# Acceptability of classical groups
in non-zero characteristic

Saikat Panja<EMAIL_ADDRESS>Harish-Chandra Research Institute- Main
Building, Chhatnag Rd, Jhusi, Uttar Pradesh 211019, India

###### Abstract.

A group $G$ is called to be _acceptable_ (due to M. Larsen) if for any finite
group $H$, two element-conjugate homomorphisms are globally conjugate. We
answer the acceptability question for general linear, special linear, unitary,
symplectic and all orthogonal groups over an algebraically closed field of
non-zero characteristics.

###### Key words and phrases:

acceptable groups, classical groups, element-conjugacy, global conjugacy

###### 2020 Mathematics Subject Classification:

20C33, 20C99

The author (Panja) is supported by PDF-M fellowship from HRI.

##  1\. Introduction

We start with a basic result, coming from an application of the representation
theory of finite groups over complex numbers. Let $G$ be a finite group.
Consider two homomorphisms $\phi_{1},\leavevmode\nobreak\
\phi_{2}:G\longrightarrow\textup{GL}(n,\mathbb{C})$. Suppose that for all
$g\in G$ there exists $h(g)\in G$, such that
$\phi_{1}(g)=h(g)^{-1}\phi_{2}(g)h(g)^{-1}$. In this case, these two
homomorphisms will be called _element-conjugate homomorphisms_. Note that here
$h:G\longrightarrow G$ is a function. If there exists a choice of constant
function $h$, we say that $\phi_{1}$ and $\phi_{2}$ are _globally conjugate_.
Now, note that if the above-mentioned maps $\phi_{1},\leavevmode\nobreak\
\phi_{2}$ are element-conjugate, then they are globally conjugate. Indeed if
they are element-wise conjugate, their corresponding character values are the
same on each of the conjugacy classes of $G$. Hence their characters are the
same. Thus the two homomorphisms are globally conjugate. Now suppose we change
$\textup{GL}(n,\mathbb{C})$ by $\textup{SL}(n,\mathbb{C})$ or
$\textup{SO}(n,\mathbb{C})$. Then it is natural to ask whether all element-
conjugate homomorphisms are globally conjugate. At this point, we define a
group $\Gamma$ to be _acceptable_ (or _strongly acceptable_) if for all finite
groups (resp. group) $G$, two element-conjugate homomorphisms are globally
conjugate.

The general question of the relation between element-conjugacy and global
conjugacy arises in many contexts such as algebra, number theory, and
geometry. For example, the multiplicity-one question in the theory of
automorphic forms (see [2]). A motivating question, related to multiplicity-
one questions in the theory of automorphic forms, is when a compact group can
be the common covering space of a pair of non-isometric isospectral manifolds
(See [17]). Motivated by this M. Larsen has dealt with many of the compact or
complex simple Lie groups, viewed as real groups, after showing that the
question reduces to the semisimple case in [9]. In a subsequent paper [10] he
proved that a connected, simply connected, compact Lie group $G$ is acceptable
if and only if it has no direct factors of type $B_{n} (n\geq 4),D_{n} (n\geq
4),\leavevmode\nobreak\ E_{n},\text{ or }F_{4}$. This carries over to the
complex groups as well. Moreover, the group of type $G_{2}$ is the only
exceptional group (simply connected or not) which is acceptable. In the year
2016, Y. Fang, G. Han and B. Sun proved that the following two statements are
equivalent:

  1. (1)

For all connected compact Lie groups $H$ and all continuous homomorphisms
$\phi,\psi:H\longrightarrow G$, if $\phi(h)$ and $\psi(h)$ are conjugate in
$G$ for all $h\in H$, then $\phi$ and $\psi$ are globally conjugate.

  2. (2)

The Lie algebra of $G$ contains no simple ideal of type $D_{n}(n\geq
4),\leavevmode\nobreak\ E_{6},\leavevmode\nobreak\ E_{7},\text{ or }E_{8}$.

These results have strong applications in many parts of mathematics. For
example how to determine whether a given representation is distinguished or
not in terms of the Langlands parameters (see [1]). Another question being
that do the multiplicities of the trivial representation of $H$ in finite-
dimensional irreducible representations of $G$ determine the conjugacy class
of $H$, where $H$ is a closed subgroup of a complex reductive group $G$ (see
[22]).

Consider a prime $p$ and the target group is considered to be one of the
general linear groups, special linear groups, unitray groups, symplectic
groups or the orthogonal groups when the field is algebraically closed field
$\overline{\mathbb{F}_{q}}$ of characteristic $p$. We prove that all these
groups are not acceptable. We are further looking into the cases for
exceptional cases, which will be carried out in future work.

### Methodology

To work with the above-mentioned groups, we make use of their conjugacy
classes. This has been adapted from the work of Green ([5]) and Macdonald
([11]) for the general linear groups and from the work of Wall ([21]) in the
cases of unitary, symplectic and orthogonal groups. We will briefly mention
these results in due course. The idea is to work with unipotent classes, for
which we need to know the class representative for the same (see [3]). The
work [4] of Gonshaw, Liebeck, and O’Brien has been a rescuer to this. After
understanding the conjugacy classes, we move to construct particular
homomorphisms from particular groups to the target groups. For most of the
cases, we have used the group
$\mathbb{Z}/p\mathbb{Z}\oplus\mathbb{Z}/p\mathbb{Z}$ to be the source group.
Without further delay, we will present the organization of the paper.

### Organization of the paper

We start with some of the general results in Section 2. This section contains
two very important results, viz. Lemma 2.1 and Lemma 2.2 which have been the
lifeline for most of the proofs. In Section 3 we briefly mention the groups
and their conjugacy classes. Thereafter Section 4, Section 5 and Section 6 are
devoted to the main proof of the results concerning general and special linear
groups, unitary groups, symplectic and orthogonal groups, respectively. Our
main results are Theorem 4.3, Theorem 5.3, Theorem 6.4, Theorem 6.7, Theorem
6.10. We finish the paper by addressing a few future questions in Section 7.

### Acknowledgment

The author would like to thank professor Anupam Singh from IISER Pune for his
helpful discussions. A part of this work was finished during his visit IIT-
Jodhpur in December 2022. I would like to take this opportunity to thank the
institute for its hospitality during the stay.

##  2\. General results

######  Lemma 2.1 (Proposition 1.1, [9]).

Let $G=G_{1}\times G_{2}$ for two groups $G_{1},\leavevmode\nobreak\ G_{2}$.
Then $G$ is acceptable if and only if $G_{1}$ and $G_{2}$ are acceptable.

###### Proof.

Let $G_{1}$ and $G_{2}$ be two acceptable groups. Assume
$\phi_{1},\phi_{2}:H\longrightarrow G_{1}\times G_{2}$ be two homomorphisms
which are element-conjugate. Then the maps
$\pi_{1}\circ\phi_{1},\pi_{1}\circ\phi_{2}:H\longrightarrow G_{1}$ are element
conjugate, hence globally conjugate, via say $g_{1}$. Similarly there exists
$g_{2}\in G_{2}$, via which
$\pi_{2}\circ\phi_{1},\pi_{2}\circ\phi_{2}:H\longrightarrow G_{2}$ are
globally conjugate. Hence $\phi_{1},\phi_{2}$ are globally conjugate.

On the other hand considering the monomorphisms $i_{1}:G_{1}\longrightarrow
G_{1}\times G_{2}$ and $i_{2}:G_{2}\longrightarrow G_{1}\times G_{2}$, we get
that if $G_{1}\times G_{2}$ is acceptable, then so are $G_{1}$ and $G_{2}$. ∎

######  Lemma 2.2.

Let $G$ be an acceptable group. Then for all $a\in G$ the subgroup
$\mathscr{C}_{G}(a)$, the centralizer of $a$ is acceptable.

###### Proof.

Let $\varphi_{1},\varphi_{2}:H\longrightarrow\mathscr{C}_{G}(a)$ be two
element conjugate homorphisms. Construct the following homomorphisms

$\displaystyle\widetilde{\varphi_{i}}:H\times\langle a\rangle\longrightarrow
G,\text{ defined as }\widetilde{\varphi_{i}}(h,a^{m})=\varphi(h)a^{m}.$ (2.1)

Then the homomorphisms are element-conjugate and hence globally conjugate
since $G$ is acceptable. Thus there exists $g\in G$ such that

$\displaystyle g\varphi_{1}(h)a^{m}g^{-1}=\varphi_{2}(g)\text{ for all
}m\in\mathbb{Z},h\in H.$

On plugging $m=0$ and $h=e_{H}$, we get that $g\in\mathscr{C}_{G}(a)$. Hence
we get the result by putting $m=0$. ∎

######  Lemma 2.3 (See Lemma 2.3 of [23] for Lie group set-up).

Let $G$ be an acceptable group and $A$ be an abelian subgroup of $G$. Then
$\mathscr{C}_{G}(A)$ is an acceptable subgroup.

###### Proof.

Let $\varphi_{1},\varphi_{2}:H\longrightarrow\mathscr{C}_{G}(A)$ be two
element conjugate homomorphisms. Construct the following homomorphisms

$\displaystyle\widetilde{\varphi_{i}}:H\times A\longrightarrow G,\text{
defined as }\widetilde{\varphi_{i}}(h,a)=\varphi(h)a.$ (2.2)

Then the homomorphisms are element-conjugate and hence globally conjugate
since $G$ is acceptable. Thus there exists $g\in G$ such that

$\displaystyle g\varphi_{1}(h)ag^{-1}=\varphi_{2}(g)\text{ for all }a\in
A,h\in H.$

Assuming $h=e_{H}$, we get that $g\in\mathscr{C}_{G}(A)$. Then taking
$a=e_{A}$, we get that $\varphi_{1}$ and $\varphi_{2}$ are globally conjugate
via $g$. ∎

##  3\. Conjugacy classes in the groups

In this section, we will discuss the classical groups in brief and present the
description of their conjugacy classes. This will be based on the works [5],
[11] and [21].

###  3.1. General linear group

The general linear group $\textup{GL}(n,\overline{\mathbb{F}_{q}})$ is defined
to be the set of all $n\times n$ matrices defined over
$\overline{\mathbb{F}_{q}}$. The conjugacy class of this group was firstly
determined in works [5] and [11] in the case of finite fields. Let $\Phi$
denote the set of all non-constant, monic, irreducible polynomials $f(x)$ with
coefficients in $\overline{\mathbb{F}_{q}}$ which is not equal to the
polynomial $x$. Let $\Lambda$ denote the set of all partitions
$\lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{k})$ with
$\lambda_{i}\geq\lambda_{i+1}\geq 0$ and each $\lambda$ are integers. The
conjugacy class of $\textup{GL}(n,\overline{\mathbb{F}_{q}})$ is in one-one
correspondence with a set of all functions $\alpha:\Phi\longrightarrow\Lambda$
which takes the value empty partition on all but finitely many polynomials in
$\Phi$, and satisfies

$\displaystyle\sum\limits_{f\in\Phi}|\alpha(f)|\deg f=n.$

Hence the conjugacy classes of $\textup{GL}(n,q)$ are determined by
combinatorial data consisting of tuples of the form $(f,\lambda_{f})$ where
$f\in\Phi$. We note that two matrices in $\textup{GL}(n,q)$ are conjugate if
and only if they are conjugate in $\textup{GL}(n,\overline{\mathbb{F}_{q}})$,
which we will use in section 7.

###  3.2. Unitary group

For the field $\overline{\mathbb{F}_{q}}$, consider an involution $\sigma$ of
$\overline{\mathbb{F}_{q}}$. This induces an automorphism on the ring
$\mathbb{\overline{\mathbb{F}_{q}}}[x]$ and the group
$\textup{GL}(n,\overline{\mathbb{F}_{q}})$. The _unitary_ group is the group
of all matrices in $\textup{GL}(n,\overline{\mathbb{F}_{q}})$ which satisfy
${}^{t}\overline{A}JA=J$, where $J$ is given by
$\begin{pmatrix}\leavevmode\nobreak\ &&&1\\\ &&1&\\\
&\reflectbox{$\ddots$}&&\\\ 1&&&\end{pmatrix}$ (matrix of a hermitian form)
and this group will be denoted by $\textup{U}(n,\overline{\mathbb{F}_{q}})$.

######  Definition 3.1.

The twisted dual of a monic degree $r$ polynomial
$f(x)\in\overline{\mathbb{F}_{q}}[x]$ satisfying $f(0)\neq 0$, is the
polynomial given by
$\widetilde{f}(x)=\overline{f(0)}^{-1}x^{r}\overline{f}(x^{-1})$. The
polynomial $f$ will be called _$\sim$ -symmetric_ if $f=\widetilde{f}$. A
monic polynomial $f(x)\in\overline{\mathbb{F}_{q}}[x]$, will be called to be
$\sim$-irreducible if and only if it does not have any proper $\sim$-symmetric
factor. We will denote the set of all $\sim$-irreducible monic polynomials by
$\widetilde{\Phi}$

We know that elements $g,h\in\textup{U}(n,\overline{\mathbb{F}_{q}})$ are
conjugate in $\textup{U}(n,q)$ if and only if they are conjugate in
$\textup{GL}(n,\overline{\mathbb{F}_{q}})$ (see [3]) and the conjugacy classes
in $\textup{U}(n,\overline{\mathbb{F}_{q}})$ are parametrized by sequences
$\\{(f,\lambda)|f\in\widetilde{\Phi},\lambda\in\Lambda\\}$ is the set of
irreducible polynomials. The class representative for the conjugacy classes
can be found in [20].

