<\infty$, such that (a) the $X_{k}$ belong
to the Boolean algebra generated by the $X_{i,j}$, (b) the $Y_{l}$ belong to
the Boolean algebra generated by the $Y_{i,j}$, and (c) each $\mathscr{Z}_{i}$
is a union of Cartesian product sets of the form $X_{k}\times Y_{l}$. ∎

The proof of Theorem 1.1 has the following consequence, which we will need
when considering $m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{R})$ for finite
truncated valuation rings $R$.

######  Corollary 2.6.

Let $\mathfrak{g}$, $\\{\mathscr{P}_{i}\\}_{i=1}^{r}$,
$\\{\mathscr{F}_{i}\\}_{i=1}^{s}$, and the $g_{ij}(T)$ be as in Theorem 1.1.
Fix $1\leq i_{\circ}\leq r$ and $1\leq j_{\circ}\leq s$. Set
$\mathscr{P}:=\mathscr{P}_{i_{\circ}}$,
$\mathscr{F}:=\mathscr{F}_{j_{\circ}}$, and $g(T):=g_{i_{\circ}j_{\circ}}(T)$.
Then $g(T)=l_{2}+\hat{g}(T)$, where $\hat{g}(T)\in\mathbb{Z}[T]$, such that
the following statements hold.

  * (a)

$\hat{g}(T)=\sum_{i=0}^{\deg(\hat{g})}\hat{g}_{i}T^{i}$ where the
$\hat{g}_{i}$ are non-negative integers and
$\sum_{i=0}^{\deg(\hat{g})}\hat{g}_{i}=l_{1}$.

  * (b)

For any $q:=p^{f}$ where $(p,f)\in\mathscr{P}\times\mathscr{F}$, there exist
$l_{1}$ vectors $\mathbf{a}_{i}\in\mathbb{F}_{q}^{\oplus m}$, $1\leq i\leq
l_{1}$, such that their projections on the first $l_{1}$ components form an
$\mathbb{F}_{q}$-basis of $\mathbb{F}_{q}^{\oplus l_{1}}$ and

$\\#\left\\{i\,:\,1\leq i\leq l_{1}\text{ and
}\mathrm{rk}_{\mathbb{F}_{q}}(F(\mathbf{a}_{i}))=2j\right\\}=\hat{g}_{j}\quad\text{for
}0\leq j\leq\deg(\hat{g}).$

##  3\. Bounds for $m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{R})$

In this section we turn our attention to
$m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{R})$, where $R$ is a finite truncated
valuation ring. It is known (see for example [McL73]) that any such $R$ is
isomorphic to $\mathcal{O}/\mathfrak{p}^{d}$ for some $d\geq 1$, where
$\mathcal{O}$ is the ring of integers of a $p$-adic field and $\mathfrak{p}$
is the maximal ideal of $\mathcal{O}$. Thus, henceforth we assume that
$R=\mathcal{O}/\mathfrak{p}^{d}$.

By analogy with the special case where $R$ is a finite field, one expects that
a suitable variation of Theorem 2.5 holds for
$m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{R})$. Indeed the method of proof of
Theorem 2.5 can be adapted to prove Proposition 3.1 below. Recall that
$p=\mathrm{char}(\mathcal{O}/\mathfrak{p})$,
$\\#\mathcal{O}/\mathfrak{p}=p^{f}$, and $p\mathcal{O}=\mathfrak{p}^{e}$. Then
$\mathcal{O}/\mathfrak{p}^{e}$ is an $\mathbb{F}_{p}$-vector space. In
Proposition 3.1, we define the map

(3.1) $\mathsf{proj}:(\mathcal{O}/\mathfrak{p}^{d})^{\oplus
m}\to(\mathcal{O}/\mathfrak{p}^{e})^{\oplus l_{1}}$

to be the natural projection on the first $l_{1}$ components, that is
$\mathsf{proj}(a_{1},\ldots,a_{m}):=(\bar{a}_{1},\ldots,\bar{a}_{l_{1}})$,
where $a\mapsto\bar{a}$ denotes the natural projection
$\mathcal{O}/\mathfrak{p}^{d}\to\mathcal{O}/\mathfrak{p}^{e}$.

######  Proposition 3.1.

Let $\mathfrak{g}$ be as in Theorem 2.5 and let $R$ be a finite truncated
local ring. Let $\mathcal{O}(d,m,e,l_{1})$ denote the set of $fel_{1}$-tuples
$(\mathbf{a}_{1},\ldots,\mathbf{a}_{fel_{1}})$ such that
$\mathbf{a}_{i}\in(\mathcal{O}/\mathfrak{p}^{d})^{\oplus m}$ for all $1\leq
i\leq fel_{1}$, and the projections
$\\{\mathsf{proj}(\mathbf{a}_{i})\\}_{i=1}^{fel_{1}}$ form a basis of the
$\mathbb{F}_{p}$-vector space $(\mathcal{O}/\mathfrak{p}^{e})^{\oplus l_{1}}$.
Then there exists a constant $M>0$, depending only on $\mathfrak{g}$, such
that for $p>M$ we have

$m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{R})=fl_{2}e+\min\left\\{\sum_{i=1}^{fel_{1}}\left(\frac{p^{fd(\mathrm{rk}_{\mathbb{Z}}(\mathfrak{g})-l_{1}-l_{2})}}{\\#\ker_{\mathcal{O}/\mathfrak{p}^{d}}(F(\mathbf{a}_{i}))}\right)^{\frac{1}{2}}\
:\
(\mathbf{a}_{1},\ldots,\mathbf{a}_{fel_{1}})\in\mathcal{O}(d,m,e,l_{1})\right\\}.$

###### Proof.

Since the argument is nearly identical to the proof of Theorem 2.5, we will
only explain the points of divergence from [BMKS19]. In what follows, we will
refer to the notation of [BMKS19]. We will not give an explicit value of $M$,
but it can be computed similar to [BMKS19]. A short exact sequence of finitely
generated abelian groups will remain exact upon tensoring with a finite
truncated valuation ring $R$ as long as $p$ is sufficiently large (depending
on the exact sequence). Using this observation, and some elementary arguments,
one can verify that
$Z(\mathfrak{g}_{R})=Z(\mathscr{G}_{R})=Z(\mathfrak{g})_{R}$,
$[\mathfrak{g}_{R},\mathfrak{g}_{R}]=[\mathfrak{g},\mathfrak{g}]_{R}$, and
$Z(\mathfrak{g})_{R}\cap[\mathfrak{g}_{R},\mathfrak{g}_{R}]=(Z(\mathfrak{g})\cap[\mathfrak{g},\mathfrak{g}])_{R}$.
In addition, the latter $R$-modules embed as free submodules of
$\mathfrak{g}_{R}$. We choose a primitive additive character
$\psi:\mathcal{O}/\mathfrak{p}^{d}\to\mathbb{C}^{*}$ as follows. Let
$\psi_{\circ}:\mathbb{Q}_{p}\to\mathbb{C}^{*}$ be the additive character
defined by $\psi_{\circ}(x):=e^{2\pi i\\{x\\}_{p}}$, where $\\{x\\}_{p}$
denotes the $p$-adic fractional part of $x$. Note that
$\ker\psi_{\circ}=\mathbb{Z}_{p}$. Now let $\mathbb{K}$ denote the fraction
field of $\mathcal{O}$, and set

$\mathfrak{D}:=\left\\{x\in\mathcal{O}\ :\
\mathrm{Tr}_{\mathbb{K}/\mathbb{Q}_{p}}(x\mathcal{O})\subseteq\mathbb{Z}_{p}\right\\}.$

Then $\mathfrak{D}=\mathfrak{p}^{-\ell}$ for some $\ell\geq 0$. Fixing a
uniformizer $\varpi\in\mathfrak{p}$, we obtain an isomorphism of the character
group $\widehat{\mathcal{O}/\mathfrak{p}^{d}}$ with
$\mathcal{O}/\mathfrak{p}^{d}$ via the map
$b+\mathfrak{p}^{d}\mapsto\psi_{b}\in\widehat{\mathcal{O}/\mathfrak{p}^{d}}$,
where
$\psi_{b}(x):=\psi_{\circ}(\mathrm{Tr}_{\mathbb{K}/\mathbb{Q}_{p}}(\varpi^{-d-\ell}bx))$.
Note that $\psi_{b}(x)=\psi(bx)$ where $\psi:=\psi_{1}$. From this it follows
that $\ker(\psi)$ does not contain any nontrivial ideal of
$\mathcal{O}/\mathfrak{p}^{d}$. Choose
$\mathbf{u}_{1},\ldots,\mathbf{u}_{l_{3}}\in\mathfrak{g}$ to be
representatives of a semibasis of
$\mathfrak{g}/(Z(\mathfrak{g})+[\mathfrak{g},\mathfrak{g}])$. Then by the
above remarks for $p$ sufficiently large $\mathfrak{g}_{R}$ is a free
$R$-module with generators
$\mathbf{w}_{1},\ldots,\mathbf{w}_{m},\mathbf{z}_{1},\ldots,\mathbf{z}_{l_{2}},\mathbf{u}_{1},\ldots,\mathbf{u}_{l_{3}}$.
Consequently, additive characters of $\mathfrak{g}_{R}$ are of the form

$\psi_{\mathbf{a}}:=\psi\left(\sum_{i=1}^{m}a_{i}\mathbf{w}_{i}+\sum_{i=1}^{l_{2}}b_{i}\mathbf{z}_{i}+\sum_{i=1}^{l_{3}}c_{i}\mathbf{u}_{i}\right),$

where $\mathbf{a}:=(a_{1},\ldots,a_{m},b_{1},\ldots,b_{l_{2}},c_{1},\ldots
c_{l_{3}})\in R^{\oplus(m+l_{2}+l_{3})}$. The dimension of the irreducible
representation of $\mathscr{G}_{R}$ corresponding to $\psi_{\mathbf{a}}$ is
equal to

$\left(\frac{q^{d(\mathrm{rk}_{\mathbb{Z}}(\mathfrak{g})-l_{1}-l_{2})}}{\\#\ker_{\mathcal{O}/\mathfrak{p}^{d}}F_{\mathfrak{g}}(a_{1},\ldots,a_{m})}\right)^{\frac{1}{2}}.$

The rest of the argument is essentially the same as the proof of Theorem 2.5.
We remark that in the proof given in [BMKS19] we used Rado-Horn’s Theorem to
reduce the mimimum from being over $\mathbb{F}_{p}$-bases to over
$\mathbb{F}_{q}$-bases. We are unable to extend this step to the setting of
finite truncated valuation rings, which is why in the statement of Proposition
3.1 the minimum is taken over a family of $\mathbb{F}_{p}$-bases. ∎

######  Definition 3.2.

Let $\\{\phi_{i}\\}_{i\in\mathcal{I}}$ be a finite family of polynomials in
$\mathbb{Z}[T_{1},\ldots,T_{m}]$. A solution
$(z_{1},\ldots,z_{m})\in(\mathcal{O}/\mathfrak{p})^{\oplus m}$ to the system
of equations $\phi_{i}=0$, $i\in\mathcal{I}$, is called refinable to
$\mathcal{O}$ if there exists a solution
$(\tilde{z}_{1},\ldots,\tilde{z}_{m})\in\mathcal{O}^{\oplus m}$ to this system
whose reduction modulo $\mathfrak{p}$ equals $(z_{1},\ldots,z_{m})$.

In the proof of Theorem 1.3 we need a variation of Hensel’s Lemma, which has
no constraint on the derivative, but assumes that the residual characteristic
is sufficiently large. This result (see Proposition 3.3 below) has been proved
by Ax and Kochen [AK65] over $\mathbb{Q}_{p}$ using ultrafilters, and by
Greenleaf [Gre65] and Greenberg [Gre74] over arbitrary $p$-adic fields (with a
quantitative lower bound on $p$) using algebraic geometry. An elementary
argument for Greenberg’s theorem was given by Kneser in [Kne78]. We remark
that a minor adaptation of the proof of Ax and Kochen establishes Proposition
3.3.