###  3.3. Symplectic group and Orthogonal group

For a vector space $V$ of dimension $2n$ over $\overline{\mathbb{F}_{q}}$,
consider the unique (up to similarity) non-degenerate alternating bilinear
form on $V$ given by

$\left<(x_{i})_{i=1}^{2n},(y_{j})_{j=1}^{2n}\right>=\sum_{j=1}^{n}x_{j}y_{2n+1-j}-\sum_{i=0}^{n-1}x_{2n-i}y_{i+1}.$

The symplectic group is the subgroup of
$\textup{GL}(2n,\overline{\mathbb{F}_{q}})$ consisting of elements preserving
this alternating form on $V$. By fixing an appropriate basis, the matrix of
the form can be chosen to be $J=\begin{pmatrix}0&\Lambda_{n}\\\
-\Lambda_{n}&0\end{pmatrix}$ where
$\Lambda_{n}=\begin{pmatrix}0&0&\cdots&0&1\\\ 0&0&\cdots&1&0\\\
\vdots&\vdots&\reflectbox{$\ddots$}&\vdots&\vdots\\\
1&0&\cdots&0&0\end{pmatrix}$ and

$\textup{Sp}(2n,\overline{\mathbb{F}_{q}})=\\{A\in\textup{GL}(2n,\overline{\mathbb{F}_{q}})\mid{}^{t}\\!AJA=J\\}.$

Now consider $V$ to be an $m$-dimensional vector space over the field
$\overline{\mathbb{F}_{q}}$. Then there a unique (up to similarity) non-
degenerate quadratic forms on $V$. The orthogonal group consists of elements
of $\textup{GL}(m,\overline{\mathbb{F}_{q}})$ which preserve a non-degenerate
quadratic form $Q$. With respect to an appropriate basis, we will fix the
matrices of the symmetric bilinear forms (associated with the quadratic forms)
as follows:

$J_{o}=\begin{pmatrix}0&0&\Lambda_{n}\\\ 0&\alpha&0\\\
\Lambda_{n}&0&0\end{pmatrix},J_{e}=\begin{pmatrix}0&\Lambda_{n}\\\
\Lambda_{n}&0\end{pmatrix}$

where $\alpha\in\mathbb{F}_{q}^{\times}$, and
$\Lambda_{l}=\begin{pmatrix}0&0&\cdots&0&1\\\ 0&0&\cdots&1&0\\\
\vdots&\vdots&\reflectbox{$\ddots$}&\vdots&\vdots\\\
1&0&\cdots&0&0\end{pmatrix}$, an $l\times l$ matrix. Then, the orthogonal
group in matrix form is

$\textup{O}(2m,\overline{\mathbb{F}_{q}})=\\{A\in\textup{GL}(2m,\overline{\mathbb{F}_{q}})\mid{}^{t}\\!{A}J_{e}A=J_{e}\\},$
$\textup{O}(2m+1,\overline{\mathbb{F}_{q}})=\\{A\in\textup{GL}(2m,\overline{\mathbb{F}_{q}})\mid{}^{t}\\!{A}J_{o}A=J_{o}\\}.$

To describe the conjugacy classes of finite symplectic and orthogonal groups,
we will need the concept of _self-reciprocal polynomials, symplectic and
orthogonal partitions_ , which are defined below. For examples of the same,
the readers are suggested to have a look at [3].

######  Definition 3.2.

The dual of a monic degree $r$ polynomial $f(x)\in k[x]$ satisfying $f(0)\neq
0$, is the polynomial given by $f^{*}(x)=f(0)^{-1}x^{r}f(x^{-1})$. The
polynomial $f$ will be called _self reciprocal_ if $f=f^{*}$. A monic
polynomial $f(x)\in\mathbb{F}_{q}[x]$, will be called to be $*-$irreducible if
and only if it does not have any proper self-reciprocal factor.

######  Definition 3.3.

A symplectic partition is a partition of a number $k$, such that the odd parts
have even multiplicity. The set of all symplectic partitions will be denoted
as $\mathcal{D}^{\prime}_{\textup{Sp}}$.

######  Definition 3.4.

An orthogonal partition is a partition of a number $k$, such that all even
parts have even multiplicity. The set of all orthogonal signed partitions will
be denoted as $\mathcal{D}^{\prime}_{\textup{O}}$.

It can be shown that the characteristic polynomial of a symplectic or
orthogonal matrix is self-reciprocal. We follow J. Milnor’s terminology [12]
to distinguish between the $*$-irreducible factors of the characteristic
polynomials. We call a $*$-irreducible polynomial $f$ to be

(1) Type $1$ if $f=f^{*}$ and $f$ is irreducible polynomial of even degree;

(2) Type $2$ if $f=gg^{*}$ and $g$ is irreducible polynomial satisfying $g\neq
g^{*}$;

(3) Type $3$ if $f(x)=x\pm 1$.

According to [21] the conjugacy classes of
$\textup{Sp}(2n,\overline{\mathbb{F}_{q}})$ are parameterized by the functions
$\lambda:\Phi\rightarrow\mathcal{P}^{2n}\cup\mathcal{D}_{\textup{Sp}}^{2n}$,
where $\Phi$ denotes the set of all monic, non-constant, irreducible
polynomials, $\mathcal{P}^{2n}$ is the set of all partitions of $1\leq k\leq
2n$ and $\mathcal{D}_{\textup{Sp}}^{2n}$ is the set of all symplectic
partitions of $1\leq k\leq 2n$. Such a $\lambda$ represent a conjugacy class
of Sp if and only if

  1. (1)

$\lambda(x)=0$,

  2. (2)

$\lambda_{\varphi^{*}}=\lambda_{\varphi}$,

  3. (3)

$\lambda_{\varphi}\in\mathcal{D}^{n}_{\textup{Sp}}$ iff $\varphi=x\pm 1$,

  4. (4)

$\displaystyle\sum_{\varphi}|\lambda_{\varphi}|\textup{deg}(\varphi)=2n$.

Also from [21], we find out that a similar kind of statement is true for the
groups $\textup{O}(n,\overline{\mathbb{F}_{q}})$. The conjugacy classes of
$\textup{O}(n,\overline{\mathbb{F}_{q}})$ are parameterized by the functions
$\lambda:\Phi\rightarrow\mathcal{P}^{n}\cup\mathcal{D}_{\textup{O}}^{n}$,
where $\Phi$ denotes the set of all monic, non-constant, irreducible
polynomials, $\mathcal{P}^{n}$ is the set of all partitions of $1\leq k\leq n$
and $\mathcal{D}_{\textup{O}}^{n}$ is the set of all orthogonal partitions of
$1\leq k\leq n$. Such a $\lambda$ represent a conjugacy class of
$\textup{O}(n,\overline{\mathbb{F}_{q}})$ if and only if

  1. (1)

$\lambda(x)=0$,

  2. (2)

$\lambda_{\varphi^{*}}=\lambda_{\varphi}$,

  3. (3)

$\lambda_{\varphi}\in\mathcal{D}^{n}_{\textup{O}}$ iff $\varphi=x\pm 1$,

  4. (4)

$\displaystyle\sum_{\varphi}|\lambda_{\varphi}|\textup{deg}(\varphi)=n$.

Class representative corresponding to given data can be found in [19], [18],
[4] and we will mention them whenever needed.

##  4\. General and special linear groups

######  Lemma 4.1.

For a field $\overline{\mathbb{F}_{q}}$, if
$\textup{GL}(n+1,\overline{\mathbb{F}_{q}})$ is acceptable, then so is
$\textup{GL}(n,\overline{\mathbb{F}_{q}})$.

###### Proof.

We are going to use Lemma 2.2. Take the following element

$\displaystyle a=\begin{pmatrix}I_{n}&0\\\
0&-1\end{pmatrix}\in\textup{GL}(n+1,\overline{\mathbb{F}_{q}}).$

Then we know that

$\displaystyle\mathscr{C}_{\textup{GL}(n+1,\overline{\mathbb{F}_{q}})}(a)=$
$\displaystyle\left\\{\begin{pmatrix}A&0\\\
0&\alpha\end{pmatrix}\middle|\begin{subarray}{c}A\in\textup{GL}(n,\overline{\mathbb{F}_{q}})\\\
\alpha\in\overline{\mathbb{F}_{q}}^{*}\end{subarray}\right\\}$
$\displaystyle\cong$
$\displaystyle\textup{GL}(n,\overline{\mathbb{F}_{q}})\times\overline{\mathbb{F}_{q}}^{*}.$

Hence using Lemma 2.1, we conclude that
$\textup{GL}(n,\overline{\mathbb{F}_{q}})$ is acceptable. ∎

######  Proposition 4.2.

For $q\geq 5$ the group $\textup{GL}(2,\overline{\mathbb{F}_{q}})$ is not
acceptable.

###### Proof.

Let $a\neq b$ be two nonzero elements of $\overline{\mathbb{F}_{q}}$ of order
$p$, where $q=p^{f}$. Further, assume that $a^{2}\neq b^{2}$. Note that such a
choice exists since $x^{2}=1$ has less than or equal to two solutions in
$\overline{\mathbb{F}_{q}}$. Consider the following two homomorphisms

$\displaystyle\varphi_{1},\varphi_{2}:\mathbb{Z}/p\mathbb{Z}\oplus\mathbb{Z}/p\mathbb{Z}\longrightarrow\textup{GL}(2,\overline{\mathbb{F}_{q}})$
defined as $\displaystyle\varphi_{1}(1,0)=\begin{pmatrix}1&a\\\
0&1\end{pmatrix},\varphi_{1}(0,1)=\begin{pmatrix}1&b\\\ 0&1\end{pmatrix}$
$\displaystyle\varphi_{2}(1,0)=\begin{pmatrix}1&b\\\
0&1\end{pmatrix},\varphi_{2}(0,1)=\begin{pmatrix}1&a\\\ 0&1\end{pmatrix}.$

Then we know that these two homomorphisms are element-conjugate. We now show
that these two homomorphisms are not globally conjugate. On the contrary,
assume that they are. So there is a matrix $\begin{pmatrix}p&q\\\
r&s\end{pmatrix}\in\textup{GL}(2,\overline{\mathbb{F}_{q}})$ acting as a
global conjugator. Hence we have the following equations:

$\displaystyle\begin{pmatrix}p&q\\\ r&s\end{pmatrix}\begin{pmatrix}1&a\\\
0&1\end{pmatrix}=\begin{pmatrix}1&b\\\ 0&1\end{pmatrix}\begin{pmatrix}p&q\\\
r&s\end{pmatrix}$ and $\displaystyle\begin{pmatrix}p&q\\\
r&s\end{pmatrix}\begin{pmatrix}1&b\\\ 0&1\end{pmatrix}=\begin{pmatrix}1&a\\\
0&1\end{pmatrix}\begin{pmatrix}p&q\\\ r&s\end{pmatrix}.$

Solving the polynomial equations obtained by comparing the coefficients, we
get that $r=s=0$. This implies that there doesn’t exist any matrix playing the
role of a global conjugator. Hence the conclusion follows. ∎

######  Theorem 4.3.

Let $p$ be a prime and $q=p^{f}$. For $q>5$ the group
$\textup{GL}(n,\overline{\mathbb{F}_{q}})$ is not acceptable for all
$n\in\mathbb{N}$.

###### Proof.

Follows from Proposition 4.2 and Lemma 4.1. ∎

######  Corollary 4.4.

The group $\textup{SL}(2n+1,\overline{\mathbb{F}_{q}})$ is not acceptable for
all $n\in\mathbb{N}$.

###### Proof.

We are going to use Lemma 2.2. Assume if possible
$\textup{SL}(2n+1,\overline{\mathbb{F}_{q}})$ to be acceptable. Choose

$\displaystyle a=\begin{pmatrix}-I_{2n}&0\\\
0&1\end{pmatrix}\in\textup{SL}(2n+1,\overline{\mathbb{F}_{q}}).$

Then we get that

$\displaystyle\mathscr{C}_{\textup{SL}(2n+1,\overline{\mathbb{F}_{q}})}(a)$
$\displaystyle=\left\\{\begin{pmatrix}A&0\\\
0&\alpha\end{pmatrix}\middle|\begin{subarray}{c}A\in\textup{GL}(n,\overline{\mathbb{F}_{q}})\\\
\alpha\in\overline{\mathbb{F}_{q}}^{*}\\\ \det
A\cdot\alpha=1\end{subarray}\right\\}$ $\displaystyle\cong$
$\displaystyle\left\\{\begin{pmatrix}A&0\\\ 0&\det
A^{-1}\end{pmatrix}\middle|A\in\textup{GL}(n,\overline{\mathbb{F}_{q}})\right\\}$
$\displaystyle\cong$ $\displaystyle\textup{GL}(2n,\overline{\mathbb{F}_{q}}).$

Hence the result follows from Lemma 2.2 and Theorem 4.3. ∎

######  Proposition 4.5.

Let the power map
$\psi_{n}:\overline{\mathbb{F}_{q}}\longrightarrow\overline{\mathbb{F}_{q}}$
defined as $\psi_{n}(x)=x^{n}$ be a surjection. Then
$\textup{SL}(n,\overline{\mathbb{F}_{q}})$ is not acceptable.

###### Proof.

On the contrary assume that $\textup{SL}(n,\overline{\mathbb{F}_{q}})$ is
acceptable. We will show that in this case, it will imply
$\textup{GL}(n,\overline{\mathbb{F}_{q}})$ to be acceptable. Note that the map
$\psi_{k}$ is a bijection. Let
$\varphi_{1},\varphi_{2}:G\longrightarrow\textup{GL}(n,\overline{\mathbb{F}_{q}})$
be two element conjugate homomorphisms. Consider the following two maps

$\displaystyle\widetilde{\varphi_{1}},\widetilde{\varphi_{2}}:G\longrightarrow\textup{SL}(n,\overline{\mathbb{F}_{q}})$
defined as
$\displaystyle\widetilde{\varphi_{i}}(x)=\psi_{n}^{-1}(\det\varphi_{i}(x))\varphi_{i}(x).$

Note that $\widetilde{\varphi_{1}}$ and $\widetilde{\varphi_{2}}$ are element
conjugate homomorphisms. Hence by assumption, they are globally conjugate.
Since $\psi_{n}$ is a bijection and conjugate matrices have the same
determinant, we get that $\varphi_{1}$ and $\varphi_{2}$ are globally
conjugate. This contradicts Theorem 4.3 and finishes the proof. ∎

######  Corollary 4.6.

Let $(q-1,n)=1$. Then $\textup{SL}(n,\overline{\mathbb{F}_{q}})$ is not
acceptable.

######  Proposition 4.7.

For $n\geq 4$, the group $\textup{SL}(n,\overline{\mathbb{F}_{q}})$ is not
acceptable.

###### Proof.

Fix $\lambda\in\overline{\mathbb{F}_{q}}$ such that $\lambda^{2}\neq 1$. Fix
the element $a=\begin{pmatrix}\lambda&&\\\ &\dfrac{1}{\lambda}&\\\
&&I\end{pmatrix}\in\textup{SL}(n,\overline{\mathbb{F}_{q}})$. Then we have
that

$\displaystyle\mathscr{C}_{\textup{SL}(n,\overline{\mathbb{F}_{q}})}(a)=$
$\displaystyle\left\\{\begin{pmatrix}\alpha&&\\\ &\beta&&\\\
&&A\end{pmatrix}\middle|\begin{subarray}{c}\alpha,\beta\in\overline{\mathbb{F}_{q}}^{*}\\\
A\in\textup{GL}(n-2,\overline{\mathbb{F}_{q}})\\\ \alpha\beta\det
A=1\end{subarray}\right\\}$ $\displaystyle\cong$
$\displaystyle\overline{\mathbb{F}_{q}}^{*}\times\textup{GL}(n-2,\overline{\mathbb{F}_{q}}).$

Since $\textup{GL}(n-2,\overline{\mathbb{F}_{q}})$ is not acceptable, using
Lemma 2.1, we get that $\textup{SL}(n,\overline{\mathbb{F}_{q}})$ is not
acceptable. ∎

##  5\. Unitary groups

######  Lemma 5.1.

For a field $\overline{\mathbb{F}_{q}}$, if
$\textup{U}(n+1,\overline{\mathbb{F}_{q}})$ is acceptable, then so is
$\textup{U}(n,\overline{\mathbb{F}_{q}})$.

###### Proof.

We work with the hermitian form, which is given by the matrix

$\displaystyle\begin{pmatrix}1&&&\\\ &1&&\\\ &&\ddots&\\\ &&&1\end{pmatrix}.$

Consider the element $a=\begin{pmatrix}I_{n}&\\\ &-1\end{pmatrix}$. Then by
Lemma 2.2, we get that the
$\mathscr{C}_{\textup{U}(n+1,\overline{\mathbb{F}_{q}})}(a)$ is acceptable, if
$\textup{U}(n+1,\overline{\mathbb{F}_{q}})$ is. Now we have that

$\displaystyle\mathscr{C}_{\textup{U}(n+1,\overline{\mathbb{F}_{q}})}(a)$
$\displaystyle=\left\\{\begin{pmatrix}A&\\\
&\alpha\end{pmatrix}\middle|\begin{subarray}{c}A\in\textup{U}(n,\overline{\mathbb{F}_{q}})\\\
\alpha\sigma(\alpha)=1\end{subarray}\right\\}$
$\displaystyle=\textup{U}(n,\overline{\mathbb{F}_{q}})\times\\{\alpha\in\overline{\mathbb{F}_{q}}^{*}|\alpha\sigma(\alpha)=1\\}.$

Hence the result follows from Lemma 2.1. ∎

######  Proposition 5.2.

The group $\textup{U}(4,\overline{\mathbb{F}_{q}})$ is not acceptable.

###### Proof.

We will work with the hermitian form given by the matrix
$\begin{pmatrix}&&&1\\\ &&1&\\\ &\reflectbox{$\ddots$}&&\\\
1&&&\end{pmatrix}$. Firstly fix $1\neq a\in\overline{\mathbb{F}_{q}}^{*}$.
Then consider the following two matrices

$\displaystyle X_{1}=\begin{pmatrix}1&1&&\\\ &1&&\\\ &&1&\\\
&&-1&1\end{pmatrix}\text{ and }X_{a}=\begin{pmatrix}1&a&&\\\ &1&&\\\ &&1&\\\
&&-a&1\end{pmatrix}.$

Then we know that these two matrices commute and are conjugate to each other.
Consider the following two homomorphisms

$\displaystyle\varphi_{1},\varphi_{2}:\mathbb{Z}/p\mathbb{Z}\oplus\mathbb{Z}/p\mathbb{Z}\longrightarrow\textup{U}(4,\overline{\mathbb{F}_{q}})$
defined as $\displaystyle\varphi_{1}((1,0))=X_{1},\varphi_{1}((0,1))=X_{a},$
$\displaystyle\varphi_{2}((1,0))=X_{a},\varphi_{2}((0,1))=X_{1}.$

Then we know that these two homomorphisms are element conjugate but not
globally conjugate. Hence $\textup{U}(4,\overline{\mathbb{F}_{q}})$ is not
acceptable. ∎

######  Theorem 5.3.

For and $n\geq 4$, the groups $\textup{U}(n,\overline{\mathbb{F}_{q}})$ is not
acceptable.

###### Proof.

Follows directly from Lemma 5.1 and Proposition 5.2. ∎

##  6\. Symplectic and Orthogonal groups

######  Proposition 6.1.

For a field $\overline{\mathbb{F}_{q}}$ if
$\textup{Sp}(2m+2,\overline{\mathbb{F}_{q}})$ is an acceptable group, then so
is $\textup{Sp}(2m,\overline{\mathbb{F}_{q}})$.

###### Proof.

Fix an element $\lambda\in\overline{\mathbb{F}_{q}}^{*}$ such that
$\lambda^{2}\neq 1$. Then fix the element

$\displaystyle a=\begin{pmatrix}\lambda&&&&&\\\ &\dfrac{1}{\lambda}&&&&\\\
&&&1&&\\\ &&-1&&&\\\ &&&\ddots&&\\\ &&&&&1\\\
&&&&-1&\end{pmatrix}\in\textup{Sp}(2m+2,\overline{\mathbb{F}_{q}}).$

Then we know that

$\displaystyle\mathscr{C}_{\textup{Sp}(2m+2,\overline{\mathbb{F}_{q}})}(a)$
$\displaystyle=\mathscr{C}_{\textup{GL}(2m+2,\overline{\mathbb{F}_{q}})}(a)\cap\textup{Sp}(2m+2,\overline{\mathbb{F}_{q}})$
$\displaystyle=\left\\{\begin{pmatrix}\alpha&&\\\ &\beta&\\\ &&A\\\
\end{pmatrix}\middle|\begin{subarray}{c}\alpha,\beta\in\overline{\mathbb{F}_{q}}^{*}\\\
A\in\textup{GL}(2m,\overline{\mathbb{F}_{q}})\end{subarray}\right\\}\cap\textup{Sp}(2m+2,\overline{\mathbb{F}_{q}})$
$\displaystyle\cong\left\\{\begin{pmatrix}\alpha&\\\
&\frac{1}{\alpha}\end{pmatrix}\middle|\alpha\in\overline{\mathbb{F}_{q}}^{*}\right\\}\times\textup{Sp}(2m,\overline{\mathbb{F}_{q}}).$

Hence the result follows from Lemma 2.1. ∎

######  Proposition 6.2.

The group $\textup{Sp}(2,\overline{\mathbb{F}_{q}})$ is not acceptable.

###### Proof.

Fix $a\in(\overline{\mathbb{F}_{q}}^{*})^{2}\setminus\\{1\\}$. It is easy to
see that

$\displaystyle\begin{pmatrix}1&1\\\ &1\end{pmatrix},\begin{pmatrix}1&a\\\
&1\end{pmatrix}\in\textup{Sp}(2,\overline{\mathbb{F}_{q}})$

are conjugate to each other by an element of
$\textup{Sp}(2,\overline{\mathbb{F}_{q}})$. Consider the following two
homomorphisms

$\displaystyle\varphi_{1},\varphi_{2}:\mathbb{Z}/p\mathbb{Z}\oplus\mathbb{Z}/p\mathbb{Z}\longrightarrow\textup{Sp}(2,\overline{\mathbb{F}_{q}})$
given by $\displaystyle\varphi_{1}\left((1,0)\right)=\begin{pmatrix}1&1\\\
&1\end{pmatrix},\varphi_{1}\left((0,1)\right)=\begin{pmatrix}1&a\\\
&1\end{pmatrix},$
$\displaystyle\varphi_{2}\left((1,0)\right)=\begin{pmatrix}1&a\\\
&1\end{pmatrix},\varphi_{2}\left((0,1)\right)=\begin{pmatrix}1&1\\\
&1\end{pmatrix}.$

Then using the same argument as Proposition 4.2 we get that these two
homomorphisms are not globally-conjugate. Hence the result follows. ∎

######  Corollary 6.3.

The group $\textup{SL}(2,\overline{\mathbb{F}_{q}})$ is not acceptable.

###### Proof.

Since
$\textup{SL}(2,\overline{\mathbb{F}_{q}})\cong\textup{Sp}(2,\overline{\mathbb{F}_{q}})$
(see [6]), the result follows immediately. ∎

######  Theorem 6.4.

The groups $\textup{Sp}(2n,\overline{\mathbb{F}_{q}})$ are not acceptable.

###### Proof.

Follows easily from Proposition 6.1 and Proposition 6.2. ∎

######  Proposition 6.5.

The acceptability of $\textup{O}(2m+1,\overline{\mathbb{F}_{q}})$ implies the
acceptability of $\textup{O}(2m-1,\overline{\mathbb{F}_{q}})$.

###### Proof.

Fix the element $a=\begin{pmatrix}-I_{2}&\\\
&&I_{2m-1}\end{pmatrix}\in\textup{O}(2m+1,\overline{\mathbb{F}_{q}})$. Then by
Lemma 2.2 we get that
$\mathscr{C}_{\textup{O}(2m+1,\overline{\mathbb{F}_{q}})}(a)$ is acceptable.
Now we have that

$\displaystyle\mathscr{C}_{\textup{O}(2m+1,\overline{\mathbb{F}_{q}})}(a)$
$\displaystyle=\left\\{\begin{pmatrix}A&\\\
&B\end{pmatrix}\middle|\begin{subarray}{c}A\in\textup{GL}(2,q)\\\
B\in\textup{GL}(2m-1,q)\end{subarray}\right\\}\cap\textup{O}(2m+1,\overline{\mathbb{F}_{q}})$
$\displaystyle\cong\left\\{A\in\textup{GL}(2,\overline{\mathbb{F}_{q}})|AA^{t}=I\right\\}\times\textup{O}(2m-1,\overline{\mathbb{F}_{q}}).$

Hence the result follows from Lemma 2.1. ∎

######  Proposition 6.6.

The group $\textup{O}(5,\overline{\mathbb{F}_{q}})$ is not acceptable.