######  Proposition 3.3.

Let $\\{\phi_{i}\\}_{i\in\mathcal{I}}$ be a finite family of multivariate
polynomials with integer coefficients. Then there exists a constant $M>0$ such
that for every $p$-adic field $\mathbb{K}$ with ring of integers
$\mathcal{O}$, such that the characteristic of the residue field
$\mathcal{O}/\mathfrak{p}$ is bigger than $M$, every solution in
$\mathcal{O}/\mathfrak{p}$ of the system of equations $\phi_{i}=0$,
$i\in\mathcal{I}$, is refinable to a solution in $\mathcal{O}$.

###### Proof of Theorem 1.3.

As in Corollary 2.6, we express $g(T)$ as $g(T)=l_{2}+\hat{g}(T)$ where
$\hat{g}(T):=\sum_{i=0}^{\deg(\hat{g})}\hat{g}_{i}T^{i}$. First we prove the
upper bound. Set $\tilde{g}_{k}:=\sum_{i=0}^{k}\hat{g}_{i}$ for $0\leq
k\leq\deg(\hat{g})$ and $\tilde{g}_{-1}:=0$. Recall that
$n:=\mathrm{rk}_{\mathbb{Z}}(\mathfrak{g}/\mathrm{Z}(\mathfrak{g}))$ is the
size of the commutator matrix $F_{\mathfrak{g}}$. For $0\leq
k\leq\lfloor\frac{n}{2}\rfloor$ let $D^{(k)}$ be the $n\times n$ diagonal
matrix whose diagonal entries are defined by $D^{(k)}_{i,i}=1$ for $i\leq 2k$
and $D^{(k)}_{i,i}=0$ for $i>2k$. Consider the system of polynomial equations
with integer coefficients in variables $\mathsf{x}_{a,b}^{(i)}$,
$\mathsf{y}_{a,b}^{(i)}$, $\mathsf{r}^{(i)}$, $\mathsf{s}^{(i)}$, and
$\mathsf{a}_{j}^{(i)}$, for $1\leq a,b\leq n$, $1\leq i\leq fl_{1}$, and
$1\leq j\leq m:=\mathrm{rk}_{\mathbb{Z}}([\mathfrak{g},\mathfrak{g}])$, that
is defined by

(3.2)
$\begin{cases}X^{(i)}F(\mathsf{a}_{1}^{(i)},\ldots,\mathsf{a}_{m}^{(i)})Y^{(i)}=D^{(k)}\quad&\text{
for }f\tilde{g}_{k-1}+1\leq i\leq f\tilde{g}_{k}\text{ and }0\leq
k\leq\deg(\hat{g}),\\\ \mathsf{r}^{(i)}\det X^{(i)}=1&\text{ for }1\leq i\leq
fl_{1},\\\ \mathsf{s}^{(i)}\det Y^{(i)}=1&\text{ for }1\leq i\leq fl_{1},\\\
\end{cases}$

where $X^{(i)}$ and $Y^{(i)}$ are $n\times n$ matrices with entries
$\mathsf{x}_{a,b}^{(i)}$ and $\mathsf{y}_{a,b}^{(i)}$, respectively. Now fix
$(p,f)\in\mathscr{P}\times\mathscr{F}$, and let
$\mathbf{a}_{1},\ldots,\mathbf{a}_{l_{1}}\in(\mathcal{O}/\mathfrak{p})^{\oplus
m}$ be chosen as in Corollary 2.6. Next choose any $\mathbb{F}_{p}$-basis
$\beta_{1},\ldots,\beta_{f}$ for $\mathcal{O}/\mathfrak{p}$, and consider the
vectors $\mathbf{a}^{\prime}_{i,j}:=\beta_{j}\mathbf{a}_{i}$ for $1\leq i\leq
l_{1}$ and $1\leq j\leq f$. The projections of the $\mathbf{a}^{\prime}_{i,j}$
on the first $l_{1}$ coordinates form a basis for the $\mathbb{F}_{p}$-vector
space $(\mathcal{O}/\mathfrak{p})^{\oplus l_{1}}$. Furthermore,

$\\#\left\\{(i,j)\,:\,\mathrm{rk}_{\mathcal{O}/\mathfrak{p}}(F(\mathbf{a}^{\prime}_{i,j}))={2k}\right\\}=f\hat{g}_{k},$

because $\\#\ker_{\mathcal{O}/\mathfrak{p}}(F(\beta_{j}\mathbf{a}_{i}))$ does
not depend on $j$. It follows that for all
$(p,f)\in\mathscr{P}\times\mathscr{F}$, the polynomial system (3.2) has a
solution in $\mathcal{O}/\mathfrak{p}$, where the vectors
$(\mathsf{a}_{1}^{(u)},\ldots,\mathsf{a}_{m}^{(u)})$ for
$f\tilde{g}_{k-1}+1\leq u\leq f\tilde{g}_{k}$ are equal to the
$\mathbf{a}^{\prime}_{i,j}$ satisfying
$\mathrm{rk}_{\mathcal{O}/\mathfrak{p}}(F(\mathbf{a}^{\prime}_{i,j}))=2k$. By
Proposition 3.3, for $p$ sufficiently large (depending only on the system
(3.2), hence only on $\mathfrak{g}$) this solution is refinable to a solution
in $\mathcal{O}$. In particular there exist vectors
$\tilde{\mathbf{a}}_{i}\in\mathcal{O}^{\oplus m}$ for $1\leq i\leq fl_{1}$
such that $X_{i}F(\tilde{\mathbf{a}}_{i})Y_{i}=D^{(k)}$ for
$f\tilde{g}_{k-1}+1\leq i\leq f\tilde{g}_{k}$ and $0\leq k\leq\deg(\hat{g})$,
where $X_{i}$ and $Y_{i}$ are invertible $n\times n$ matrices with entries in
$\mathcal{O}$. If $\varpi\in\mathfrak{p}$ is a uniformizer then for $0\leq
j\leq e-1$ we have
$\\#\ker_{\mathcal{O}/\mathfrak{p}^{d}}F(\varpi^{j}\tilde{\mathbf{a}}_{i})=p^{fd(n-2k)+2kfj}$.
Since $\mathrm{rk}_{\mathbb{Z}}(\mathfrak{g})-l_{1}-l_{2}=n$, we obtain

$\left(\frac{p^{fd(\mathrm{rk}_{\mathbb{Z}}(\mathfrak{g})-l_{1}-l_{2})}}{\\#\ker_{\mathcal{O}/\mathfrak{p}^{d}}(F(\varpi^{j}\tilde{\mathbf{a}}_{i}))}\right)^{\frac{1}{2}}=p^{kf(d-j)}.$

Furthermore, the projections to the first $l_{1}$ coordinates of the list of
vectors

$\\{\varpi^{j}\tilde{\mathbf{a}}_{i}\ :\ 0\leq j\leq e-1\text{ and }1\leq
i\leq fl_{1}\\}$

forms a basis of the $\mathbb{F}_{p}$-vector space
$(\mathcal{O}/\mathfrak{p}^{e})^{\oplus l_{1}}$. The upper bound part of the
theorem follows from Proposition 3.1.

For the lower bound, it suffices to assume that
$p>\max\left\\{l_{1}+l_{2},l_{2}+\sum_{i=0}^{\deg(\hat{g})}\hat{g}_{i}\right\\}$.
Suppose that the $fel_{1}$-tuple
$(\mathbf{a}_{1},\ldots,\mathbf{a}_{fel_{1}})$ of vectors in
$(\mathcal{O}/\mathfrak{p}^{d})^{\oplus m}$ corresponds as in Proposition 3.1
to the value of $m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{R})$. Let
$\overline{\mathsf{proj}}:(\mathcal{O}/\mathfrak{p}^{d})^{\oplus
m}\to(\mathcal{O}/\mathfrak{p})^{\oplus l_{1}}$ be the natural projection map
defined similar to $\mathsf{proj}$ (see (3.1)). The vectors
$\overline{\mathsf{proj}}(\mathbf{a}_{i})$ form a spanning set of the
$\mathbb{F}_{p}$-vector space $(\mathcal{O}/\mathfrak{p})^{\oplus l_{1}}$,
hence without loss of generality we can assume that
$\\{\overline{\mathsf{proj}}(\mathbf{a}_{i})\\}_{i=1}^{fl_{1}}$ is an
$\mathbb{F}_{p}$-basis of $(\mathcal{O}/\mathfrak{p})^{\oplus l_{1}}$. For
$d\geq 1$ and $1\leq i\leq fl_{1}$ set

$x_{i,d}:=\left(\frac{p^{f(\mathrm{rk}_{\mathrm{Z}}(\mathfrak{g})-l_{1}-l_{2})}}{\\#\ker_{\mathcal{O}/\mathfrak{p}^{d}}(F(\mathbf{a}_{i}))}\right)^{\frac{1}{2}}.$

Note that $x_{i,d}^{2}$ is the cardinality of the subgroup of
$(\mathcal{O}/\mathfrak{p}^{d})^{\oplus m}$ that is generated by the columns
of $F(\mathbf{a}_{i})$, and since $F(\mathbf{a}_{i})$ is a skew symmetric
matrix, we also have $x_{i,d}=p^{fy_{i,d}}$ for a non-negative integer
$y_{i,d}$. It follows that $x_{i,d}\geq x_{i,1}^{d}$. By Proposition 3.1 (for
$R=\mathbb{F}_{p^{f}}$) and Theorem 1.1,

(3.3) $fl_{2}+\sum_{i=1}^{fl_{1}}p^{fy_{i,1}}\geq fg(p^{f}).$

Setting $y_{i,d}=0$ for $fl_{1}+1\leq i\leq f(l_{1}+l_{2})$ and $d\geq 1$,
from (3.3) it follows that

(3.4) $\sum_{i=1}^{f(l_{1}+l_{2})}p^{fy_{i,1}}\geq fg(p^{f}).$

Our strategy is to prove that

(3.5) $\sum_{i=1}^{f(l_{1}+l_{2})}p^{fdy_{i,1}}\geq fg(p^{fd}).$

Without loss of generality we assume that $y_{1,1}\geq\cdots\geq
y_{f(l_{1}+l_{2}),1}$. Set $N:=\deg(\hat{g})$. There are two cases to
consider:

_Case 1._ $y_{i,1}\leq N$ for all $i$. We claim that $y_{i,1}=N$ for $1\leq
i\leq f\hat{g}_{N}$: otherwise since $p\geq 1+l_{1}+l_{2}$ and the
coefficients of $\hat{g}(T)$ are non-negative, we have

$\sum_{i=1}^{f(l_{1}+l_{2})}p^{fy_{i,1}}\leq(f\hat{g}_{N}-1)p^{fN}+f(l_{1}+l_{2})p^{f(N-1)}<f\hat{g}_{N}p^{fN}\leq
f\hat{g}(p^{f})\leq fg(p^{f}),$

which is a contradiction. It follows from above that $y_{i,1}=N$ for $1\leq
i\leq f\hat{g}_{N}$. If $y_{f\hat{g}_{N}+1,1}=N$ then from $p\geq
l_{2}+\sum_{i=0}^{N}\hat{g}_{i}$ it follows that

$\displaystyle m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{R})$
$\displaystyle\geq\sum_{i=1}^{f(l_{1}+l_{2})}x_{i,d}\geq\sum_{i=1}^{f(l_{1}+l_{2})}x_{i,1}^{d}=\sum_{i=1}^{f(l_{1}+l_{2})}p^{fdy_{i,1}}\geq(f\hat{g}_{N}+1)p^{fdN}$
$\displaystyle=f\hat{g}_{N}p^{fdN}+p^{fd(N-1)}p^{fd}\geq
f\hat{g}_{N}p^{fdN}+p^{fd(N-1)}f\left(l_{2}+\sum_{i=0}^{N-1}\hat{g}_{i}\right)\geq
fg(p^{fd}).$

If $y_{f\hat{g}_{N}+1,1}<N$, then by cancelling out the summands $p^{fN}$ from
both sides of (3.4) we obtain a similar relation for a polynomial of lower
degree on the right hand side of (3.5), and we can repeat the above argument
(leading to either Case 1 above, or Case 2 below).