###### Proof.

We will work with the non-degenerate quadratic form of the form

$\displaystyle\begin{pmatrix}0&0&I_{2}\\\ 0&1&0\\\ I_{2}&0&0\end{pmatrix},$

where $I_{2}$ is the identity matrix of size $2\times 2$. Consider the
matrices

$\displaystyle\alpha_{1}=\begin{pmatrix}1&1\\\ &1\end{pmatrix}\text{ and
}\alpha_{a}=\begin{pmatrix}1&a\\\ &1\end{pmatrix},$

where $a\neq 1$. Then the matrices

$\displaystyle X_{1}=\begin{pmatrix}\alpha_{1}&&\\\ &1&\\\
&&{}^{t}\alpha_{1}^{-1}\end{pmatrix}\text{ and
}X_{a}=\begin{pmatrix}\alpha_{a}&&\\\ &1&\\\
&&{}^{t}\alpha_{a}^{-1}\end{pmatrix}.$

Then $X_{1}\cdot X_{a}=X_{a}\cdot X_{1}$ and both are elements of order $p$
and are conjugate in $\textup{O}(5,\overline{\mathbb{F}_{q}})$. Consider the
following two homomorphisms:

$\displaystyle\varphi_{1},\varphi_{2}:\mathbb{Z}/p\mathbb{Z}\oplus\mathbb{Z}/p\mathbb{Z}\longrightarrow\textup{O}(5,\overline{\mathbb{F}_{q}})$
given by $\displaystyle\varphi_{1}((1,0))=X_{1},\varphi_{1}((0,1))=X_{a}$
$\displaystyle\varphi_{2}((1,0))=X_{a},\varphi_{2}((0,1))=X_{1}.$

Then although these two homomorphisms are element-conjugate, they are not
globally conjugate, using the same argument as of Proposition 4.2. ∎

######  Theorem 6.7.

For $m\geq 2$ none of the groups $\textup{O}(2m+1,\overline{\mathbb{F}_{q}})$
are acceptable.

###### Proof.

Follows from Proposition 6.6 and Proposition 6.5. ∎

######  Proposition 6.8.

The acceptability of the group $\textup{O}(2m+2,\overline{\mathbb{F}_{q}})$
implies the acceptability of $\textup{O}(2m,\overline{\mathbb{F}_{q}})$.

###### Proof.

We will work with the quadratic form

$\displaystyle\begin{pmatrix}0&1&&&\\\ 1&0&&&\\\ &&&\ddots&&\\\ &&&&0&1\\\
&&&&1&0\end{pmatrix}.$

Consider the element $a=\begin{pmatrix}I_{2m}&\\\
&-I_{2}\end{pmatrix}\in\textup{O}(2m+2,\overline{\mathbb{F}_{q}})$. Then by
Lemma 2.2, we have that
$\mathscr{C}_{\textup{O}(2m+2,\overline{\mathbb{F}_{q}})}(a)$ is acceptable.
Note that

$\displaystyle\mathscr{C}_{\textup{O}^{+}(2m+2,\overline{\mathbb{F}_{q}})(a)}$
$\displaystyle=\left\\{\begin{pmatrix}A&\\\
&B\end{pmatrix}\middle|\begin{subarray}{c}A\in\textup{GL}(2m,\overline{\mathbb{F}_{q}})\\\
B\in\textup{GL}(2,\overline{\mathbb{F}_{q}})\end{subarray}\right\\}\cap\textup{O}(2m+2,\overline{\mathbb{F}_{q}})$
$\displaystyle=\textup{O}(2m,q)\times\textup{O}(2,\overline{\mathbb{F}_{q}}).$

Hence the result follows from Lemma 2.1. ∎

######  Proposition 6.9.

For $q\geq 5$, the group $\textup{O}(4,\overline{\mathbb{F}_{q}})$ is not
acceptable.

###### Proof.

In this proof we will work with the quadratic form $\begin{pmatrix}0&I_{2}\\\
I_{2}&0\end{pmatrix}$. The proof is similar to that of Proposition 6.6, but we
will replicate it with proper modification for completeness. Consider the
matrices

$\displaystyle\alpha_{1}=\begin{pmatrix}1&1\\\ &1\end{pmatrix}\text{ and
}\alpha_{a}=\begin{pmatrix}1&a\\\ &1\end{pmatrix},$

where $a\neq 1$. Then the matrices

$\displaystyle X_{1}=\begin{pmatrix}\alpha_{1}&\\\
&{}^{t}\alpha_{1}^{-1}\end{pmatrix}\text{ and
}X_{a}=\begin{pmatrix}\alpha_{a}&\\\ &{}^{t}\alpha_{a}^{-1}\end{pmatrix}.$

Then $X_{1}\cdot X_{a}=X_{a}\cdot X_{1}$ and both are element of order $p$ and
are conjugate in $\textup{O}(4,\overline{\mathbb{F}_{q}})$. Now consider the
following two homomorphisms:

$\displaystyle\varphi_{1},\varphi_{2}:\mathbb{Z}/p\mathbb{Z}\oplus\mathbb{Z}/p\mathbb{Z}\longrightarrow\textup{O}(4,\overline{\mathbb{F}_{q}})$
given by $\displaystyle\varphi_{1}((1,0))=X_{1},\varphi_{1}((0,1))=X_{a}$
$\displaystyle\varphi_{2}((1,0))=X_{a},\varphi_{2}((0,1))=X_{1}.$

Then although these two homomorphisms are element-conjugate, they are not
globally conjugate, using the same argument as of Proposition 4.2. ∎

######  Theorem 6.10.

For $m\geq 2$, the groups $\textup{O}(2m,q)$ are not acceptable.

###### Proof.

This is an immediate consequence of Proposition 6.8 and Proposition 6.9. ∎

##  7\. Concluding remarks and future questions

###  7.1. Finite groups of Lie type

Consider the finite general linear group $\textup{GL}(n,q)$, where $q$ is a
power of prime. Following the proof of Theorem 4.3 we easily see that these
groups are not acceptable, for all $n$ and $q$. The same result holds for
$\textup{U}(n,q)$ as well. For finite fields, in case of symplectic and
orthgonal (three different) groups the unipotent conjugacy class further
breaks into several conjugacy classes. Hence the method adopted here will not
be applicable to decide the acceptability of these groups. At this point we
believe that these groups are unacceptable as well.

###  7.2. Exceptional cases

We have seen that the groups discussed in this article are all unacceptable.
On the contrary, the analogue of these groups (defined over $\mathbb{C}$ or
$\mathbb{R}$) are acceptable (see [9, pp. 254] for precise table). In the same
paper M. Larsen proved that $F_{4}(\mathbb{C})$ and its compact form are
unacceptable (see [9, Proposition 3.11]). Hence it will be interesting to see
which exceptional groups (defined over $\overline{\mathbb{F}_{p}}$) are
acceptable.

###  7.3. Application to invariant theory

Invariant theory studies the ring of those regular functions on an affine
variety $X$ which are constant on the orbits of an action of a linear
algebraic group $G$ on $X$, where the action is given by a morphism $G\times
X\longrightarrow X$. Consider a field $\mathbb{F}$ and a subgroup $G$ of
$\textup{GL}(V)\cong\textup{GL}(n,\mathbb{F})$. Take an action of $G$ on a
variety $X$. The algebra $\mathbb{F}[X]$ of regular functions on $X$, can be
equipped with a $G$-module structure by defining $g\cdot f(v)=f(g^{-1}\cdot
v)$ for all $g\in G$, $f\in V^{*}$ and $v\in V$. The ring of invariant

$\displaystyle\mathbb{F}[X]^{G}=\\{f\in\mathbb{F}[X]|g\cdot f=f\text{ for all
}g\in G\\}$

is finitely generated, due to classical results of Noether (see [13], [14]). A
consequence of seminal works [7] and [8] of the French mathematician Vincent
Lafforgue is that: when $\mathbb{F}=\mathbb{C}$ and $G$ is a connected linear
reductive group and if $\mathbb{C}[G^{n}]^{G}$ (where the action is diagonal
conjugation) is generated by $1$-argument invariants for all $n\geq 1$, then
$G$ is acceptable. The Italian mathematician Claudio Procesi has proved that
this is true for classical groups general linear groups, symplectic groups,
and orthogonal groups over $\mathbb{C}$ in his works [15] and [16]. Given that
we have proved the unacceptability of
$\textup{GL}(n,\overline{\mathbb{F}_{q}})$,
$\textup{SL}(n,\overline{\mathbb{F}_{q}})$,
$\textup{U}(n,\overline{\mathbb{F}_{q}})$,
$\textup{Sp}(2n,\overline{\mathbb{F}_{q}})$,
$\textup{O}^{\epsilon}(n,\overline{\mathbb{F}_{q}})$, we would like to finish
by posing the question that does there exist $n\geq 1$ such that
$\overline{\mathbb{F}_{q}}[G^{n}]^{G}$ is not generated by $1$-argument
invariants in case $G$ is one of the groups discussed in this paper.

## References

  * [1] U. K. Anandavardhanan and D. Prasad, A local-global question in automorphic forms, Compos. Math., 149 (2013), pp. 959–995.
  * [2] D. Blasius, On multiplicities for ${\rm SL}(n)$, Israel J. Math., 88 (1994), pp. 237–251.
  * [3] T. C. Burness and M. Giudici, Classical groups, derangements and primes, vol. 25 of Australian Mathematical Society Lecture Series, Cambridge University Press, Cambridge, 2016.
  * [4] S. Gonshaw, M. W. Liebeck, and E. A. O’Brien, Unipotent class representatives for finite classical groups, J. Group Theory, 20 (2017), pp. 505–525.
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  * [6] L. C. Grove, Classical groups and geometric algebra, vol. 39 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 2002.
  * [7] V. Lafforgue, Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale, J. Amer. Math. Soc., 31 (2018), pp. 719–891.
  * [8] , Shtukas for reductive groups and Langlands correspondence for function fields, in Proceedings of the International Congress of Mathematicians—Rio de Janeiro 2018. Vol. I. Plenary lectures, World Sci. Publ., Hackensack, NJ, 2018, pp. 635–668.
  * [9] M. Larsen, On the conjugacy of element-conjugate homomorphisms, Israel J. Math., 88 (1994), pp. 253–277.
  * [10] , On the conjugacy of element-conjugate homomorphisms. II, Quart. J. Math. Oxford Ser. (2), 47 (1996), pp. 73–85.
  * [11] I. G. Macdonald, Numbers of conjugacy classes in some finite classical groups, Bull. Austral. Math. Soc., 23 (1981), pp. 23–48.
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  * [13] E. Noether, Der Endlichkeitssatz der Invarianten endlicher Gruppen, Math. Ann., 77 (1915), pp. 89–92.
  * [14] , Der Endlichkeitssatz der Invarianten endlicher Gruppen der charakteristik p, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, (1926), pp. 28–35.
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  * [16] , Lie groups, Universitext, Springer, New York, 2007.  An approach through invariants and representations.
  * [17] T. Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math. (2), 121 (1985), pp. 169–186.
  * [18] D. E. Taylor, Conjugacy classes in finite orthogonal groups, (2020).
  * [19] , Conjugacy classes in finite symplectic groups, (2020).
  * [20] , Conjugacy classes in finite unitray groups, (2020).
  * [21] G. E. Wall, On the conjugacy classes in the unitary, symplectic and orthogonal groups, J. Austral. Math. Soc., 3 (1963), pp. 1–62.
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# High frequency non-gyrokinetic turbulence at tokamak edge parameters

M. Raeth<EMAIL_ADDRESS>Max Planck Institute for Plasma Physics,
Boltzmannstr. 2, 85748 Garching, Germany  K. Hallatschek
<EMAIL_ADDRESS>Max Planck Institute for Plasma Physics,
Boltzmannstr. 2, 85748 Garching, Germany

###### Abstract

First of a kind 6D-Vlasov computer simulations of high frequency ion Bernstein
wave turbulence for parameters relevant to the tokamak edge show transport
comparable to sub-Larmor-frequency gyrokinetic turbulence. The customary
restriction of magnetized plasma turbulence studies to the gyrokinetic
approximation may not be based on physics but only a practical constraint due
to computational cost.

Five-dimensional gyrokinetic turbulence simulations have become the gold
standard for theoretical transport studies in the core of magnetic confinement
systems. Yet, their descriptive power still falls short of the tokamak edge-
plasma and particularly the high confinement mode (H-mode) transition, likely
due to the presence of high gradients ($\rho_{i}/L_{n}=O(1)$) and elevated
fluctuation levels ($\delta n/n=O(1)$). On the other hand, the axioms of
gyrokinetics, that the distribution function is nearly constant on a Larmor
circle and the turbulence time scales long compared to the Larmor period, are
ultimately just assumptions. Time and again, confined plasma has proven to
evade preconceptions, such as with the unexpected finding of the importance of
“hyperfine turbulence” [1] on the extremely small electron Larmor radius
scale. Nevertheless, owing to the many extremely disparate time scales for
strongly magnetized plasmas (e.g. for transport, gyration, Alfven-waves),
there have been only steps in the direction of full 6D simulations, consisting
of only two space dimensions [2] and/or linear verification with standard
gyro-kinetic ion temperature gradient (ITG) modes [3], or the study of waves
[4] in a gyrokinetic background. (The situation is very different from
astrophysics or inertial fusion, where 6D simulations are common place.)