_Case 2._ $y_{1,1}\geq N+1$. Then from $p\geq l_{2}+\sum_{i=0}^{N}\hat{g}_{i}$
it follows that

(3.6) $\displaystyle m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{R})\geq
p^{fdy_{1,1}}\geq p^{fd(N+1)}$
$\displaystyle\geq(1+l_{2}+\sum_{i=0}^{N}\hat{g}_{i})^{f}p^{fdN}$ (3.7)
$\displaystyle\geq fl_{2}+fp^{fdN}\sum_{i=0}^{N}\hat{g}_{i}\geq
fg(p^{fd}).\qed$

##  4\. Polynomiality over rings for pattern groups

In this section we prove Theorem 1.5, which in particular establishes
Conjecture 1.2 for pattern groups. Throughout this section we set
$\mathfrak{g}:=\mathfrak{g}_{\prec}$. We begin with recalling some notation
and general facts from [BMKS19] and [BMKS16]. For an abelian $p$-group
$\Gamma$, we set

$\Omega_{1}(\Gamma):=\\{g\in\Gamma:g^{p}=\mathbf{1}\\}.$

Note that $\Omega_{1}(\Gamma)$ is an $\mathbb{F}_{p}$-vector space. For an
abelian group $A$ we set $\widehat{A}:=\mathrm{Hom}(A,\mathbb{C}^{\ast})$.
When $A$ is an elementary abelian $p$-group, $\widehat{A}$ is canonically an
$\mathbb{F}_{p}$-vector space and there exists an isomorphism
$\widehat{A}\cong\mathrm{Hom}(A,\mathbb{Z}/p\mathbb{Z})$ obtained by
identifying $\mathbb{Z}/p\mathbb{Z}$ with the subgroup of $p$-th roots of
unity in $\mathbb{C}^{\ast}$.

Recall that $R\cong\mathcal{O}/\mathfrak{p}^{d}$ for $d\geq 1$. Fix a
primitive character $\psi:R\to\mathbb{C}^{*}$ (for example the character
described in the proof of Proposition 3.1), and also set
$\psi_{b}(x):=\psi(bx)$ for $b\in R$ and $x\in R$, so that the map

(4.1) $R\to\widehat{R}\ ,\ b\mapsto\psi_{b}$

is a group isomorphism. Now let $\mathfrak{a}$ be an ideal in $R$ and set
$\operatorname{Ann}(\mathfrak{a}):=\\{r\in R:r\mathfrak{a}=0\\}$. From
surjectivity of the restriction $\widehat{R}\to\widehat{\mathfrak{a}}$ it
follows that the map

$R/\operatorname{Ann}(\mathfrak{a})\to\widehat{\mathfrak{a}}\ ,\
b+\operatorname{Ann}(\mathfrak{a})\mapsto{\psi_{b}}\big{|}_{\mathfrak{a}}$

is a group isomorphism. Notice that
$\operatorname{Ann}(\mathfrak{p}^{k}/\mathfrak{p}^{d})=\mathfrak{p}^{(d-k)}/\mathfrak{p}^{d}$
for $0\leq k\leq d$, so that we obtain the following lemma.

######  Lemma 4.1.

For integers $0\leq k\leq d$, every additive character of the abelian group
$\mathfrak{p}^{k}/\mathfrak{p}^{d}$ is of the form

$\psi_{b}:\mathfrak{p}^{k}/\mathfrak{p}^{d}\to\mathbb{C}^{*}\ ,\
x\mapsto\psi(bx),$

for a unique
$b+\mathfrak{p}^{(d-k)}/\mathfrak{p}^{d}\in(\mathcal{O}/\mathfrak{p}^{d})/(\mathfrak{p}^{(d-k)}/\mathfrak{p}^{d})\cong\mathcal{O}/\mathfrak{p}^{d-k}$.
In particular
$\widehat{\mathfrak{p}^{k}/\mathfrak{p}^{d}}\cong\mathcal{O}/\mathfrak{p}^{(d-k)}$.

Hereafter, for a finite $p$-group $\mathrm{G}$ the
$\mathbb{Z}/p\mathbb{Z}$-vector space
$\mathrm{Hom}(\Omega_{1}(\mathrm{Z}(\mathrm{G})),\mathbb{C}^{*})$ will be
denoted by $\widehat{\Omega}_{1}(\mathrm{Z}(\mathrm{G}))$. In the following
lemma, by the central character of an irreducible representation $(\rho,V)$ of
$\mathrm{G}$ we mean the group homomorphism
$\chi:\mathrm{Z}(\mathrm{G})\to\mathbb{C}^{*}$ satisfying
$\rho(g)=\chi(g)1_{V}$ for $g\in\mathrm{Z}(\mathrm{G})$.

######  Lemma 4.2.

Let $\mathrm{G}$ be a finite $p$-group.

  * (i)

Let $(\rho_{i},V_{i})_{1\leq i\leq k}$ be a family of irreducible
representations of $\mathrm{G}$ with central characters $\chi_{i}$. Suppose
that $\\{{\chi_{i}}_{|_{\Omega_{1}(\mathrm{Z}(\mathrm{G}))}}:1\leq i\leq k\\}$
spans $\widehat{\Omega}_{1}(\mathrm{Z}(\mathrm{G}))$. Then $\oplus_{1\leq
i\leq k}\rho_{i}$ is a faithful representation of $\mathrm{G}$.

  * (ii)

Let $\rho$ be a faithful representation of $\mathrm{G}$ of dimension
$m_{{}_{\mathrm{faithful}}}(\mathrm{G})$. Then $\rho$ decomposes as a direct
sum of exactly $r:=r_{G}$ irreducible representations, where $r_{G}$ denotes
the minimum number of generators of the abelian group
$\mathrm{Z}(\mathrm{G})$. Furthermore, the restrictions to
$\Omega_{1}(\mathrm{Z}(\mathrm{G}))$ of the central characters of these
representations form a $\mathbb{Z}/p\mathbb{Z}$-basis for
$\widehat{\Omega}_{1}(\mathrm{Z}(\mathrm{G}))$.

###### Proof.

For (i), see [BMKS19, Lem. 3.4] or [MR10]. For (ii), see [BMKS19, Lem. 3.5] ∎

######  Remark 4.3.

For a finite abelian group $A$, the minimum number of generators is equal to
the number of invariant factors of $A$. If $A$ is a finite $p$-group, then the
minimum number of generators of $A$ is equal to
$\dim_{\mathbb{Z}/p\mathbb{Z}}(A\otimes_{\mathbb{Z}}\mathbb{Z}/p\mathbb{Z})$.
In particular, the minimum number of generators of the abelian group
$\mathcal{O}/\mathfrak{p}^{d}$ is equal to $fe$.

Next we establish a general combinatorial lemma about bases of direct sums.

######  Lemma 4.4.

Let $V_{1}$ and $V_{2}$ be two vector spaces of dimensions $d_{1}$ and $d_{2}$
over an arbitrary field. Let $S=\\{(v_{i},v^{\prime}_{i}):1\leq i\leq
d_{1}+d_{2}\\}$ be a basis for $V_{1}\oplus V_{2}$. Then $S$ can be
partitioned into two sets $B_{1}$ and $B_{2}$, of sizes $d_{1}$ and $d_{2}$
respectively, such that the $V_{i}$-components of the vectors in $B_{i}$ form
a basis of $V_{i}$ for $i=1,2$.

###### Proof.

The following argument is communicated to us by I. Bogdanov. Let
$\\{e_{1},\dots,e_{d_{1}}\\}$ and
$\\{e^{\prime}_{1},\dots,e^{\prime}_{d_{2}}\\}$ denote bases of $V_{1}$ and
$V_{2}$, respectively. Then for $1\leq j\leq d_{1}+d_{2}$ we obtain

$v_{j}=\sum_{i=1}^{d_{1}}a_{ij}e_{i}\quad\text{and}\quad
v^{\prime}_{j}=\sum_{i=1}^{d_{2}}b_{ij}e^{\prime}_{j}.$

Consider the square matrix $A$ of size $d_{1}+d_{2}$ whose $i$-th column, for
$1\leq i\leq d_{1}+d_{2}$, is the transpose of the row vector

$(a_{1,i},\cdots,a_{d_{1},i},b_{1,i},\ldots,b_{d_{2},i}).$

Since $S$ is a basis, $A$ is invertible. Set $I:=\\{1,\dots,d_{1}\\}$ and
consider the generalized Laplace expansion

$\det(A)=\sum_{\begin{subarray}{c}J\subseteq\\{1,\dots,d_{1}+d_{2}\\}\\\
|J|=d_{1}\end{subarray}}(-1)^{\sum_{i\in I}i+\sum_{j\in J}j}\det A_{I,J}\det
A_{I^{\prime},J^{\prime}},$

where $I^{\prime}:=\\{d_{1}+1,\ldots,d_{1}+d_{2}\\}$, $J^{\prime}$ is the
complement of $J$ in $\\{1,\ldots,d_{1}+d_{2}\\}$, and $A_{I,J}$
(respectively, $A_{I^{\prime},J^{\prime}}$) is the minor of $A$ corresponding
to rows and columns indexed by $I$ and $J$ (respectively, by $I^{\prime}$ and
$J^{\prime}$). Since $\det(A)\neq 0$, there exists $J$ with $\det A_{I,J}\det
A_{I^{\prime},J^{\prime}}\neq 0$. The desired sets are
$B_{1}:=\\{(v_{j},v^{\prime}_{j}):j\in J\\}$ and
$B_{2}:=\\{(v_{j},v^{\prime}_{j}):j\in J^{\prime}\\}$. ∎

Lemma 4.4 and induction yield the following.

######  Lemma 4.5.

Let $V$ be a finite dimensional vector space over an arbitrary field and let
$S$ be a basis for $V^{\oplus k}$ where $k\geq 1$. Then $S$ can be partitioned
into subsets $B_{1},\dots,B_{k}$ such that each of the sets
$\\{\pi_{\ell}(w):w\in B_{\ell}\\}$ is a basis of $V$ for $1\leq\ell\leq k$,
where $\pi_{\ell}(v_{1},\dots,v_{k}):=v_{\ell}$.

The following simple lemma was also used in [BMKS16].

######  Lemma 4.6.

Let $a_{0},\dots,a_{m-1}$ be non-negative real numbers with
$\sum_{\ell=0}^{m-1}a_{\ell}=m$. Assume that for any $0\leq\ell\leq m-1$ we
have $a_{\ell}+\cdots+a_{m-1}\leq m-\ell$. Then for any decreasing sequence
$x_{0}\geq\cdots\geq x_{m-1}$, we have
$\sum_{\ell=0}^{m-1}a_{\ell}x_{\ell}\geq\sum_{\ell=0}^{m-1}x_{\ell}$.

###### Proof.

Define $A_{\ell}:=\sum_{j=\ell}^{m-1}a_{j}$ for $0\leq\ell\leq m-1$ and set
$A_{m}:=0$. Then $A_{\ell}\leq m-\ell$ and $A_{0}=m$. Set $\Delta
A_{\ell}:=A_{\ell}-A_{\ell+1}$ and $\Delta x_{\ell}:=x_{\ell}-x_{\ell-1}$.
Since $\Delta x_{\ell}\leq 0$, we have

$\sum_{\ell=0}^{m-1}a_{\ell}x_{\ell}=\sum_{\ell=0}^{m-1}\Delta
A_{\ell}x_{\ell}=A_{0}x_{0}+\sum_{\ell=1}^{m-1}A_{\ell}\Delta x_{\ell}\geq
mx_{0}+\sum_{\ell=1}^{m-1}(m-\ell)\Delta
x_{\ell}=\sum_{\ell=0}^{m-1}x_{\ell}.\qed$

For the proof of Theorem 1.5 we need some standard facts from the _orbit
method_ for describing irreducible representations of finite $p$-groups. The
tools that we need from the orbit method can be found in various references,
including [Kaz77], [How77], and [BS08]. For the reader’s convenience, we
briefly review the pertinent results from the orbit method (we follow [BS08]
closely; see also [BMKS19, Sec. 3.2]). Let $\mathfrak{f}$ and
$\exp(\mathfrak{f})$ be as in Section 1. Then we can define a coadjoint action
of $\exp(\mathfrak{f})$ on
$\widehat{\mathfrak{f}}:=\mathrm{Hom}_{\mathbb{Z}}(\mathfrak{f},\mathbb{C}^{*})$
as follows. For $x\in\exp(\mathfrak{f})=\mathfrak{f}$ and
$\theta\in\widehat{\mathfrak{f}}$, we define
$\theta^{x}\in\widehat{\mathfrak{f}}$ by

$\theta^{x}(y):=\theta\left(\sum_{k\geq
0}\frac{\mathrm{ad}_{x}^{k}(y)}{k!}\right),\quad\text{ for }y\in\mathfrak{f}.$

Note that only finitely many summands are nonzero.