Using the novel 6D turbulence code BSL6D, which is particularly efficient in
simulating ion gyrations [5], we demonstrate that indeed genuinely non-
gyrokinetic instabilities and turbulence have to be taken into account for
high gradients such as those typical for a tokamak edge.

The target of our 6D simulations is the ion kinetic equation in dimensionless
($\rho_{i}=v_{\text{th}}=n=T=1$) variables

$\partial_{t}f+\bm{v}\cdot\nabla
f+(-\nabla\phi+\bm{v}\times\bm{B})\cdot\nabla_{\bm{v}}f=0$ (1)

where for simplicity we restrict ourselves to electrostatic perturbations in
“slab geometry”, i.e., to a homogeneous constant magnetic field
$\bm{B}=\hat{z}$. Furthermore, the system is assumed to be quasineutral
($n_{i}=n_{e}$) and the electrons to be adiabatic, $n_{e}=\phi$. The initial
state is chosen to be in (unstable) equilibrium, which is why it has to be a
function of the gyrocenter coordinate $\bm{r}_{gc}=\bm{r}-\bm{\rho}$,
$\bm{\rho}=\hat{z}\times\bm{v}$, in particular, $x_{gc}=x+v_{y}$. We choose
the gradients to be in x direction, such as

$f_{0}:=f|_{t=0}=g(x_{gc},v),\quad g(x,v)=f_{M}(n(x),T(x)),$ (2)

with the Boltzmann distribution $f_{M}(n,T)=n(2\pi
T)^{-3/2}e^{-\frac{v^{2}}{2T}}$, where $n(x),T(x)$ represent the background
density and temperature profiles as function of the gyrocenter position.

## Linear equations

To study the stability of gyrokinetic and non-gyrokinetic modes for given
density and temperature gradients we linearize (1) around $h=0$ with $f=h-\phi
g_{0}+f_{0}$, $g_{0}=f_{M}(1,1)$, resulting in

$\partial_{t}h+\bm{v}\cdot\nabla
h+\bm{v}\times\hat{z}\cdot\nabla_{\bm{v}}h=\partial_{t}\phi
g_{0}+\nabla\phi\cdot\nabla_{v}(f_{0}-g_{0}).$ (3)

Using the Boussinesq approximation

$\nabla_{v}(f_{0}-g_{0})\approx\nabla_{v}(f_{0}-g_{0})|_{x_{gc}=0}=:\bm{v}^{*}g_{0},$
(4) $\text{with}\;\bm{v}^{*}=\hat{y}\left[\kappa_{T}\partial_{T}(\ln
f_{M})+\kappa_{n}\right],$ (5)

where $\kappa_{n}=L_{n}^{-1}=(\ln n)^{\prime}$, $\kappa_{T}=L_{T}^{-1}=(\ln
T)^{\prime}$, $\partial_{T}\ln f_{M}=(v^{2}-3)/2$, the linearized equation
becomes translation invariant and the response of the velocity integral over
$h$ to an electric potential on the right hand side of (3) can be analyzed in
Fourier space.

At $\bm{v}^{*}=0$ the response is the well-known Gordeyev integral [6, 7]

$\int\frac{h}{\phi}d^{3}v=G_{\omega}(k_{\perp}\sqrt{T},k_{z}\sqrt{T})=$ (6)
$=-i\omega\int_{0}^{\infty}e^{-2Tk_{\perp}^{2}\sin^{2}\frac{t^{\prime}}{2}-\frac{k_{z}^{2}Tt^{\prime
2}}{2}+i\omega t^{\prime}}dt^{\prime},$ (7)

where the $T$ dependence has been explicitly indicated. Noting the structure
of the right hand side, the response for finite gradient term $\bm{v}^{*}$ can
be expressed using a $T$-derivative,

$\int\frac{h}{\phi}d^{3}v=\left[1-\frac{k_{y}}{\omega}(\kappa_{T}\partial_{T}+\kappa_{n})\right]G_{\omega}(k_{\perp}\sqrt{T},k_{z}\sqrt{T})=$
(8)
$=\left[-\omega+{k_{y}}(\kappa_{T}\partial_{T}+\kappa_{n})\right]\sum_{p\in\mathds{Z}}\frac{\Gamma_{p}(T{k_{\perp}}^{2})}{|{k_{z}}|\sqrt{2T}}Z\left(\frac{\omega-p}{|{k_{z}}|\sqrt{2T}}\right),$
(9)

where the series expansion [7] of $G_{\omega}$ has been inserted,
$\Gamma_{p}(x)=I_{p}(x)e^{-x}$,
$Z(x)=e^{-x^{2}}(i\sqrt{\pi}-2\int_{0}^{x}e^{x^{\prime 2}}dx^{\prime})$ is the
Fried-Conte plasma dispersion function, and the expressions are to be
evaluated at $T=1$. The electrostatic dispersion relation follows from quasi-
neutrality $n_{i}=n_{e}$, where $n_{i}=\int hd^{3}v-\phi$ and $n_{e}=\phi$, as

$2=\int hd^{3}v\,/\,\phi.$ (10)

In the absence of gradients this equation describes non-gyrokinetic
“neutralized” ion Bernstein waves (IBW) close to the harmonics of the Larmor
frequency with a frequency shift depending on the wavenumbers [8]. Already for
relatively small gradient parameters $\kappa_{n},\kappa_{T}<1$ IBW for many
harmonics are destabilized, as depicted by the numerical solutions of (10) in
Fig. 1, easily exceeding the growth rate of the (gyrokinetic) ITG mode. The
IBWs at higher harmonics tend to have a higher wavenumber of maximum growth
rate.

Figure 1: Growth rate $\Im\omega$ and frequency shift $\Re(\omega-p)$ relative
to the associated harmonic $p$ of the Larmor frequency for various eigenmodes
of (10) for $\kappa_{T}=0.34$, $\kappa_{n}=0.23$, ${k_{\perp}}=k_{y}$,
$k_{z}=0.025$; $p=0$ represents the gyrokinetic slab ITG mode, $p>0$ non-
gyrokinetic unstable IBWs.

## Critical gradients of ion Bernstein waves

In principle, unstable non-gyrokinetic modes can exist for arbitrary complex
$\omega$, since the real and imaginary part of (10) can always be satisfied by
choosing suitable real parameters $\kappa_{n},\kappa_{T}$, which in general
turn out to be of order of the inverse Larmor radius. However, we would like
to know under which conditions there are also low gradient non-gyrokinetic
instabilities with $|k_{y}\kappa_{n}|,|k_{y}\kappa_{T}|\ll 1$ and $\omega$
arbitrarily close to the Larmor resonances – similar to the well-known
gradient threshold for the (slab) gyrokinetic ITG modes at very small
frequencies. Cauchy’s argument principle [9] implies that threshold parameters
for the instability fulfill the dispersion relation exactly at real $\omega$.
Conversely, stability is implied, if no such threshold parameters can be
found.

Due to the Gaussian dependence of the imaginary part of $Z$, it can be
neglected in all but one Larmor resonance $p$ for frequencies close to the
gyro resonances, i.e., if $\delta\omega=(\omega-p)\ll 1$ and $|k_{z}|\ll 1$,
the latter of which must be verified a posteriori. The imaginary part of
(10,9) then turns into

$[-\omega+k_{y}(\kappa_{n}+\kappa_{T}\partial_{T})]\frac{\Gamma_{p}(T{k_{\perp}}^{2})}{\sqrt{T}}e^{-\frac{\delta\omega^{2}}{2{k_{z}}^{2}T}}=0,$
(11)

equivalent to

$-\frac{\omega}{k_{y}}+\kappa_{n}+\left(\frac{\delta\omega^{2}}{k_{z}^{2}}+\frac{d\ln\Gamma_{p}({k_{\perp}}^{2})}{d\ln{k_{\perp}}}-1\right)\frac{\kappa_{T}}{2}=0,$
(12)

while the real part of the dispersion relation is

$\chi_{\text{nr}}+\left[-\omega+{k_{y}}(\kappa_{T}\partial_{T}+\kappa_{n})\right]\frac{\Gamma_{p}(T{k_{\perp}}^{2})}{|{k_{z}}|\sqrt{2T}}Z\left(\frac{\delta\omega}{|{k_{z}}|\sqrt{2T}}\right)=0$
(13)

with the weakly $\omega$-dependent non-resonant charge density response
defined as

$\chi_{\text{nr}}=-2+\left[-\omega+{k_{y}}(\kappa_{T}\partial_{T}+\kappa_{n})\right]\sum_{q\in\mathds{Z},q\neq
p}\frac{\Gamma_{q}(T{k_{\perp}}^{2})}{|{k_{z}}|\sqrt{2T}}Z\left(\frac{\omega-q}{|{k_{z}}|\sqrt{2T}}\right).$
(14)

In the limit $\omega=p$, $k_{z}\rightarrow 0$ and small gradients, the
constant term dominates the expression for $\chi_{\text{nr}}$. Direct
numerical inspection for varying $p$ and arbitrary ${k_{\perp}}$ shows that
$|\chi_{\text{nr}}+2|<1.29|1-k_{y}\kappa_{n}/\omega|+0.52|k_{y}\kappa_{T}/\omega|$
for $p\neq 0$, and $|\chi_{\text{nr}}+2|<|k_{y}\kappa_{T}|$ for $p=0$. Hence,
for sufficiently small gradients $\chi_{\text{nr}}<0$, and more specifically
$-2.3\lesssim\chi_{\text{nr}}\lesssim-0.7$. Solving (11) for $\kappa_{n}$ and
inserting it into (13) cancels the plasma dispersion function,

$\chi_{\text{nr}}+\frac{1}{2}\frac{\delta\omega}{k_{z}^{2}}\kappa_{T}k_{y}\Gamma_{p}({k_{\perp}}^{2})=0,$
(15)

where the identity $Z^{\prime}(x)=-2(1+xZ(x))$ has been used.

Solving (15) for $k_{z}^{2}$ one can replace $k_{z}$ as a parameter by
$\delta\omega$ provided that
$\delta\omega\,k_{y}\kappa_{T}\chi_{\text{nr}}<0$. With this, (11) results in

$-\frac{p}{k_{y}}+\kappa_{n}+\left(\frac{d\ln\Gamma_{p}({k_{\perp}}^{2})}{d\ln{k_{\perp}}}-1\right)\frac{\kappa_{T}}{2}-\delta\omega\left(1+\frac{\chi_{\text{nr}}}{\Gamma_{p}}\right)=0.$
(16)

Fixing the signs of $k_{y},\kappa_{T}$ to be positive, the sign of the term
proportional to $\delta\omega$ is always positive for low gradients, since
$\chi_{\text{nr}}<-\Gamma_{p}$, because $\Gamma_{0}\leq 1$,
$\Gamma_{p}<0.25/p$ for any ${k_{\perp}}$. Thus, the existence of a threshold
requires the rest of (16) to be negative, in other words,

$\kappa_{n}<\frac{p}{k_{y}}+\left(1-\frac{d\ln\Gamma_{0}({k_{\perp}}^{2})}{d\ln{k_{\perp}}}\right)\frac{\kappa_{T}}{2}.$
(17)

For the gyrokinetic ITG mode $p=0$, and the standard threshold condition
results. For the instability of non-gyrokinetic modes (17) requires $p>0$ and
no conditions on the gradients themselves besides their assumed smallness, and
the condition that $\kappa_{T}\neq 0$, since otherwise (15) cannot be solved
for $k_{z}^{2}$. Then, for an arbitrarily small $\delta\omega$ a marginally
stable threshold mode exists at small $k_{y},{k_{\perp}}$, since

$\frac{k_{y}\chi_{\text{nr}}}{\Gamma_{p}({k_{\perp}}^{2})}=-O(k_{y}^{1-2p})\rightarrow-\infty\mbox{
for }k_{y}={k_{\perp}}\rightarrow 0$ (18)

and another one at large ${k_{\perp}}$ at fixed $k_{y}$, since

$\frac{k_{y}\chi_{\text{nr}}}{\Gamma_{p}({k_{\perp}}^{2})}=-O({k_{\perp}})\rightarrow+\infty\mbox{
for }{k_{\perp}}\rightarrow\infty,k_{y}=const.$ (19)

Since according to (15) in both limits ${k_{z}}\rightarrow 0$, the initial
assumption of a single resonance contributing to the imaginary part of the ion
response is justified. On the other hand, if (17) does not hold, (16) has no
valid solutions in ${k_{\perp}}$ and $\delta\omega$, and hence there are no
real solutions for any $\omega$, $k_{z}$, $k_{y}$, and ${k_{\perp}}$ for the
system of equations (11,13) and therefore according to Cauchy’s argument
theorem [9] there are no low-gradient instabilities close to the resonance in
the system.

Since the condition $p>0$ only concerns the sign of the phase velocity and not
the gradients themselves, the non-gyrokinetic high frequency IBW instabilities
occur even when the gyrokinetic ITG modes are ruled out by the thresholds –
the IBWs just require a non-zero temperature gradient.

Solving the equations (15,16) numerically for $\delta\omega$ and $\kappa_{T}$
without restriction to weak gradients (Fig. 2) confirms the theory. As
expected, low frequency slab ITG modes at $p=0$ show an increasing temperature
gradient threshold with increasing density gradient. The ratio of temperature
to density gradient has to be $\eta=\frac{\kappa_{T}}{\kappa_{n}}\gtrapprox
0.9$ at the most unstable wavenumber. In contrast, the graph for $p=1$, but
not $p<0$, shows that the Bernstein instability can occur for very small
temperature gradients. Interestingly, in the H-mode typically $\eta=0.6-1.2$
[10], which should largely suppress the gyrokinetic ion temperature gradient
mode. In all cases the critical temperature gradient increases with the
density gradient.

Figure 2: Critical temperature gradient for low frequency ITG mode (middle)
and first harmonic IBWs with $p=-1$ (left) and $p=1$ (right) for varying
$\kappa_{n}$ at $k_{z}=0.025$.

As for the growth rates at fixed temperature gradient (Fig. 3) one sees that
in contrast to $p=0$ the modes with $p>0$ are further destabilized with
increasing density gradient. The ITG instability is stabilized for $\eta<1.1$,
while the high frequency IBW have the maximum growth rate at $\eta<1$, again
suggesting their relevance in the H-mode regime, possibly as part of the
residual transport.

Figure 3: Growth rate of ITG ($p=0$) and IBW ($p\neq 0$) instability as a
function of $\frac{1}{\eta}=\frac{\kappa_{n}}{\kappa_{T}}$ for fixed
temperature gradient $\kappa_{T}=0.36$ and $k_{y}={k_{\perp}}=2$,
$k_{z}=0.025$.

## 6D turbulence simulations

Having executed extensive linear and nonlinear benchmarking simulations [11,
12] with BSL6D to solve (1), confirming the dispersion relation and
reproducing results analogous to [3], for brevity we report only on a
characteristic turbulence run showing the most relevant features of the high-
frequency turbulence.

Since there is no equivalent to the gyrokinetic flux-tube scenario for the
non-gyrokinetic equations without the additional approximation (4), we use
periodic boundary conditions. The ion distribution function
$f({\bm{x}},{\bm{v}})$ is initialized with a Maxwellian $f_{0}=f_{M}(n,T)$
with a background density and temperature profile similar to (2),

$n=1+g_{n}\sin\frac{2\pi
x_{gc}}{L_{x}}+\text{ran}(\bm{r}_{gc}),\;T=1+g_{T}\sin\frac{2\pi
x_{gc}}{L_{x}},$ (20)

to eliminate spurious transient gyro-oscillations, with the addition of a
small white noise density perturbation $\text{ran}(\bm{r}_{gc})$. The profile
amplitudes have been set to $g_{T}=0.6$ and $g_{n}=0.49$, resulting in the
maximum inverse gradient lengths $\max(\kappa_{T})=0.375$ and
$\max(\kappa_{n})=0.28$, to specifically render both slab ITG and IBWs
unstable. (For lower temperature gradient the ITG would be completely
suppressed.)

The computation has been carried out in a phase space region covering the
spatial domain $L_{x}\times L_{y}\times L_{z}=4\pi\times 2\pi\times 80\pi$ and
the velocity domain $[-4,4]^{3}$ using a computational grid of dimensions
$64\times 32\times 16\times 32\times 32\times 16$ for the phase space
coordinates $(x,y,z,v_{x},v_{y},v_{z})$. $N_{t}=1250000$ steps have been
carried out with the time step $\Delta t=0.008$ up to the end time
$t_{e}=10000$, using $150000$ core hours on the Intel Xeon 8160 CPUs of the
Cineca Marconi cluster. In the following, band pass filters have been used
diagnostically to separate fluctuation frequencies in the range
$|\omega-p|<1/2$ for integer $p$.

Figure 4: Amplitude of dominant Fourier modes in $y$-direction at their local
maxima, $x=5.6$ and $x=3.6$ at $k_{z}=0.025$ for gyrokinetic modes $p=0$ and
the first Larmor harmonic $p=1$.

The simulation starts with the linear growth followed by the turbulent
saturation of the instabilities. The amplitude of the dominant gyrokinetic and
non-gyrokinetic modes are shown in Fig. 4. Their respective growth rates
(indexed “BSL6D”),

$\gamma_{\mathrm{BSL6D}}$ $\gamma_{\mathrm{ana}}$ $p=0$ 0.0040 0.0024 $p=1$
0.0062 0.0070

are consistent with the ones (indexed “ana”) computed from the dispersion
relation (10) using the local gradients $\kappa_{n}=0.25$, $\kappa_{T}=0.34$,
and $(k_{x},k_{y},k_{z},p)$ equal to $(0.5,2,0.025,0)$ and $(3,4,0.025,1)$,
respectively. An exact agreement cannot be expected, since the inherently non-
local setup of the simulation prohibits eigenmodes with a well defined
wavenumber $k_{x}$.

Figure 5: Cross sections of potential fluctuations $\phi$ for $p=0$ (left
column) and $p=1$ (right column) at fixed $z$ and $t=1500,5500$ (rows).

The progression to turbulence is also observable in 2D cross sections of the
electric potential (Fig. 5) (for a movie see [13]). At $t=1500$ the simulation
is in the linear phase and a clear mode structure of the ITG and IBW (with
larger wavenumbers) is visible. At $t=5500$ the potential indicates nonlinear
ITG/IBW turbulence.

While the total energy flux across the magnetic field in the 6D turbulence
simulations necessarily contains contributions from the Poynting flux [14] and
genuinely non-gyrokinetic terms, the dominant component in the case at hand
still comes from the $\bm{E}\times\bm{B}$ flow of the internal ion energy
density $\epsilon=\int fv^{2}d^{3}v/2$, namely
$\bm{q}_{\perp}\approx\epsilon\hat{z}\times\nabla\phi.$ Despite their high
wavenumbers, the non-gyrokinetic $p\neq 0$ IBW components of the potential
fluctuations are significant in comparison to the ITG fluctuations. The
transport of the former even exceeds the one of the latter (Fig. 6) in the
turbulent phase, although the mixing length argument would suggest a much
lower transport due to the high wavenumber. We believe this to be at least
partially due to the (necessarily) non-local setup, which tends to damp long
wavelengths.

Figure 6: Comparison between low frequency modes $p=0$ and IBWs ($p\neq 1$)
for potential amplitude $|\phi|$ (top) and internal ion energy flux $q_{x}$
(bottom), spatially averaged at $x=7.3$ and box-car averaged over $380$ time
units.

## Discussion

Using a novel 6D kinetic simulation code specifically designed for strongly
magnetized plasmas we have shown that high frequency non-gyrokinetic
instabilities (ion Bernstein waves, IBWs) can produce transport-relevant
turbulence, even in competition to gyrokinetic ITG turbulence, which is
usually exclusively considered in those plasmas. The simulations have been
simplified by assuming electrostatic fields, adiabatic electrons and a
homogeneous magnetic field. They refer therefore mostly to the ion turbulence
component of low-$\beta$, high-gradient plasmas, where the magnetic
fluctuations are small and the magnetic field inhomogeneities are unimportant
compared to the plasma gradients. Nevertheless, the IBWs can be analytically
shown to be unstable at arbitrarily low gradients and wavenumbers. They
require only the presence of a temperature gradient, and not the excess of a
temperature gradient threshold like the ITG modes.

Gradient lengths such as in the simulation detailed above, $\sim 3\rho_{i}$,
are regularly encountered in the edge of tokamak H-mode discharges [10], where
also the temperature gradients are typically below the ITG instability
threshold, since $\eta=L_{n}/L_{T}\sim 0.6-1.2$. This, together with the
potential breakdown of gyrokinetic theory for short gradient lengths and the
lack of a satisfying explanation for the L/H-transition, suggests a full 6D
treatment of turbulence to be very relevant to the edge region of magnetic
confinement devices. In future work, more detailed electron physics and
inhomogeneous, fluctuating magnetic fields will be included in such
simulations. It seems likely that the inclusion of additional degrees of
freedom would rather increase than decrease the intensity of the non-
gyrokinetic modes.

###### Acknowledgements.

This work has been carried out partly within the framework of the EUROfusion
Consortium, funded by the European Union via the Euratom Research and Training
Programme (Grant Agreement No 101052200 – EUROfusion). Support has also been
received by the EUROfusion High Performance Computer (Marconi-Fusion). Views
and opinions expressed are however those of the author(s) only and do not
necessarily reflect those of the European Union or the European Commission.
Neither the European Union nor the European Commission can be held responsible
for them. Numerical simulations were performed at the MARCONI-Fusion
supercomputer at CINECA, Italy and at the HPC system at the Max Planck
Computing and Data Facility (MPCDF), Germany.

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# Extended Path Partition Conjecture for Semicomplete and Acyclic Compositions

Jiangdong Ai Department of Computer Science. Royal Holloway University of
London<EMAIL_ADDRESS>Stefanie Gerke Department of
Mathematics. Royal Holloway University of London<EMAIL_ADDRESS>Gregory Gutin Department of Computer Science. Royal Holloway University of
London<EMAIL_ADDRESS>Yacong Zhou Department of Computer Science. Royal
Holloway University of London<EMAIL_ADDRESS>
###### Abstract

Let $D$ be a digraph and let $\lambda(D)$ denote the number of vertices in a
longest path of $D$. For a pair of vertex-disjoint induced subdigraphs $A$ and
$B$ of $D$, we say that $(A,B)$ is a partition of $D$ if $V(A)\cup V(B)=V(D).$
The Path Partition Conjecture (PPC) states that for every digraph, $D$, and
every integer $q$ with $1\leq q\leq\lambda(D)-1$, there exists a partition
$(A,B)$ of $D$ such that $\lambda(A)\leq q$ and $\lambda(B)\leq\lambda(D)-q.$
Let $T$ be a digraph with vertex set $\\{u_{1},\dots,u_{t}\\}$ and for every
$i\in[t]$, let $H_{i}$ be a digraph with vertex set
$\\{u_{i,j_{i}}\colon\,j_{i}\in[n_{i}]\\}$. The composition
$Q=T[H_{1},\dots,H_{t}]$ of $T$ and $H_{1},\ldots,H_{t}$ is a digraph with
vertex set $\\{u_{i,j_{i}}\colon\,i\in[t],j_{i}\in[n_{i}]\\}$ and arc set

$A(Q)=\cup^{t}_{i=1}A(H_{i})\cup\\{u_{i,j_{i}}u_{p,q_{p}}\colon\,u_{i}u_{p}\in
A(T),j_{i}\in[n_{i}],q_{p}\in[n_{p}]\\}.$

We say that $Q$ is acyclic (semicomplete, respectively) if $T$ is acyclic
(semicomplete, respectively). In this paper, we introduce a conjecture
stronger than PPC using a property first studied by Bang-Jensen, Nielsen and
Yeo (2006) and show that the stronger conjecture holds for wide families of
acyclic and semicomplete compositions.

##  1 Introduction

Let $G$ be a directed or undirected graph. The order of $G$ is the number of
vertices in $G.$ Let $\lambda(G)$ be the maximum order of a path in $G.$ (In
this paper all paths and cycles in digraphs are directed.) A partition of $G$
is a pair $(A,B)$ of vertex-disjoint induced subgraphs of $G$ such that
$V(A)\cup V(B)=V(G).$ Terminology and notation for digraphs not introduced in
this section follows those in [2, 3].

The Path Partition Conjecture (PPC) can be stated as follows.

######  Conjecture 1.

[Path Partition Conjecture (PPC)] Let $G$ be a directed or undirected graph.
For every natural number $q$ with $0<q<\lambda(G)$ there is a partition
$(A,B)$ of $G$ such that $\lambda(A)\leq q$ and $\lambda(B)\leq\lambda(G)-q.$

Note that the PPC trivially holds for every directed and undirected graph $G$
with a Hamilton path $P$ as one can choose any induced subgraph on $q$
vertices for $A$. In general, the PPC remains far from proven. Frick [10]
mentioned that the PPC for undirected graphs was discussed by Lovász and Mihók
in $1981$ in Szeged. Several results supporting the conjecture for undirected
graphs have been proved in the literature, see e.g. [6, 7, 8, 12, 13, 16]. In
1982, Laborde et al. [14] extended Conjecture 1 to digraphs. (Note that the
PPC for undirected graphs is equivalent to the PPC for symmetric digraphs; a
digraph $D$ is symmetric if it can be obtained from an undirected graph $G$ by
replacing every edge $uv$ by the pair $uv,vu$ of arcs.) There is a number of
papers in which the PPC is proven for special classes of digraphs, see e.g.
[1, 5, 9, 14, 17].