######  Theorem 4.7.

Assume that $p$ is an odd prime that is strictly larger than the nilpotency
class of $\mathfrak{f}$. Then there exists a bijection between the orbits of
the coadjoint action on $\widehat{\mathfrak{f}}$ and characters of irreducible
representations of $\exp(\mathfrak{f})$. Furthermore, if $\rho_{\Theta}$
denotes the irreducible representation corresponding to the coadjoint orbit
$\Theta\subseteq\widehat{\mathfrak{f}}$, then the following statements hold.

  * (a)

The character of $\rho_{\Theta}$ is given by
$\displaystyle\chi_{\Theta}(x):=|\Theta|^{-1/2}\sum_{\theta\in\Theta}\theta(x)$
for $x\in\exp(\mathfrak{f})=\mathfrak{f}$.

  * (b)

$\displaystyle\dim(\rho_{\Theta})=\left(\frac{\\#\mathfrak{f}}{\\#\mathrm{Stab}_{\mathfrak{f}}(\theta_{0})}\right)^{1/2}$
for $\theta_{0}\in\Theta$, where
$\mathrm{Stab}_{\mathfrak{f}}(\theta_{0}):=\left\\{x\in\mathfrak{f}\,:\,\theta_{0}([x,\mathfrak{f}])=1\right\\}.$

  * (c)

$\rho_{\Theta}(g)$ is multiplication by the scalar $\theta(g)$ for
$g\in\mathrm{Z}(\exp(\mathfrak{f}))$.

Recall that in this section $\mathfrak{g}:=\mathfrak{g}_{\prec}$. Our next
goal is to use the orbit method to study irreducible representations of
$\mathscr{G}_{R}:=\exp(\mathfrak{g}_{R})$.

######  Remark 4.8.

The nilpotency class of $\mathfrak{g}$ is $\max\\{\alpha(i,j):i\prec j\\}+1$,
where $\alpha(i,j)$ is defined as in (1.1). To see this, note that the $l$-th
term in the lower central series of $\mathfrak{g}$ is the $\mathbb{Z}$-span of
the $e_{ij}$ where $\alpha(i,j)\geq l$. It is also straightforward to verify
that $\mathrm{Z}(\mathfrak{g})$ is the $\mathbb{Z}$-span of the $e_{ij}$ where
$(i,j)\in I_{\mathrm{ex}}$ (see [BMKS19, Lemma 4.2]). The same argument proves
that $\mathrm{Z}(\mathfrak{g}_{R})$ is a free $R$-module generated by the
$e_{ij}$ for $(i,j)\in I_{\mathrm{ex}}$.

By the isomorphism (4.1), elements of $\widehat{\mathfrak{g}_{R}}$ are of the
form $\psi_{\bf b}$ for a vector ${\bf b}:=(b_{ij})_{i\prec j}$ with entries
in $R$, where

(4.2) $\psi_{{\bf b}}\left(\sum_{i\prec
j}x_{ij}e_{ij}\right):=\psi\left(\sum_{i\prec j}b_{ij}x_{ij}\right).$

######  Definition 4.9.

We define the level of $b:=\tilde{b}+\mathfrak{p}^{d}\in R$ to be the smallest
$k$ such that $\tilde{b}\not\in\mathfrak{p}^{k+1}$, and we denote it by
$\mathbf{lev}(b)$.

######  Proposition 4.10.

Let $b_{ij}=\tilde{b}_{ij}+\mathfrak{p}^{d}$ for $i\prec j$ be elements of
$R$. Set ${\bf b}:=(b_{ij})_{i\prec j}$, and let $\rho_{{}_{\bf b}}$ be the
irreducible representation of $\mathscr{G}_{R}$ corresponding to the orbit of
$\psi_{\bf b}$ under the coadjoint action. Then $\dim\rho_{{}_{\bf b}}\geq
p^{f(d-\mathbf{lev}(b_{i_{1}j_{1}}))\alpha(i_{1},j_{1})}$ for all
$(i_{1},j_{1})\in I_{\mathrm{ex}}$.

###### Proof.

We use Theorem 4.7(b). Let $x=\sum_{i\prec j}x_{ij}e_{ij}\in\mathfrak{g}_{R}$
be an element of $\mathrm{Stab}_{\mathfrak{g}_{R}}(\psi_{{\bf b}})$. Then for
any $y=\sum_{i\prec j}y_{ij}e_{ij}\in\mathfrak{g}_{R}$ we have

$\begin{split}1=\psi_{{\bf b}}([x,y])&=\psi_{{\bf b}}\left(\sum_{i\prec
j,\,k\prec l}x_{ij}y_{kl}[e_{ij},e_{kl}]\right)\\\ &=\psi\left(\sum_{i\prec
j}\sum_{i\prec k\prec
j}b_{ij}(x_{ik}y_{kj}-x_{kj}y_{ik})\right)=\psi\left(\sum_{i\prec
j}\left(\sum_{k\prec i}b_{kj}x_{ki}-\sum_{j\prec
l}b_{il}x_{jl}\right)y_{ij}\right).\end{split}$

Since the $y_{ij}\in R$ are arbitrary and $\psi$ is primitive, the stabilizer
of $\psi_{{}_{\bf b}}$ is equal to the solution set of the system of linear
equations $\mathscr{L}:=\left\\{L_{ij}(x_{st})=0\ :\ i,j\in[n]\text{ and
}i\prec j\right\\}$, where

(4.3) $L_{ij}(x_{st}):=\sum_{k\prec i}b_{kj}x_{ki}-\sum_{j\prec
l}b_{il}x_{jl}.$

In particular since $(i_{1},j_{1})\in I_{\mathrm{ex}}$, the linear forms
$L_{i_{1}i}(x_{st})$ and $L_{jj_{1}}(x_{st})$, for $i_{1}\prec i\prec j_{1}$
and $i_{1}\prec j\prec j_{1}$, yield $2\alpha(i_{1},j_{1})$ linear equations
that can be written as

(4.4) $b_{i_{1}j_{1}}x_{ij_{1}}=-\sum_{i\prec k\neq
j_{1}}b_{i_{1}k}x_{ik}\quad\text{ and }\quad
b_{i_{1}j_{1}}x_{i_{1}j}=-\sum_{i_{1}\neq l\prec j}b_{lj_{1}}x_{lj}.$

The cardinality of the kernel of the map $R\to R,\,x\mapsto b_{i_{1}j_{1}}x$,
is $p^{f\mathbf{lev}(b_{i_{1}j_{1}})}$. It follows that for any choice of
values for the variables on the right hand side of (4.4), there exists at most
$p^{f\mathbf{lev}(b_{i_{1}j_{1}})}$ choices for each of the
$2\alpha(i_{1},j_{1})$ variables $x_{i_{1}i}$ and $x_{jj_{1}}$ such that the
corresponding equation in (4.4) is satisfied. Therefore the number of
solutions of the linear system $\mathscr{L}$ is at most

(4.5)
$p^{fd(|I|-2\alpha(i_{1},j_{1}))+2f\alpha(i_{1},j_{1})\mathbf{lev}(b_{i_{1}j_{1}})},$

where $I:=\\{(i,j):i\prec j\\}$. Using Theorem 4.7(b) it is now
straightforward to verify that $\dim\rho_{{}_{\bf b}}\geq
p^{f(d-\mathbf{lev}(b_{i_{1}j_{1}}))\alpha(i_{1},j_{1})}$. ∎

######  Lemma 4.11.

Let $b=\tilde{b}+\mathfrak{p}^{d}\in R$. Fix $(i_{1},j_{1})\in
I_{\mathrm{ex}}$ and let ${\bf b}:=(b_{ij})_{i\prec j}$ where
$b_{i_{1}j_{1}}=b$ and $b_{ij}=0$ for all other pairs $(i,j)$. Then the
dimension of the irreducible representation of $\mathscr{G}_{R}$ that
corresponds to the coadjoint orbit of $\psi_{\bf
b}\in\widehat{\mathfrak{g}_{R}}$ is equal to
$q^{(n-\mathbf{lev}(b))\alpha(i_{1},j_{1})}$.

###### Proof.

We write the linear system in the proof of Proposition 4.10 explicitly, and
determine the number of its solutions. In this case the only nonzero equations
in the linear system $\mathscr{L}$ are the $L_{ij_{1}}$ and the $L_{i_{1}j}$
for $i_{1}\prec i,j\prec j_{1}$, and they are of the form
$b_{i_{1}j_{1}}x_{i_{1}j}=0$ and $b_{i_{1}j_{1}}x_{ij_{1}}=0$. Thus the number
of solutions of the linear system is equal to (4.5). The assertion of the
lemma follows from Theorem 4.7(b). ∎

In the next lemma we prove that
$m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{R})\leq\sum_{\ell=0}^{e-1}\sum_{(i,j)\in
I_{\mathrm{ex}}}fp^{f(d-\ell)\alpha(i,j)}$.

######  Lemma 4.12.

The group $\mathscr{G}_{R}$ has a faithful representation of dimension

$\sum_{\ell=0}^{e-1}\sum_{(i,j)\in I_{\mathrm{ex}}}fp^{f(d-\ell)\alpha(i,j)}.$

###### Proof.

Let $\omega_{1},\dots,\omega_{f}$ be units in $\mathcal{O}$ such that
$\\{\omega_{1}+\mathfrak{p},\dots,\omega_{f}+\mathfrak{p}\\}$ forms a basis
for $\mathcal{O}/\mathfrak{p}$ over $\mathbb{F}_{p}$. Set
$\tilde{b}_{k\ell}:=\omega_{k}\varpi^{\ell}$ and
$b_{k\ell}:=\tilde{b}_{k\ell}+\mathfrak{p}^{e}$ for $1\leq k\leq f$ and
$0\leq\ell\leq e-1$. The ${b}_{k\ell}$ form a basis of the
$\mathbb{F}_{p}$-vector space $\mathcal{O}/\mathfrak{p}^{e}$. Since
$\mathrm{Z}(\mathscr{G}_{R})\cong\mathrm{Z}(\mathfrak{g}_{\mathrm{R}})$ as
abelian groups, from Remark 4.8 it follows that

(4.6)
$\widehat{\Omega}_{1}(\mathrm{Z}(\mathscr{G}_{R}))\cong\widehat{\Omega}_{1}(\mathrm{Z}(\mathfrak{g}_{\mathrm{R}}))=\bigoplus_{(i,j)\in
I_{\mathrm{ex}}}\widehat{\Omega}_{1}(\mathcal{O}/\mathfrak{p}^{d}).$

Note that
$\Omega_{1}(\mathcal{O}/\mathfrak{p}^{d})=\mathfrak{p}^{d-e}/\mathfrak{p}^{d}$,
so that Lemma 4.1 yields an isomorphism of abelian groups
$\mathcal{O}/\mathfrak{p}^{e}\cong\widehat{\Omega}_{1}(\mathcal{O}/\mathfrak{p}^{d}).$
Given $i_{\circ},j_{\circ}\in[n]$ such that $i_{\circ}\prec j_{\circ}$, we
define ${\bf b}({i_{\circ},j_{\circ},k,\ell})$, for $1\leq k\leq f$ and
$0\leq\ell\leq e-1$, to be the vector in $\bigoplus_{i\prec
j}\mathcal{O}/\mathfrak{p}^{d}$ with exactly one nonzero component, at the
$(i_{\circ},j_{\circ})$-position, equal to $\tilde{b}_{k\ell}$.