In this paper we consider a stronger notion which was introduced in [5] by
Bang-Jensen, Nielsen and Yeo which we call the BNY property. To state the
property we need the following notation. A $k$-path subdigraph of a digraph
$D$ is a collection of $k$ vertex-disjoint paths in $D$. The maximum order of
a $k$-path subdigraph $D$ will be denoted by $\lambda_{k}(D)$; clearly
$\lambda(D)=\lambda_{1}(D).$

Property BNY. A digraph $D$ satisfies BNY if for every integer $q$ with $1\leq
q\leq\lambda(D)-1$ there is a partition $(A,B)$ of $D$ such that
$\lambda(A)\leq q$ and for all $k\in\\{1,2,\dots,|V(B)|\\}$,
$\lambda_{k}(B)\leq\lambda_{k}(D)-q.$

We conjecture that one can strengthen the PPC in the following way.

######  Conjecture 2.

Property BNY holds for every digraph.

Bang-Jensen, Nielsen and Yeo [5] proved that Conjecture 2 holds for several
generalizations of tournaments, see also Corollary 8 in Section 3. In this
paper we show that Conjecture 2 holds for some compositions of digraphs. Let
$T$ be a digraph with vertex set $\\{u_{1},\dots,u_{t}\\}$ and for every
$i\in[t]$, let $H_{i}$ be a digraph with vertex set
$\\{u_{i,j_{i}}\colon\,j_{i}\in[n_{i}]\\}$. The _composition_ of $T$ and
$H_{1},\dots,H_{t}$ is the digraph $Q=T[H_{1},\dots,H_{t}]$ with vertex set
$\\{u_{i,j_{i}}\colon\,i\in[t],j_{i}\in[n_{i}]\\}$ and arc set

$A(Q)=\cup^{t}_{i=1}A(H_{i})\cup\\{u_{i,j_{i}}u_{p,q_{p}}\colon\,u_{i}u_{p}\in
A(T),j_{i}\in[n_{i}],q_{p}\in[n_{p}]\\}.$

See Figure 1 for an example of a composition, $Q=T_{3}[H_{1},H_{2},H_{3}]$
where $V(H_{1})=\\{a,d\\}$, $V(H_{2})=\\{b\\}$, $V(H_{3})=\\{c\\}$ and $T_{3}$
is a 3-cycle.

$a$$b$$c$$d$ Figure 1: $Q=T_{3}[H_{1},H_{2},H_{3}]$

A composition $Q$ is acyclic if $T$ is acyclic, that is, $T$ contains no
directed cycles. Acyclic digraphs form a well-studied family of digraphs of
great interest in graph theory, algorithms and applications (e.g. see Chapter
3 of [3]). The main result of [1] states that if $Q=T[H_{1},\dots,H_{t}]$ is
an acyclic composition and the PPC holds for every digraph $H_{i}$, $i\in[t]$,
then the PPC holds for $Q$. We show that analogous result for the BNY
property, see Section 2.

######  Theorem 3.

Let $Q=T[H_{1},\dots,H_{t}]$ be an acyclic composition. If for every
$i\in[t]$, $H_{i}$ satisfies property BNY, then $Q$ satisfies property BNY.

Note that a single vertex satisfies the property BNY and it follows from
Theorem 3 that every acyclic digraph satisfies property BNY.

We also consider semicomplete compositions. A digraph $D$ is semicomplete if
for every pair $x,y$ of distinct vertices of $D$, there is at least one arc
between $x$ and $y.$ In particular, a tournament is a semicomplete digraph,
where there is exactly one arc between $x,y$ for every pair $x,y$ of distinct
vertices. A composition $Q=T[H_{1},\ldots,H_{t}]$ is called _semicomplete_ if
$T$ is a semicomplete digraph. We show (a strengthening of) the following
theorem in Section 3.

######  Theorem 4.

Let $Q=T[H_{1},\dots,H_{t}]$ be a semicomplete composition. If for every
$i\in[t]$, $H_{i}$ satisfies property BNY, then $Q$ satisfies property BNY.

##  2 Acyclic Compositions

We restate Theorem 3 for the convenience of the reader. Note that the main
idea of the proof of Theorem 3 is similar to that of the main acyclic
composition result of [1].

Theorem 3 Let $Q=T[H_{1},\dots,H_{t}]$ be an acyclic composition. If for every
$i\in[t]$, $H_{i}$ satisfies property BNY, then $Q$ satisfies property BNY.

###### Proof.

For $i=1,\ldots,t$, define $p_{i}^{in}$ as the maximum order of a path in
$Q-H_{i}$ whose terminal vertex has an arc into $H_{i}$, and $p_{i}^{end}$ as
the maximum order of a path in $Q$ such that the terminal vertex of this path
belongs to $H_{i}$. Note that $T$ is acyclic and thus no path can enter and
leave $H_{i}$ more than once and hence $p_{i}^{end}=p_{i}^{in}+\lambda(H_{i})$
and in particular $p_{i}^{end}>p_{i}^{in}$.

We partition $Q$ as follows

  1. 1.

If $p_{i}^{end}\leq q$, then the entire graph $H_{i}$ belongs to $A$.

  2. 2.

If $p_{i}^{in}\geq q$, then the entire graph $H_{i}$ belongs to $B$.

  3. 3.

If $p_{i}^{in}<q<p_{i}^{end}$, we let $q_{i}=q-p^{in}_{i}$ and partition
$H_{i}$ into $(A_{i},B_{i})$ such that for all $k=1,\ldots,V(B_{i})$, we have
$\lambda(A_{i})\leq q_{i}$ and
$\lambda_{k}(B_{i})\leq\lambda_{k}(H_{i})-q_{i}$. The digraph $A_{i}$ belongs
to $A$ and the digraph $B_{i}$ belongs to $B$.

We first show that $\lambda(A)\leq q$. Consider a longest path $P$ in $A$ and
let $x\in V(H_{i})$ be the last vertex of this path. If $p_{i}^{end}\leq q$
then by definition of $p_{i}^{end}$ and the fact that $A$ is a subgraph of
$Q$, the longest path ending in $x$ has order at most $q$ and thus $P$ has
order at most $q$. If $p_{i}^{end}>q$ then $p_{i}^{in}<q$ and the longest path
in the intersection of $A$ and $H_{i}$ has order at most $q-p_{i}^{in}$. As
$T$ is acyclic $P$ can enter $H_{i}$ at most once and thus $P$ has order at
most $p_{i}^{in}+q-p_{i}^{in}=q$.

Now we show that for $k=1,\ldots,|V(B)|$,
$\lambda_{k}(B)\leq\lambda_{k}(Q)-q$. Let $W_{k}=P_{1}\cup P_{2}\cdots\cup
P_{k}$ be a $k$-path subdigraph with maximum order in $B$, and let $x_{1}$,
$x_{2}$, $\dots$, $x_{k}$ be the first vertices of $P_{1}$, $P_{2}$, $\dots$,
$P_{k}$, respectively. Without loss of generality, we assume that $x_{1}$ is
in the graph $H_{l}$ that has the minimum $p^{in}$-value among all the graphs
with starting vertices in $W_{k}$. Suppose that $P^{\prime}$ is a path of
order $p_{l}^{in}$ in $Q-H_{l}$, such that the terminal vertex of $P^{\prime}$
has an arc into $H_{l}$. We will show that $P^{\prime}$ does not intersect
$W_{k}$. In fact we claim something stronger namely that for every path from
$x\in V(H_{i})$ to $y\in V(H_{j})$, either $i=j$ or $p_{i}^{end}\leq
p_{j}^{in}$. Since there is a path from every vertex in $P^{\prime}$ to
$x_{1}$ and $P^{\prime}$ avoids $H_{l}$, it follows that the $p^{in}$-values
of all their host graphs $H_{i}$ must be smaller than $p_{l}^{in}$.

To prove our claim assume that $i\not=j$ and consider a longest path $P^{*}$
of length $p_{i}^{end}$ with a terminal vertex in $H_{i}$. This path $P^{*}$
cannot use a vertex in $H_{j}$ as by the definition of compositions and the
existence of a path from $x\in V(H_{i})$ to $y\in V(H_{j})$ there is a path
from every vertex in $H_{i}$ to every vertex in $H_{j}$ and thus (a part of)
$P^{*}$ together with a path from $H_{i}$ to $H_{j}$ would form a closed walk
and therefore would induce a cycle in $T$. Hence we can extend $P^{*}$ to a
path to $H_{j}$ and thus $p_{i}^{end}\leq p_{j}^{in}$.

Now, if $p_{l}^{in}\geq q$ then $|V(P^{\prime})|=p_{l}^{in}\geq q$. We can
obtain a new $k$-path subdigraph $W_{k}^{\prime}\subseteq Q$ from $W_{k}$ by
appending $P_{1}$ to $P^{\prime}$. Observe that
$\lambda_{k}(Q)\geq|W_{k}^{\prime}|=|W_{k}|+p_{l}^{in}\geq\lambda_{k}(B)+q$.
If $p_{l}^{in}<q$ then $p_{l}^{end}>q$. Assume that $W_{k}$ intersects $B_{l}$
in $b$ paths. By construction, we know
$\lambda_{b}(B_{l})\leq\lambda_{b}(H_{l})-q_{l}$. We substitute the $b$ paths
in $B_{l}$ by a maximum $b$-path subdigraph in $H_{l}$ and adding $P^{\prime}$
to one of the paths. This gives

$\displaystyle\lambda_{k}(Q)$
$\displaystyle\geq|W_{k}|+(\lambda_{b}(H_{l})-\lambda_{b}(B_{l}))+|V(P)|$
$\displaystyle\geq\lambda_{k}(B)+q_{l}+p_{l}^{in}$
$\displaystyle=\lambda_{k}(B)+q,$

which completes the proof. ∎

##  3 Semicomplete Compositions

Recall that composition $Q=T[H_{1},\dots,H_{t}]$ is semicomplete if $T$ is a
semicomplete digraph, that is, for every pair $x,y$ of distinct vertices in
$T$, there is at least one arc between $x$ and $y$. In addition, if
$H_{1},\dots,H_{t}$ are digraphs without arcs, we call $Q$ an extended
semicomplete digraph and denote it by $Q=T[E_{n_{1}},\dots,E_{n_{t}}]$ where
$n_{i}=|V(H_{i})|$ for $1\leq i\leq t.$

Bang-Jensen et al. [5] proved the following somewhat surprising result.

######  Lemma 5.

[5] Let $Q^{\prime}=T[E_{n_{1}},\dots,E_{n_{t}}]$ be an extended semicomplete
digraph and let $v_{i,k}$ denote the maximum number of vertices of $E_{n_{i}}$
that can be covered by a $k$-path subdigraph in $Q^{\prime}$. Then for each
$i\in[t]$ every maximum $k$-path subdigraph in $Q^{\prime}$ covers exactly
$v_{i,k}$ vertices of $E_{n_{i}}.$

Note that in the previous lemma $v_{i,k}$ is defined as the maximum number of
vertices in $E_{n_{j}}$ in _any_ $k$-path subdigraph not only the maximal
ones. In particular this implies that every semicomplete graph has a Hamilton
path, which is a much easier statement of course. With Lemma 5 we can prove
the following theorem which is a generalisation of a result in [5] and Sun
[15]. Our proof is similar to that in [15].

######  Theorem 6.

Let $Q=T[H_{1},\dots,H_{t}]$ be a semicomplete composition and let $l_{i,k}$
denote the maximum number of vertices of $H_{i}$ that can be covered by a
$k$-path subdigraph in $Q$. Then for each $i\in[t]$ every maximum $k$-path
subdigraph in $Q$ covers exactly $l_{i,k}$ vertices of $H_{i}$.

###### Proof.

Let $Q^{\prime}=T[E_{n_{1}},\dots,E_{n_{t}}]$ be the extended semicomplete
digraph obtained by removing all arcs in $H_{i}$ for $1\leq i\leq t$ and
define $v_{i,k}$ as in Lemma 5. According to Lemma 5, every maximum $k$-path
subdigraph $L$ in $Q^{\prime}$ contains exactly $v_{i,k}$ vertices from
$E_{n_{i}}.$ For each $i=1,\dots,t$ consider a (maximum) $v_{i,k}$-path
subdigraph $L_{i}.$ Now for each $i\in[t]$, substitute the $v_{i,k}$ vertices
of $E_{n_{i}}$ in $L$ by a path from $L_{i}$ to obtain a $k$-path subdigraph
in $Q$ containing $|L_{i}|$ vertices in $H_{i}$. Note that this operation is
possible as each vertex in $H_{i}$ has the same in- and out-neighbours outside
$H_{i}.$ Thus $l_{i,k}\geq|L_{i}|.$

We will now show that $l_{i,k}\leq|L_{i}|$ and thus $l_{i,k}=|L_{i}|.$ Assume
for a contradiction that there exists a $k$-path subdigraph $P_{k}$ in $Q$
that covers in, say $H_{1}$, more than $|L_{1}|$ vertices. Note that $P_{k}$
can enter or visit $H_{1}$ at most $v_{1,k}$ times because $P_{k}$ translates
to a $k$-path subdigraph in $Q^{\prime}$ (for example by replacing each path
in $H_{1}$ by the first vertex of the corresponding path in $E_{n_{1}}$.) Thus
the graph induced by $P_{k}$ in $H_{1}$ is a $b$-paths subdigraph of $H_{1}$
with $b\leq v_{1,k}$ and thus this graph has no more than $|L_{1}|$ vertices.