Since $\mathrm{Z}(\mathfrak{g}_{R})\cong\bigoplus_{(i,j)\in
I_{\mathrm{ex}}}\mathcal{O}/\mathfrak{p}^{d}$ (see Remark 4.8), from (4.6) it
follows that the restrictions of the characters
$\psi_{\mathbf{b}(i_{\circ},j_{\circ},k,l)}$ to $\mathrm{Z}(\mathscr{G}_{R})$
form an $\mathbb{F}_{p}$-basis of
$\widehat{\Omega}_{1}(\mathrm{Z}(\mathscr{G}_{R}))$. Let
$\rho_{\mathbf{b}(i_{\circ},j_{\circ},k,\ell)}$ denote the irreducible
representation of $\mathscr{G}_{R}$ that corresponds to the coadjoint orbit of
$\psi_{\mathbf{b}(i_{\circ},j_{\circ},k,\ell)}$ (see Theorem 4.7). Then by
Lemma 4.2, the representation

$\rho:=\bigoplus_{\begin{subarray}{c}1\leq k\leq f\\\ 0\leq\ell\leq
e-1\end{subarray}}\bigoplus_{(i,j)\in I_{\mathrm{ex}}}\rho_{{\bf
b}(i,j,k,\ell)}$

is faithful. The set $\\{b_{k\ell}\,:\,1\leq k\leq f,\ 0\leq\ell\leq e-1\\}$
contains exactly $f$ elements of any given level $\ell$ where $0\leq\ell\leq
e-1$, and by Lemma 4.11 we have
$\dim\rho_{\mathbf{b}(i,j,k,\ell)}=p^{f(d-\ell)\alpha(i,j)}$ for any $(i,j)\in
I_{\mathrm{ex}}$. These imply that $\dim\rho=\sum_{\ell=0}^{e-1}\sum_{(i,j)\in
I_{\mathrm{ex}}}fq^{(n-\ell)\alpha(i,j)}.$ ∎

###### Proof of the Theorem 1.5.

By Lemma 4.12 it suffices to prove that

$m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{R})\geq\sum_{\ell=0}^{e-1}\sum_{(i,j)\in
I_{\mathrm{ex}}}fp^{f(d-\ell)\alpha(i,j)}.$

Let $\rho$ be a faithful representation of $\mathscr{G}_{R}$ of dimension
$m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{R})$. By Lemma 4.2(ii) and Remark
4.3, $\rho$ is a direct sum of $N:=ef(\\#I_{\mathrm{ex}})$ irreducible
representations. Thus we can express $\rho$ as

$\rho=\bigoplus_{k=1}^{N}\rho_{\psi_{\mathbf{a}_{k}}},$

where each $\rho_{\psi_{\mathbf{a}_{k}}}$ is the irreducible representation of
$\mathscr{G}_{R}$ corresponding (according to Theorem 4.7) to the coadjoint
orbit of the character $\psi_{\mathbf{a}_{k}}$ of $\mathfrak{g}_{R}$ (and
$\mathbf{a}_{k}$ is a vector in $\bigoplus_{i\prec
j}\mathcal{O}/\mathfrak{p}^{d}$). Furthermore, by Theorem 4.7(c) and Lemma
4.2(ii) the restrictions of the $\psi_{\mathbf{a}_{k}}$ to
$\Omega_{1}(\mathrm{Z}(\mathscr{G}_{R}))$ form a
$\mathbb{Z}/p\mathbb{Z}$-basis of
$\widehat{\Omega}_{1}(\mathrm{Z}(\mathscr{G}_{R}))$. Since
$\Omega_{1}(\mathcal{O}/\mathfrak{p}^{d})=\mathfrak{p}^{d-e}/\mathfrak{p}^{d}$,
Lemma 4.1 implies that
$\widehat{\Omega}_{1}(\mathcal{O}/\mathfrak{p}^{d})\cong\mathcal{O}/\mathfrak{p}^{e}$
as $\mathbb{Z}/p\mathbb{Z}$-vector spaces, from which it follows that the
projections of the vectors $\mathbf{a}_{k}$ onto $\bigoplus_{(i,j)\in
I_{\mathrm{ex}}}\mathcal{O}/\mathfrak{p}^{e}$ form a
$\mathbb{Z}/p\mathbb{Z}$-basis. Using Lemma 4.5 we can partition the set
$\\{\mathbf{a}_{k}\\}_{k=1}^{N}$ into $\\#I_{\mathrm{ex}}$ sets of cadinality
$ef$, say $\\{\mathscr{A}_{(r,s)}\,:\,(r,s)\in I_{\mathrm{ex}}\\}$, such that
the $(r,s)$-components of the elements of $\mathscr{A}_{(r,s)}$ form a basis
of $\mathcal{O}/\mathfrak{p}^{e}$. To complete the proof, it suffices to
verify that

(4.7)
$\sum_{\mathbf{a}\in\mathscr{A}_{(r,s)}}\dim(\rho_{\psi_{\mathbf{a}}})\geq
f\sum_{\ell=0}^{e-1}p^{f(d-\ell)\alpha(r,s)}\quad\text{ for }(r,s)\in
I_{\mathrm{ex}}.$

Fix $(r,s)\in I_{\mathrm{ex}}$, and denote the $(r,s)$-component of any
$\mathbf{a}\in\mathscr{A}_{(r,s)}$ by $\mathbf{a}_{(r,s)}$. For $0\leq\ell\leq
e-1$ set

$N_{\ell}:=\\#\\{\mathbf{a}\in\mathscr{A}_{(r,s)}\,:\,\mathbf{lev}(\mathbf{a}_{(r,s)})=\ell\\},$

where $\mathbf{lev}(\cdot)$ is the level as in Definition 4.9. From
Proposition 4.10 it follows that if $\mathbf{lev}(\mathbf{a}_{(r,s)})=\ell$
then $\dim(\rho_{\psi_{\mathbf{a}}})\geq p^{f(d-\ell)\alpha(r,s)}$. This
implies that

(4.8)
$\sum_{\mathbf{a}\in\mathscr{A}_{(r,s)}}\dim(\rho_{\psi_{\mathbf{a}}})\geq\sum_{\ell=0}^{e-1}N_{\ell}p^{f(d-\ell)\alpha(r,s)}.$

Note that $\sum_{\ell=0}^{e-1}N_{\ell}=ef$, and

$\sum_{k=\ell}^{e-1}N_{k}\leq\dim_{\mathbb{Z}/p\mathbb{Z}}(\mathfrak{p}^{\ell}/\mathfrak{p}^{e})=(e-\ell)f.$

Thus by Lemma 4.6 for $m:=e$, $a_{\ell}:=N_{\ell}/f$ and
$x_{\ell}:=p^{f(d-\ell)\alpha(r,s)}$ we obtain

(4.9) $\sum_{\ell=0}^{e-1}N_{\ell}p^{f(d-\ell)\alpha(r,s)}\geq
f\sum_{\ell=0}^{e-1}p^{f(d-\ell)\alpha(r,s)}.$

Inequality (4.7) now follows from (4.8) and (4.9). ∎

##  5\. The faithful dimension in the case $\mathfrak{g}:=\mathfrak{m}_{n,c}$

In this section we prove Theorem 1.6. We begin by introducing some notation.
Recall that $[n]:=\\{1,\ldots,n\\}$ for $n\in\mathbb{N}$. For $k\geq 1$, the
set of sequences

$\mathbf{i}=(i_{1},\dots,i_{k}),\quad i_{1},\dots,i_{k}\in[n]$

is denoted by $\mathscr{A}(n,k)$. We say that $\mathbf{i}$ is _decreasing_ if
$i_{1}\geq\dots\geq i_{k}$. The set of decreasing sequences in
$\mathscr{A}(n,k)$ is denoted by $\mathscr{D}(n,k)$. There is an obvious
sorting map

$\bar{\,}:\mathscr{A}(n,k)\to\mathscr{D}(n,k).$

For example $\overline{(5,4,4,2,3,1)}=(5,4,4,3,2,1)$. We say that
$\mathbf{i}\in\mathscr{A}(n,k)$ is a _Hall sequence_ if the initial sequence
$(i_{1},\dots,i_{k-1})$ is decreasing and $i_{k-1}<i_{k}$ (by convention,
every element of $\mathscr{A}(n,1)$ is a Hall sequence). The subset of Hall
sequences in $\mathscr{A}(n,k)$ is denoted by $\mathscr{H}(n,k)$.

######  Lemma 5.1.

  1. (a)

For $n,k\geq 1$ we have $\displaystyle\\#\mathscr{D}(n,k)={n+k-1\choose k}.$

  2. (b)

For $n\geq 2$ and $k\geq 2$ we have

$\\#\mathscr{H}(n,k)=\sum_{m=1}^{n-1}m{k+m-2\choose k-2}=(k-1){k+n-2\choose
k}.$

###### Proof.

For part (a), note that cardinality of $\mathscr{D}(n,k)$ is equal to the
number of solutions of the equation $t_{1}+\cdots+t_{n}=k$ in non-negative
integers, where $t_{j}=\\#\\{\ell\in[k]:i_{\ell}=j\\}$.

For part (b), fix $n,k\geq 2$ and for $m\in[n-1]$ denote by $A_{m}$ the subset
of $\mathscr{H}(n,k)$ consisting of those sequences
$(i_{1},\dots,i_{k})\in\mathscr{H}(n,k)$ such that $i_{k-1}=m$. Note that
since $i_{k-1}<i_{k}\leq n$, we must have $i_{k-1}\in[n-1]$. Once $m$ is
fixed, there are $n-m$ options left for $i_{k}$. Moreover, the sequence
$(i_{1},\dots,i_{k-1})$ can be any decreasing sequence in $[n]$ for which
$i_{k-1}=m$. Using the map
$(i_{1},\dots,i_{k-2})\mapsto(i_{1}-m+1,\dots,i_{k-2}-m+1)$ these sequences
stand in one-to-one correspondence with elements of the set
$\mathscr{D}(n-m+1,k-2)$, whose cardinality by part (a) is equal to
${n-m+k-2\choose k-2}$. Since the two choices can be made independently of
each other we have

$\\#\mathscr{H}(n,k)=\sum_{m=1}^{n-1}(n-m){n-m+k-2\choose k-2}.$

Replacing $m$ by $n-m$ in this expression yields the first equality in part
(b). For the second equality note that

$\displaystyle\sum_{m=1}^{n-1}m{k+m-2\choose
k-2}=\sum_{m=1}^{n-1}(k-1){k+m-2\choose
k-1}=(k-1)\sum_{m=1}^{n-1}{k+m-2\choose k-1},$

and $\sum_{m=1}^{n-1}{k+m-2\choose k-1}$ counts the number of subsets of size
$k$ of $[k+n-2]$, where the summand ${k+m-2\choose k-1}$ corresponds to
subsets of $[k+n-2]$ with maximum equal to $k+m-1$. ∎

Since $\mathfrak{f}_{n,2}=\mathfrak{m}_{n,2}$, the case $c=2$ of Theorem 1.6
follows from [BMKS16, Theorem 2.13]. Thus from now on we assume that $c\geq
3$. We denote the standard generators of $\mathfrak{m}_{n,c}$ by
$X_{n}:=\\{\mathsf{v}_{i}:i\in[n]\\}$. For each $k$-tuple
$\mathbf{i}=(i_{1},\dots,i_{k})\in\mathscr{A}(n,k)$, write

$\mathsf{v}_{\mathbf{i}}:=[\mathsf{v}_{i_{1}},[\mathsf{v}_{i_{2}},\dots,[\mathsf{v}_{i_{k-1}},\mathsf{v}_{i_{k}}]\dots].$

Set $\mathscr{H}^{\leq c-1}:=\bigcup_{j=1}^{c-1}\mathscr{H}(n,j)$ and
$\mathscr{H}^{\geq 2}:=\bigcup_{j=2}^{c}\mathscr{H}(n,j)$.

######  Lemma 5.2.

Let $n$ and $c$ be as above. For each $1\leq j\leq c$, the set
$\\{\mathsf{v}_{\mathbf{i}}:\mathbf{i}\in\mathscr{H}(n,j)\\}$ is a basis for
the vector space $\mathfrak{m}_{n,c}^{j}/\mathfrak{m}_{n,c}^{j+1}$. In
particular,

  1. (a)

$\\{\mathsf{v}_{\mathbf{i}}:\mathbf{i}\in\mathscr{H}(n,c)\\}$ forms a basis
for $\mathrm{Z}(\mathfrak{m}_{n,c})$.

  2. (b)

Elements in
$\left\\{\mathsf{v}_{\mathbf{i}}\,:\,\mathbf{i}\in\mathscr{H}^{\leq
c-1}\right\\}$ represent a basis of
$\mathfrak{m}_{n,c}/\mathrm{Z}(\mathfrak{m}_{n,c})$.

  3. (c)

Elements in
$\left\\{\mathsf{v}_{\mathbf{i}}\,:\,\mathbf{i}\in\mathscr{H}^{\geq
2}\right\\}$ form a basis for $[\mathfrak{m}_{n,c},\mathfrak{m}_{n,c}]$.

###### Proof.

This is well known. See [ABRdS18] for a proof. ∎

In the sequel, we will carry out a close study of the commutator matrix of
$\mathfrak{m}_{n,c}$ with respect to the bases for
$\mathfrak{m}_{n,c}/\mathrm{Z}(\mathfrak{m}_{n,c})$ and
$[\mathfrak{m}_{n,c},\mathfrak{m}_{n,c}]$ described in Lemma 5.2. For
$\mathbf{i},\mathbf{j}\in\mathscr{H}^{\leq c-1}$ let
$\lambda_{\mathbf{i}\mathbf{j}}^{\mathbf{k}}\in\mathbb{Z}$ be the structural
constants defined by

$[\mathsf{v}_{\mathbf{i}},\mathsf{v}_{\mathbf{j}}]=\sum_{k\in\mathscr{H}^{\geq
2}}\lambda^{\mathbf{k}}_{\mathbf{i}\mathbf{j}}\mathsf{v}_{\mathbf{k}}.$

Let $\mathbf{T}:=(T_{\mathbf{i}})_{i\in\mathscr{H}^{\geq 2}}$ be a vector of
variables, and for $\mathbf{i},\mathbf{j}\in H^{\leq c-1}$, define the linear
forms

$\Lambda_{\mathbf{i}\mathbf{j}}(\mathbf{T}):=\sum_{\mathbf{k}\in\mathscr{H}^{\geq
2}}\lambda_{\mathbf{i}\mathbf{j}}^{\mathbf{k}}T_{\mathbf{k}}.$

The commutator matrix of $\mathfrak{g}$ relative to the chosen bases is the
$N\times N$ skew-symmetric matrix of linear forms given by

$F_{\mathfrak{g}}(\mathbf{T}):=[\Lambda_{\mathbf{i}\mathbf{j}}(\mathbf{T})]_{\mathbf{i},\mathbf{j}\in\mathscr{H}^{\leq
c-1}},\quad\text{where $N:=\\#\mathscr{H}^{\geq 2}$.}$

For any $(i_{1},\ldots,i_{c})\in\mathscr{H}(n,c)$, set $\mathbf{i}:=(i_{1})$
and $\mathbf{j}:=(i_{2},\ldots,i_{c})\in\mathscr{H}(n,c-1)$. Then the
$(\mathbf{i},\mathbf{j})$ entry of $F_{\mathfrak{g}}(\mathbf{T})$ is equal to
$T_{(i_{1},\ldots,i_{c})}$. Further, for any $\mathbf{i}:=(i_{1},\dots,i_{k})$
and $\mathbf{j}:=(j_{1},\dots,j_{l})$, we have
$\Lambda_{\mathbf{i}\mathbf{j}}=0$ unless $\min(k,l)=1$.

######  Lemma 5.3.

Let $\mathbf{i}:=(i_{1})\in\mathscr{H}(n,1)$ and
$\mathbf{j}:=(i_{2},\dots,i_{c})\in\mathscr{H}(n,c-1)$. Then

${\Lambda}_{\mathbf{i}\mathbf{j}}(\mathbf{T})=\begin{cases}T_{(\overline{i_{1},\dots,i_{c-1}},i_{c})}&\text{
if }\ i_{1}\geq i_{c-1},\\\
T_{(i_{2},\dots,i_{c-1},i_{1},i_{c})}-T_{(\overline{i_{2},\dots,i_{c-2},i_{c}},i_{1},i_{c-1})}&\text{
if }\ i_{1}<i_{c-1}.\end{cases}$

###### Proof.

First observe that for $a,a^{\prime}\in\mathfrak{m}_{n,c}$ and
$a^{\prime\prime}\in[\mathfrak{m}_{n,c},\mathfrak{m}_{n,c}]$ we have

(5.1) $[a,[a^{\prime},a^{\prime\prime}]]=[a^{\prime},[a,a^{\prime\prime}]],$

because
$[a,[a^{\prime},a^{\prime\prime}]]=-[a^{\prime},[a^{\prime\prime},a]]-[a^{\prime\prime},[a,a^{\prime}]]=[a^{\prime},[a,a^{\prime\prime}]],$
where the second equality follows from the fact that $\mathfrak{m}_{n,c}$ is
metabelian and $a^{\prime\prime}$ is a commutator.

Suppose that $i_{1}\geq i_{c-1}$, and let $2\leq r\leq c-1$ be the smallest
integer with $i_{1}\geq i_{r}$. By the assumption such an $r$ exists. If
$c=3$, then $i_{1}\geq i_{2}$ and the claim is immediate. For $c>3$, set
$\eta_{k}:=[\mathsf{v}_{i_{k}},[\mathsf{v}_{i_{k+1}},\dots,[\mathsf{v}_{i_{c-1}},\mathsf{v}_{i_{c}}]\dots]$.
It follows from (5.1) that

$[\mathsf{v}_{i_{1}},[\mathsf{v}_{i_{2}},\dots,[\mathsf{v}_{i_{c-1}},\mathsf{v}_{i_{k}}]\dots]=[\mathsf{v}_{i_{1}},[\mathsf{v}_{i_{2}},\eta_{3}]]=[\mathsf{v}_{i_{2}},[\mathsf{v}_{i_{1}},\eta_{3}]].$

By repeating this process, we can keep swapping $\mathsf{v}_{i_{1}}$ with
subsequent terms $\mathsf{v}_{i_{3}},\dots$ until we arrive at
$\mathsf{v}_{i_{r}}$ where the process is terminated. Note that

$(i_{2},\dots,i_{r-1},i_{1},i_{r},\dots,i_{c-1})=\overline{(i_{1},\dots,i_{c})}.$

This proves the lemma for the case $i_{1}\geq i_{c-1}$.

Let us now consider the case $i_{1}<i_{c-1}$. In this case, the process
described above can be continued all the way until $\mathsf{v}_{i_{1}}$
arrives in the innermost commutator, that is,

$[\mathsf{v}_{i_{1}},[\mathsf{v}_{i_{2}},\dots,[\mathsf{v}_{i_{c-1}},\mathsf{v}_{i_{k}}]\dots]=[\mathsf{v}_{i_{2}},[\mathsf{v}_{i_{3}},\dots,[\mathsf{v}_{i_{c-2}},[\mathsf{v}_{i_{1}},[\mathsf{v}_{i_{c-1}},\mathsf{v}_{i_{c}}]]\dots].$

We can now use the Jacobi identity to write
$[\mathsf{v}_{i_{1}},[\mathsf{v}_{i_{c-1}},\mathsf{v}_{i_{c}}]]=[\mathsf{v}_{i_{c-1}},[\mathsf{v}_{i_{1}},\mathsf{v}_{i_{c}}]]-[\mathsf{v}_{i_{c}},[\mathsf{v}_{i_{1}},\mathsf{v}_{i_{c-1}}]]$.
Similar to the previous case, by a repeated application of (5.1) we can move
$\mathsf{v}_{i_{c}}$ inside to the location so that the resulting sequence is
decreasing. This proves the claim. ∎

From now on we set $\widetilde{F}_{\mathfrak{g}}(\mathbf{T})$ to be equal to
the submatrix of ${F}_{\mathfrak{g}}(\mathbf{T})$ that lies in the
intersection of rows $\mathbf{i}\in\mathscr{H}(n,1)$ and columns
$\mathbf{j}\in\mathscr{H}(n,c-1)$. We remark that the only variables that
appear in the entries of $\widetilde{F}_{\mathfrak{g}}(\mathbf{T})$ are the
$T_{\mathbf{i}}$ for $\mathbf{i}\in\mathscr{H}(n,c)$.

######  Lemma 5.4.

Let $K$ be any field. Let $\alpha_{1},\dots,\alpha_{n}\in K$ and
$\lambda_{0},\lambda_{1},\dots,\lambda_{n-1}\in K$. For each
$\mathbf{i}:=(i_{1},\ldots,i_{c})\in\mathscr{H}(n,c)$ define

$T_{\mathbf{i}}:=\displaystyle\left(\prod_{k=1}^{c-2}\alpha_{i_{k}}\right)\begin{vmatrix}\alpha_{i_{c-1}}&\alpha_{i_{c}}\\\
\lambda_{i_{c-1}-1}&\lambda_{i_{c}-1}\\\ \end{vmatrix}.$

Set $\mathbf{T}:=(T_{\mathbf{i}})_{\mathbf{i}\in\mathscr{H}(n,c)}$. Then the
matrix $\widetilde{F}_{\mathfrak{g}}(\mathbf{T})$ has rank at most $1$.

###### Proof.

First we show that for every $\mathbf{i}:=(i_{1})\in\mathscr{H}(n,1)$ and
$\mathbf{j}:=(i_{2},\dots,i_{c})\in\mathscr{H}(n,c-1)$,

(5.2)
$\widetilde{F}_{\mathfrak{g}}(\mathbf{T})_{\mathbf{i}\mathbf{j}}=\left(\prod_{k=1}^{c-2}\alpha_{i_{k}}\right)\begin{vmatrix}\alpha_{i_{c-1}}&\alpha_{i_{c}}\\\
\lambda_{i_{c-1}-1}&\lambda_{i_{c}-1}\\\ \end{vmatrix}.$

We consider two different cases. If $i_{1}\geq i_{c-1}$, then it follows from
Lemma 5.3 that

${\Lambda}_{\mathbf{i}\mathbf{j}}(\mathbf{T})=T_{(\overline{i_{1},\dots,i_{c-1}},i_{c})}.$

Since the sequence $(i_{2},\dots,i_{c-1})$ is decreasing, and $i_{1}\geq
i_{c-1}$, it follows that $i_{c-1}$ is also the least element of
$(\overline{i_{1},\dots,i_{c-1}},i_{c})$ and hence the first $c-2$ terms of
$(\overline{i_{1},\dots,i_{c-1}},i_{c})$ are precisely $i_{1},\dots,i_{c-2}$,
perhaps in a different order. Thus (5.2) follows from the fact that the last
two terms of the sequence are $i_{c-1}$ and $i_{c}$, in the same order.