As $|L_{i}|=l_{i,k}$ no $k$-path subdigraph in $Q$ can have more than
$|L_{1}|+\ldots+|L_{t}|$ vertices and as we have constructed a (maximum) path
with $|L_{1}|+\ldots+|L_{t}|$ vertices, it follows that in fact every maximum
$k$-path subdigraph covers exactly $l_{i,k}$ vertices in $H_{i}$. ∎

Now we can prove the following theorem, which is a strengthening of Theorem 4.

######  Theorem 7.

Let $Q=T[H_{1},\dots,H_{t}]$ be a semicomplete composition. Let $l_{i}$ denote
the maximum number of vertices of $H_{i}$ that can be covered by a longest
path in $Q$, and let $l_{m}=\max\\{l_{i}\mid i\in[t]\\}.$ If $H_{m}$ satisfies
BNY, then $Q$ satisfies BNY as well.

###### Proof.

Define $l_{i,k}$ as in Theorem 6 as the maximum number of vertices of $H_{i}$
that can be covered by a $k$-path subdigraph in $Q$. Note that $l_{i}=l_{i,1}$
and since every $k$-paths subigraph of $H_{i}$ is also a $k$-path subdigraph
in $Q$ we have

$\lambda_{k}(H_{i})\leq l_{i,k}.$ (1)

Moreover by Theorem 6, we have

$\lambda_{k}(Q)=l_{1,k}+l_{2,k}+\dots+l_{t,k}.$ (2)

In particular, $\lambda(Q)=l_{1}+\ldots+l_{t}$. We distinguish between 3 cases
depending on $q$ and $l_{i},\ldots,l_{t}$. The first case is not strictly
necessary but it shows the approach without any technical difficulties. The
third case is the most interesting and there we need that $H_{m}$ satisfies
BNY.

Case 1: $q=l_{i_{1}}+\ldots+l_{i_{s}}$ for some indices
$i_{1},\ldots,i_{s}\in\\{1,\ldots,t\\}$. Then $A$ consists of the graph
induced by the vertex set of $H_{i_{1}},H_{i_{2}},\ldots,H_{i_{s}}$ and $B$ of
the graph induced by the remaining vertices. Note that any path in $A$ is also
a path in $Q$ and thus at most $l_{i_{j}}$ vertices of $H_{i_{j}}$ can belong
to any path in $A$ and particularly to a longest path. Thus

$\lambda(A)\leq l_{i_{1}}+\dots+l_{i_{s}}=q$

and similarly

$\lambda_{k}(B)\leq\sum_{j\not\in\\{i_{1},\ldots,i_{s}\\}}l_{j,k}\stackrel{{\scriptstyle(\ref{q1})}}{{=}}\lambda_{k}(Q)-(l_{i_{1},k}+\dots+l_{i_{s},k}).$

Note that $l_{i,k+1}\geq l_{i,k}$ as one can obtain a $(k+1)$-path subgraph
from a $k$-path subdigraph by deleting an edge (or if the $k$-path subdigraph
consists of isolated vertices then adding another vertex.). In particular

$l_{i,k}\geq l_{i}$ (3)

and thus

$\lambda_{k}(B)\leq\lambda_{k}(Q)-(l_{i_{1}}+\dots+l_{i_{s}})\leq\lambda_{k}(Q)-q.$

Case 2: $l_{i_{1}}+\ldots+l_{i_{s}}<q<l_{i_{1}}+\ldots+l_{i_{s}}+l_{h}$ for
some indices $i_{1},\ldots,i_{s},h\in\\{1,\ldots,t\\}$ and $l_{h}=|H_{h}|$. In
this case $A$ consists of the graph induced by $H_{i_{1}},\ldots H_{i_{s}}$
and (any) $q^{\prime}=q-l_{i_{1}}-\ldots-l_{i_{s}}$ vertices of $H_{h}$. The
graph $B$ will be induced by the remaining vertices and in particular will
have $l_{h}-q^{\prime}$ vertices of $H_{h}$. Again, any path in $A$ is also a
path in $Q$ and thus at most $l_{i_{j}}$ vertices of $H_{i_{j}}$ can belong to
any path in $A$ and particularly to a longest path. Thus

$\lambda(A)\leq q^{\prime}+l_{i_{1}}+\dots+l_{i_{s}}=q.$

For $B$ we note that $|H_{h}|\geq l_{h,k}\geq l_{h}=|H_{h}|$ and thus

$\displaystyle\lambda_{k}(B)$ $\displaystyle\leq$ $\displaystyle
l_{h,k}-q^{\prime}+\sum_{j\not\in\\{i_{1},\ldots,i_{s},h\\}}l_{j,k}$
$\displaystyle\stackrel{{\scriptstyle(\ref{q1})}}{{=}}$ $\displaystyle
l_{h,k}-q^{\prime}+\lambda_{k}(Q)-(l_{i_{1},k}+\dots+l_{i_{s},k}+l_{h,k})$
$\displaystyle\leq$
$\displaystyle\lambda_{k}(Q)-q^{\prime}-(l_{i_{1},k}+\dots+l_{i_{s},k})$
$\displaystyle\stackrel{{\scriptstyle(\ref{lik})}}{{\leq}}$
$\displaystyle\lambda_{k}(Q)-q^{\prime}-(l_{i_{1}}+\dots+l_{i_{s}})$
$\displaystyle\leq$ $\displaystyle\lambda_{k}(Q)-q.$

Case 3: $l_{i_{1}}+\ldots+l_{i_{s}}<q<l_{i_{1}}+\ldots+l_{i_{s}}+l_{m}$ and
for all $j\in\\{i_{1},\ldots,i_{s},m\\}$ we have $l_{j}<|H_{j}|$.

We first prove that the subgraph of the semicomplete graph $T$ that is induced
by the vertices corresponding to $H_{i_{1}},\ldots,H_{i_{s}},H_{m}$ is
acyclic. So assume this graph has a cycle including the vertex corresponding
to $H_{j_{1}}\ldots H_{j_{p}}$. For each $r\in\\{1,\ldots,p\\}$, any path
visits $H_{j_{r}}$ at most $l_{j_{r}}<|H_{j_{r}}|$ times in $Q$ and therefore
can be extended along the cycle avoiding the vertices on this path. In
particular a longest path in $Q$ can be extended which means it was not the
longest after all.

Note that since the subgraph of the semicomplete graph $T$ that is induced by
the vertices corresponding to $H_{i_{1}},\ldots,H_{i_{s}},H_{m}$ is acyclic,
any (longest) path in this graph will visits $H_{m}$ (and all other
$H_{i_{j}}$) only once.

Now let $q^{\prime}:=q-l_{i_{1}}-\ldots-l_{i_{s}}$. Let $(A_{m},B_{m})$ be a
partition of $H_{m}$ such that $\lambda(A_{m})\leq q^{\prime}$ and for all
$k^{\prime}\in\\{1,\ldots,|V(B_{m})|\\}$, we have
$\lambda_{k^{\prime}}(B_{m})\leq\lambda_{k^{\prime}}(H_{m})-q^{\prime}$. Let
$A$ be the graph induced by the vertices of $H_{i_{1}},\ldots,H_{i_{s}},A_{m}$
and let $B$ be the graph induced by the remaining vertices. Because the
underlying subgraph of $T$ induced by $A$ is acyclic as we discussed above, we
have

$\displaystyle\lambda(A)$ $\displaystyle\leq$
$\displaystyle\lambda(H_{i_{1}})+\ldots+\lambda(H_{i_{s}})+\lambda(A_{m})$
$\displaystyle\stackrel{{\scriptstyle(\ref{e1})}}{{\leq}}$ $\displaystyle
l_{i_{1}}+\ldots+l_{i_{s}}+q^{\prime}$ $\displaystyle=$ $\displaystyle q.$

Let $Q^{\prime}=T[E_{n_{1}},\dots,E_{n_{t}}]$ be the extended semicomplete
digraph obtained by removing all arcs in $H_{i}$ for $1\leq i\leq t$. Fix
$k\in\\{1,\ldots,|V(B)|\\}$. Define $v_{i,k}$ as in Lemma 5 as the maximum
number of vertices of $E_{n_{i}}$ that can be covered by a $k$-path subdigraph
in Q’. Note that no $k$-path subdigraph in $Q$ can visit $H_{m}$ more than
$v_{m,k}$ times, since each $k$-path subdigraph in $Q$ visiting $H_{m}$ more
than $v_{m,k}$ times can be translated to a $k$-path subdigraph in
$Q^{\prime}$ containing more than $v_{m,k}$ vertices in $E_{n_{m}}$. Let
$u_{k}=\min\\{v_{m,k},|V(B_{m}|)\\}$. If $u_{k}\geq 1$ then

$\displaystyle\lambda_{u_{k}}(B_{m})$ $\displaystyle\leq$
$\displaystyle\lambda_{u_{k}}(H_{m})-q^{\prime}\leq l_{m,k}-q^{\prime}.$ (4)

If $u_{k}=0$ (that is, if $\lambda(H_{m})\leq q^{\prime}$ and we can choose
$A_{m}=H_{m}$ and $B_{m}$ to be empty) we set
$\lambda_{u_{k}}=l_{m,k}-q^{\prime}$.

Any (maximum) $k$-path subdigraph in $B$ is also a $k$-path subdigraph in $Q$,
thus any $k$-path subdigraph in $B$ can visit $B_{m}$ at most $u_{k}$ times.
Therefore

$\displaystyle\lambda_{k}(B)$ $\displaystyle\leq$
$\displaystyle\lambda_{u_{k}}(B_{m})+\sum_{j\not\in\\{i_{1},\ldots,i_{s},m\\}}l_{j,k}$
$\displaystyle\stackrel{{\scriptstyle(\ref{uk})}}{{\leq}}$ $\displaystyle
l_{m,k}-q^{\prime}+\lambda_{k}(Q)-\sum_{j\in\\{i_{1},\ldots,i_{s},m\\}}l_{j,k}$
$\displaystyle\stackrel{{\scriptstyle(\ref{lik})}}{{\leq}}$
$\displaystyle\lambda_{k}(Q)-q^{\prime}-\sum_{j\in\\{i_{1},\ldots,i_{s}\\}}l_{j}$
$\displaystyle\leq$ $\displaystyle\lambda_{k}(Q)-q.$

∎

A digraph is strong if there is a directed path between any two vertices. A
digraph $D$ is quasi-transitive, if the existence of the arcs $xy$ and $yz$
implies an arc between $x$ and $z$. A digraph is locally in-semicomplete, if
the in-neighbours of every vertex induce a semicomplete digraph. Important
results on the above three classes are overviewed in chapters of [3].

Strong semicomplete compositions generalize strong quasi-transitive digraphs
[4]. However, semicomplete compositions form a more complicated digraph class
than quasi-transitive digraphs. For example, it is well known that the
Hamilton Cycle problem (HCP) is polynomial-time solvable for quasi-transitive
digraphs [11]. However, HCP is $\mathcal{NP}$-complete for semicomplete
compositions. Indeed, it is well known that the Hamilton Path problem is
$\mathcal{NP}$-complete for arbitrary digraphs. Let $D$ be a digraph. Form a
new digraph $D^{\prime}$ from $D$ by adding a new vertex $u$ with all arcs
from $u$ to $D$ and a new vertex $v$ with all arcs from $D$ to $v$ and an arc
$vu.$ Observe that $D^{\prime}$ has a Hamilton cycle if and only if $D$ has a
Hamilton path. Thus, HCP is $\mathcal{NP}$-complete even for semicomplete
compositions in which $T$ is a 2-cycle. We refer the readers to [15] for the
structural properties of semicomplete compositions and some results on
connectivity, paths, cycles, strong spanning subdigraphs and acyclic spanning
subgraphs.

By Theorems 3 and 7 and results in [5] on BNY, we have the following:

######  Corollary 8.

Let $Q=T[H_{1},\dots,H_{t}]$ be an acyclic or semicomplete composition such
that each $H_{i}$ is a quasi-transitive digraph, an extended semicomplete
digraph, a locally in-semicomplete digraph, or an acyclic digraph. Then $Q$
satisfies BNY.

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