If $i_{1}<i_{c-1}$, then it follows from Lemma 5.3 that

${\Lambda}_{\mathbf{i}\mathbf{j}}(\mathbf{T})=T_{(i_{2},\dots,i_{c-1},i_{1},i_{c})}-T_{(\overline{i_{2},\dots,i_{c-2},i_{c}},i_{1},i_{c-1})}.$

Since both $(i_{2},\dots,i_{c-1},i_{1},i_{c})$ and
$(\overline{i_{2},\dots,i_{c-2},i_{c}},i_{1},i_{c-1})$ are Hall sequences, we
have

(5.3)
$\begin{split}{\Lambda}_{\mathbf{i}\mathbf{j}}(\mathbf{T})&=\alpha_{i_{c-1}}\cdot\left(\prod_{k=2}^{c-2}\alpha_{i_{k}}\right)\cdot\begin{vmatrix}\alpha_{i_{1}}&\alpha_{i_{c}}\\\
\lambda_{i_{1}-1}&\lambda_{i_{c}-1}\\\
\end{vmatrix}-\alpha_{i_{c}}\cdot\left(\prod_{k=2}^{c-2}\alpha_{i_{k}}\right)\cdot\begin{vmatrix}\alpha_{i_{1}}&\alpha_{i_{c-1}}\\\
\lambda_{i_{1}-1}&\lambda_{i_{c-1}-1}\\\ \end{vmatrix}\\\
&=\left(\prod_{k=2}^{c-2}\alpha_{i_{k}}\right)\cdot\left(\alpha_{i_{c-1}}\cdot\begin{vmatrix}\alpha_{i_{1}}&\alpha_{i_{c}}\\\
\lambda_{i_{1}-1}&\lambda_{i_{c}-1}\\\
\end{vmatrix}-\alpha_{i_{c}}\cdot\begin{vmatrix}\alpha_{i_{1}}&\alpha_{i_{c-1}}\\\
\lambda_{i_{1}-1}&\lambda_{i_{c-1}-1}\\\ \end{vmatrix}\right)\end{split}$

After applying the elementary identity

$r^{\prime}\begin{vmatrix}q&r\\\ s&t\\\
\end{vmatrix}-r\begin{vmatrix}q&r^{\prime}\\\ s&t^{\prime}\\\
\end{vmatrix}=q\begin{vmatrix}r^{\prime}&r\\\ t^{\prime}&t\\\ \end{vmatrix}$

to the expression on the right hand side and merging the prefactor
$q:=\alpha_{i_{1}}$ with the product $\prod_{k=2}^{c-2}\alpha_{i_{k}}$ we
obtain (5.2). From (5.2) it follows that for
$\mathbf{i}:=(i_{1})\in\mathscr{H}(n,1)$ and
$\mathbf{j}:=(i_{2},\ldots,i_{c})\in\mathscr{H}(n,c-1)$, we have
$\widetilde{F}_{\mathfrak{g}}(\mathbf{T})_{\mathbf{i}\mathbf{j}}=\alpha_{i_{1}}\beta_{\mathbf{j}}$
for some $\beta_{\mathbf{j}}\in K$. Thus
$\widetilde{F}_{\mathfrak{g}}(\mathbf{T})$ is expressible as the product of
the column vector $(\alpha_{i})_{i\in\mathscr{H}(n,1)}$ and the row vector
$(\beta_{\mathbf{j}})_{\mathbf{j}\in\mathscr{H}(n,c-1)}$, so that
$\mathrm{rk}(\widetilde{F}_{\mathfrak{g}}(\mathbf{T}))\leq 1$. ∎

######  Lemma 5.5.

Suppose $r,m,\delta\geq 1$. Let $K$ be any field.

  1. (a)

Suppose $f\in K[x_{1},\dots,x_{m}]$ is a polynomial in $m$ variables over $K$
such that the degree of $f$ as a polynomial in $x_{i}$ is at most $\delta_{i}$
for all $1\leq i\leq m$. Assume that $\\#K\geq
1+\max(\delta_{1},\dots,\delta_{m})$, and $f(x_{1},\dots,x_{m})=0$ for all
$(x_{1},\dots,x_{m})\in K^{m}$. Then $f$ is the zero polynomial.

  2. (b)

Suppose $f_{1},\dots,f_{r}\in K[x_{1},\dots,x_{m}]$, and the degree of each
one of $f_{1},\dots,f_{r}$ in the variable $x_{i}$ is at most $\delta_{i}$ for
$1\leq i\leq m$. Let $f:K^{m}\to K^{r}$ be the polynomial map defined by
$f=(f_{1},\dots,f_{r})$. Assume that the $K$-subspace of
$K[x_{1},\dots,x_{m}]$ spanned by $f_{1},\dots,f_{r}$ has dimension at least
$\delta$. If $\\#K\geq 1+\max(\delta_{1},\dots,\delta_{m})$, then there exist
$u_{1},\dots,u_{\delta}\in K^{m}$ for which $f(u_{1}),\dots,f(u_{\delta})$ are
linearly independent vectors in $K^{r}$.

###### Proof.

Part (a) follows by induction on $m$. For $m=1$, the statement is clear.
Assuming that (a) holds for $m-1$, write
$f(x_{1},\dots,x_{m}):=\sum_{i=0}^{\delta_{m}}c_{i}x_{m}^{i}$ where the
$c_{i}\in K[x_{1},\ldots,x_{m-1}]$. Fix $(x_{1},\dots,x_{m-1})\in K^{m-1}$ and
consider the one-variable polynomial

$g(x_{m}):=f(x_{1},\dots,x_{m-1},x_{m}).$

Since $g(x_{m})=0$ for all $x_{m}\in K$, and $\\#K>\deg g$, it follows that
all the coefficients of $g$ vanish. The claim follows by the induction
hypothesis applied to the $c_{i}$.

For (b) by choosing a maximal linearly independent subset of
$f_{1},\dots,f_{r}$, we can assume that $\delta=r$. Thus the goal is to find
$u_{1},\dots,u_{r}\in K^{m}$ for which $f(u_{1}),\dots,f(u_{r})$ are linearly
independent vectors in $K^{r}$. If this is not the case, then the image of $f$
must lie in a proper subspace of $K^{r}$. This implies that there exists a
non-zero vector $(c_{1},\dots,c_{r})\in K^{r}$ such that
$c_{1}f_{1}(u)+\cdots+c_{r}f_{r}(u)=0$ for all $u\in K^{m}$. By part (a), this
implies that the polynomial $c_{1}f_{1}+\cdots+c_{r}f_{r}$ is the zero
polynomial, which is a contradiction. ∎

We are now ready to prove Theorem 1.6 in the case where $R$ is a finite field.
In this case, we need to show that for $p$ sufficiently large we have

(5.4)
$m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{\mathbb{F}_{q}})=(c-1){n+c-2\choose
c}fq\qquad\text{ for all }f\geq 1,$

where $q:=p^{f}$. We will use Lemma 5.4, as follows. We set $\lambda_{0}:=0$
and $\lambda_{1}:=1$. Then the entries of $\widetilde{F}_{\mathfrak{g}}$ are
polynomials in variables $\alpha_{1},\dots,\alpha_{n}$ and
$\lambda_{2},\dots,\lambda_{n-1}$. Clearly $\widetilde{F}_{\mathfrak{g}}$
takes values in the vector space of matrices of size
$n\times\\#\mathscr{H}(n,c-1)$ with entries in
$\mathbb{F}_{q}[\alpha_{1},\ldots,\alpha_{n},\lambda_{2},\ldots,\lambda_{n-1}]$.
Let $\mathcal{V}$ be the subspace of
$\mathbb{F}_{q}[\alpha_{1},\ldots,\alpha_{n},\lambda_{2},\ldots,\lambda_{n-1}]$
that is spanned by the matrix entries of $\widetilde{F}_{\mathfrak{g}}$. First
we show that the set $\\{T_{\mathbf{i}}\\}_{\mathbf{i}\in\mathscr{H}(n,c)}$ is
linearly independent in
$\mathbb{F}_{q}[\alpha_{1},\ldots,\alpha_{n},\lambda_{2},\ldots,\lambda_{n-1}]$.
Suppose that

$\sum_{\mathbf{i}\in\mathscr{H}(n,c)}c_{\mathbf{i}}\cdot\left(\prod_{k=1}^{c-2}\alpha_{i_{k}}\right)\cdot\begin{vmatrix}\alpha_{i_{c-1}}&\alpha_{i_{c}}\\\
\lambda_{i_{c-1}-1}&\lambda_{i_{c}-1}\\\ \end{vmatrix}=0,$

for some coefficients $c_{\mathbf{i}}\in\mathbb{F}_{q}$. We will show that all
the coefficients must be zero. For each such linear dependence, define $r$ to
be the largest value of $i_{c}$ for which there exists
$\mathbf{i}=(i_{1},\dots,i_{c})\in\mathscr{H}(n,c)$ such that
$c_{\mathbf{i}}\neq 0$. Note that clearly $r\geq 2$. First suppose that $r=2$.
This implies that $i_{c-1}=1$ and hence the linear dependence equation
simplifies to

$\sum_{(i_{1},\dots,i_{c-2})\in\mathscr{D}(n,c-2)}c_{(i_{1},\dots,i_{c-2},1,2)}\prod_{k=1}^{c-1}\alpha_{i_{k}}=0.$

Since the map $g\mapsto(\deg_{\alpha_{1}}(g),\dots,\deg_{\alpha_{n}}(g))$ is
injective on the involved monomials, there is no possibility of cancellation
and the claim follows. Suppose that $r\geq 3$. This implies that there are
some monomials involving $\lambda_{r-1}$ and that there are no terms involving
$\lambda_{j}$ for $j>r-1$. It is also clear that the only terms involving
$\lambda_{r-1}$ correspond to those sequences $(i_{1},\dots,i_{c})$ for which
$i_{c}=r$ and hence these terms are of the form

$\prod_{k=1}^{c-2}\alpha_{i_{k}}(\alpha_{i_{c-1}}\lambda_{r-1}-\alpha_{r}\lambda_{i_{c-1}-1}).$

By considering only the terms involving $\lambda_{r-1}$ we obtain

$\sum_{(i_{1},\dots,i_{c-1})\in\mathscr{D}(n-1),i_{c-1}<r}c_{\mathbf{i}}\prod_{k=1}^{c-1}\alpha_{i_{k}}=0.$

As in the previous case the monomials corresponding to different $\mathbf{i}$
are distinct, and hence this can only happen if $c_{\mathbf{i}}=0$ for all
such terms. This contradicts the choice of $r$.

From linear independence of
$\\{T_{\mathbf{i}}\\}_{\mathbf{i}\in\mathscr{H}(n,c)}$ and Lemmas 5.5 and 5.4
it follows that for $q\geq c$ one can find $\\#\mathscr{H}(n,c)$ linearly
independent vectors
$\mathbf{T}^{\prime}:=(T_{i})_{\mathbf{i}\in\mathscr{H}(n,c)}$ with entries in
$\mathbb{F}_{q}$ such that
$\mathrm{rk}(\widetilde{F}_{\mathfrak{g}}(\mathbf{T}^{\prime}))=1$. We can
extend each such vector $\mathbf{T}^{\prime}$ to a vector
$\mathbf{T}:=(T_{\mathbf{i}})_{\mathbf{i}\in\mathscr{H}^{\geq 2}}$ by setting
$T_{\mathbf{i}}=0$ for $\mathbf{i}\in\mathscr{H}(n,k)$, where $2\leq k\leq
c-1$. For the latter vectors $\mathbf{T}$ we have
$\mathrm{rk}_{\mathbb{F}_{q}}(F_{\mathfrak{g}}(\mathbf{T}))=2$. Since
$F_{\mathfrak{g}}$ is skew symmetric and each variable $T_{\mathbf{i}}$ where
$\mathbf{i}:=(i_{1},\ldots,i_{k})$ appears in the
$(\mathbf{i}^{\prime},\mathbf{j})$-entry of $F_{\mathfrak{g}}(\mathbf{T})$,
where $\mathbf{i}^{\prime}:=(i_{1})$ and $\mathbf{j}:=(i_{2},\ldots,i_{k})$,
we have $\mathrm{rk}_{\mathbb{F}_{q}}(F_{\mathfrak{g}}(\mathbf{T}))\geq 2$
whenever $\mathbf{T}\neq 0$. In view of Theorem 2.5 this proves the assertion
in the case that $R$ is a finite field.

Let us now consider the case where $R=\mathcal{O}/\mathfrak{p}^{d}$. Using
Theorem 1.3 and equality (5.4) which was just proved, the upper bound follows
immediately. Let us now turn to establishing the lower bound. We start with
the following lemma. Recall the notion of level of an element of
$\mathcal{O}/\mathfrak{p}^{d}$ from Definition 4.9.

######  Lemma 5.6.

Let $B$ be an $r\times r$ skew-symmetric matrix with entries in
$\mathcal{O}/\mathfrak{p}^{d}$. Suppose $B$ has an entry of level $\ell$. Then

$\\#\ker B\leq p^{fd(r-2)+2f\ell}.$

###### Proof.

Write $B=(B_{ij})_{1\leq i,j\leq r}$. If $\ell=d$, there is nothing to prove.
Next assume that the entry in question is non-zero, and hence non-diagonal.
After possibly permuting rows and columns, we can assume that
$\mathbf{lev}(B_{12})=\ell$. Consider a vector
$(x_{1},\ldots,x_{r})\in(\mathcal{O}/\mathfrak{p}^{d})^{\oplus r}$ in $\ker
B$. Then the following equations hold:

$B_{12}x_{2}=-\sum_{j=3}^{r}B_{1j}x_{j}\quad\text{and}\quad
B_{21}x_{1}=-\sum_{j=3}^{r}B_{2j}x_{j}.$

Once the values of the $x_{j}$ for $3\leq j\leq r$ are set, the number of
choices for each one of $x_{1}$ and $x_{2}$ is $p^{f\ell}$. Since the number
of choices for $(x_{3},\dots,x_{r})$ is $p^{fd(r-2)}$, the claim follows
immediately. ∎

In order to avoid confusion with parameters $n$ and $c$ of
$\mathfrak{m}_{n,c}$, henceforth we will denote the parameters
$l_{1},l_{2},n,m$ that were associated to $\mathfrak{g}$ in Section 2 by
$\underline{l}_{1},\underline{l}_{2},\underline{n},\underline{m}$. Thus in
particular we have

$\underline{m}=\\#\mathscr{H}^{\geq
2}\quad,\quad\underline{l}_{1}=\\#\mathscr{H}(n,c)=(c-1){n+c-2\choose
c}\quad\text{and}\quad\underline{l}_{2}=0.$

Moreover, the commutator matrix $F_{\mathfrak{g}}(\mathbf{T})$ is an
$\underline{n}\times\underline{n}$ matrix with
$\underline{n}=\mathrm{rk}_{\mathbb{Z}}(\mathfrak{m}_{n,c})-\underline{l}_{1}$.

######  Lemma 5.7.

Let $\mathbf{a}:=(\mathbf{a}(\mathbf{i}))_{\mathbf{i}\in\mathscr{H}^{\geq
2}}\in(\mathcal{O}/\mathfrak{p}^{d})^{\oplus\underline{m}}$. Then

$\\#\ker_{\mathcal{O}/\mathfrak{p}^{d}}F_{\mathfrak{g}}(\mathbf{a})\leq\min_{\mathbf{i}\in\mathscr{H}(n,c)}p^{fd(\underline{n}-2)+2f\mathbf{lev}(\mathbf{a}(\mathbf{i}))}.$

###### Proof.

By Lemma 5.6, it suffices to find an entry of the matrix
$F_{\mathfrak{g}}(\mathbf{a})$ equal to $\mathbf{a}(\mathbf{i})$, for any
given $\mathbf{i}\in\mathscr{H}(n,c)$. Write $\mathbf{i}:=(i_{1},\dots,i_{c})$
and define $\mathbf{i}^{\prime}:=(i_{1})$ and
$\mathbf{j}:=(i_{2},\dots,i_{c})$. It follows from Lemma 5.3 that the entry
$(\mathbf{i}^{\prime},\mathbf{j})$ of $F_{\mathfrak{g}}(\mathbf{a})$ is equal
to $\mathbf{a}(\mathbf{i})$. ∎

Let
$S=\\{\mathbf{a}_{1},\ldots,\mathbf{a}_{fe\underline{l}_{1}}\\}\subseteq(\mathcal{O}/\mathfrak{p}^{d})^{\oplus\underline{m}}$
be a set of vectors that corresponds to
$m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{R})$ in Proposition 3.1 (thus we
assume that $p$ is sufficiently large accordingly). Let $\pi_{\mathbf{i}}$ for
$\mathbf{i}\in\mathscr{H}(n,c)$ denote the natural projection

$\bigoplus_{\mathbf{i}\in\mathscr{H}(n,c)}(\mathcal{O}/\mathfrak{p}^{e})\to\mathcal{O}/\mathfrak{p}^{e},$

that maps $\mathbf{a}$ to $\mathbf{a}(\mathbf{i})$. Using Lemma 4.5, we
partition $S$ into sets $S_{\mathbf{i}}$, each of cardinality $ef$, such that
$\pi_{\mathbf{i}}({\mathsf{proj}}(S_{\mathbf{i}}))$ is a basis for the
$\mathbb{F}_{p}$-vecror space $\mathcal{O}/\mathfrak{p}^{e}$. By Lemma 5.7 we
have

$\displaystyle m_{{}_{\mathrm{faithful}}}(\mathscr{G}_{R})$
$\displaystyle=\sum_{i=1}^{fe\underline{l}_{1}}\left(\frac{p^{fd\underline{n}}}{\\#\ker_{\mathcal{O}/\mathfrak{p}^{d}}(F(\mathbf{a}_{i}))}\right)^{\frac{1}{2}}$
(5.5) $\displaystyle=\sum_{\mathbf{i}\in\mathscr{H}(n,c)}\sum_{\mathbf{a}\in
S_{\mathbf{i}}}\left(\frac{p^{fd\underline{n}}}{\\#\ker_{\mathcal{O}/\mathfrak{p}^{d}}(F(\mathbf{a}))}\right)^{\frac{1}{2}}\geq\sum_{\mathbf{i}\in\mathscr{H}(n,c)}\sum_{\mathbf{a}\in
S_{\mathbf{i}}}p^{f(d-\mathbf{lev}(\mathbf{a}(\mathbf{i}))}).$

Recall that for every $0\leq\ell\leq e-1$, the set of
$x\in(\mathcal{O}/\mathfrak{p}^{e})$ with $\mathbf{lev}(x)\geq\ell$ is an
$\mathbb{F}_{p}$\- subspace of dimension $f(e-\ell)$. Fix
$\mathbf{i}\in\mathscr{H}(n,c)$. Since
$\\{\mathbf{a}(\mathbf{i}):\mathbf{a}\in S_{\mathbf{i}}\\}$ is a basis for
$\mathcal{O}/\mathfrak{p}^{e}$, it follows that for every $0\leq\ell\leq e-1$,
the cardinality of the set of $\mathbf{a}\in S_{\mathbf{i}}$ with
$\mathbf{lev}(\mathbf{a}(\mathbf{i}))\geq\ell$ is at most $f(e-\ell)$. Using
Lemma 4.6 it follows that

(5.6) $\sum_{\mathbf{a}\in
S_{\mathbf{i}}}p^{f(d-\mathbf{lev}(\mathbf{a}(\mathbf{i}))})\geq
f\sum_{\ell=0}^{e-1}p^{f(d-\ell)}.$

The assertion of Theorem 1.6 follows immediately from combining (5) and (5.6).

## References

  * [ABRdS18] Menny Aka, Emmanuel Breuillard, Lior Rosenzweig, and Nicolas de Saxcé, _Diophantine approximation on matrices and Lie groups_ , Geom. Funct. Anal. 28 (2018), no. 1, 1–57. MR 3777412
  * [AK65] James Ax and Simon Kochen, _Diophantine problems over local fields. I_ , Amer. J. Math. 87 (1965), 605–630. MR 184930
  * [AKOV13] Nir Avni, Benjamin Klopsch, Uri Onn, and Christopher Voll, _Representation zeta functions of compact $p$-adic analytic groups and arithmetic groups_, Duke Math. J. 162 (2013), no. 1, 111–197. MR 3011874
  * [Ax67] James Ax, _Solving diophantine problems modulo every prime_ , Ann. of Math. (2) 85 (1967), 161–183. MR 0209224
  * [Ax68] by same author, _The elementary theory of finite fields_ , Ann. of Math. (2) 88 (1968), 239–271. MR 229613
  * [BF03] Grégory Berhuy and Giordano Favi, _Essential dimension: a functorial point of view (after A. Merkurjev)_ , Doc. Math. 8 (2003), 279–330. MR 2029168
  * [BMKS16] Mohammad Bardestani, Keivan Mallahi-Karai, and Hadi Salmasian, _Minimal dimension of faithful representations for $p$-groups_, J. Group Theory 19 (2016), no. 4, 589–608. MR 3518394
  * [BMKS19] by same author, _Kirillov’s orbit method and polynomiality of the faithful dimension of $p$-groups_, Compos. Math. 155 (2019), no. 8, 1618–1654. MR 4043356
  * [BR97] J. Buhler and Z. Reichstein, _On the essential dimension of a finite group_ , Compositio Math. 106 (1997), no. 2, 159–179. MR 1457337
  * [BS08] Mitya Boyarchenko and Maria Sabitova, _The orbit method for profinite groups and a $p$-adic analogue of Brown’s theorem_, Israel J. Math. 165 (2008), 67–91. MR 2403615
  * [Del84] P. Deligne, _Les corps locaux de caractéristique $p$, limites de corps locaux de caractéristique $0$_, Representations of reductive groups over a local field, Travaux en Cours, Hermann, Paris, 1984, pp. 119–157. MR 771673
  * [Gre65] Newcomb Greenleaf, _Irreducible subvarieties and rational points_ , Amer. J. Math. 87 (1965), 25–31. MR 182625
  * [Gre74] Marvin J. Greenberg, _Strictly local solutions of Diophantine equations_ , Pacific J. Math. 51 (1974), 143–153. MR 351994
  * [GS84] Fritz Grunewald and Dan Segal, _Reflections on the classification of torsion-free nilpotent groups_ , Group theory, Academic Press, London, 1984, pp. 121–158. MR 780569
  * [How77] Roger E. Howe, _On representations of discrete, finitely generated, torsion-free, nilpotent groups_ , Pacific J. Math. 73 (1977), no. 2, 281–305. MR 0499004
  * [Jac85] Nathan Jacobson, _Basic algebra. I_ , second ed., W. H. Freeman and Company, New York, 1985. MR 780184
  * [Kaz77] D. Kazhdan, _Proof of Springer’s hypothesis_ , Israel J. Math. 28 (1977), no. 4, 272–286. MR 0486181
  * [Khu98] E. I. Khukhro, _$p$ -automorphisms of finite $p$-groups_, London Mathematical Society Lecture Note Series, vol. 246, Cambridge University Press, Cambridge, 1998. MR 1615819
  * [KM08] Nikita A. Karpenko and Alexander S. Merkurjev, _Essential dimension of finite $p$-groups_, Invent. Math. 172 (2008), no. 3, 491–508. MR 2393078
  * [Kne78] Martin Kneser, _Konstruktive Lösung $p$-adischer Gleichungssysteme_, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1978), no. 5, 67–69. MR 523328
  * [Lag83] J. C. Lagarias, _Sets of primes determined by systems of polynomial congruences_ , Illinois J. Math. 27 (1983), no. 2, 224–239. MR 694641
  * [Mar18] Daniel A. Marcus, _Number fields_ , Universitext, Springer, Cham, 2018, Second edition of [ MR0457396], With a foreword by Barry Mazur. MR 3822326
  * [McL73] K. R. McLean, _Commutative artinian principal ideal rings_ , Proc. London Math. Soc. (3) 26 (1973), 249–272. MR 319981
  * [Mer17] Alexander S. Merkurjev, _Essential dimension_ , Bull. Amer. Math. Soc. (N.S.) 54 (2017), no. 4, 635–661. MR 3683628
  * [MR10] Aurel Meyer and Zinovy Reichstein, _Some consequences of the Karpenko-Merkurjev theorem_ , Doc. Math. (2010), no. Extra vol.: Andrei A. Suslin sixtieth birthday, 445–457. MR 2804261
  * [OV15] E. A. O’Brien and C. Voll, _Enumerating classes and characters of $p$-groups_, Trans. Amer. Math. Soc. 367 (2015), no. 11, 7775–7796. MR 3391899
  * [SV14] A. Stasinski and C. Voll, _Representation zeta functions of nilpotent groups and generating functions for Weyl groups of type $B$_, Amer. J. Math. 136 (2014), no. 2, 501–550. MR 3188068
  * [Tie18] Elena Tielker, _Topics in the representation theory of finite p-groups_ , Master’s thesis, 2018.
  * [vdD91] Lou van den Dries, _A remark on Ax’s theorem on solvability modulo primes_ , Math. Z. 208 (1991), no. 1, 65–70. MR 1125733
  * [Vol04] Christopher Voll, _Zeta functions of groups and enumeration in Bruhat-Tits buildings_ , Amer. J. Math. 126 (2004), no. 5, 1005–1032. MR 2089080
  * [Vol05] C. Voll, _Functional equations for local normal zeta functions of nilpotent groups_ , Geom. Funct. Anal. 15 (2005), no. 1, 274–295, With an appendix by A. Beauville. MR 2140633