# On the supersolvability of a finite group by the sum of subgroup orders

Marius Tărnăuceanu

(February 14, 2021)

###### Abstract

Let $G$ be a finite group and $\sigma_{1}(G)=\frac{1}{|G|}\sum_{H\leq
G}\,|H|$. In this paper, we prove that if $\sigma_{1}(G)<2+\frac{11}{|G|}$ ,
then $G$ is supersolvable. In particular, some new characterizations of the
well-known groups $\mathbb{Z}_{2}\times\mathbb{Z}_{4}$ and $A_{4}$ are
obtained. We also show that $\sigma_{1}(G)<c$ does not imply the
supersolvability of $G$ for no constant $c\in(2,\infty)$.

MSC2000 : Primary 20D60; Secondary 20D10, 20F16, 20F17.

Key words : subgroup orders, supersolvable groups.

##  1 Introduction

Given a finite group $G$, we consider the function

$\sigma_{1}(G)=\frac{1}{|G|}\sum_{H\leq G}\,|H|$

studied in our previous papers [7, 10, 11]. Recall some basic properties of
$\sigma_{1}$:

  * -

if $G$ is cyclic of order $n$ and $\sigma(n)$ denotes the sum of all divisors
of $n$, then $\sigma_{1}(G)=\frac{\sigma(n)}{n}$ ;

  * -

$\sigma_{1}$ is multiplicative, i.e. if $G_{i}$, $i=1,2,\dots,m$, are finite
groups of coprime orders, then
$\sigma_{1}(\prod_{i=1}^{m}G_{i})=\prod_{i=1}^{m}\sigma_{1}(G_{i})$;

  * -

$\sigma_{1}(G)\geq\sigma_{1}(G/H)+\frac{1}{(G:H)}\,(\sigma_{1}(H)-1)\geq\sigma_{1}(G/H)$,
for all $H\unlhd G$.

The function $\sigma_{1}$ was used to provide some criteria for a group to
belong to one of the well-known classes of groups. For instance, if
$\sigma_{1}(G)\leq 2$ then $G$ is cyclic of deficient or perfect order by [7],
while if $\sigma_{1}(G)\leq\frac{117}{20}$ then $G$ is solvable by [4, 11].
Also, we note that there is no constant $c\in(2,\infty)$ such that
$\sigma_{1}(G)<c$ implies the nilpotency of $G$, but this can be obtained from
the condition $\sigma_{1}(G)<2+\frac{4}{|G|}$ (see [10]).

In the current paper, we will first determine the structure of finite groups
$G$ satisfying $\sigma_{1}(G)<3$. This will allow us to prove that if
$\sigma_{1}(G)<2+\frac{11}{|G|}$ then $G$ is supersolvable. Moreover, $A_{4}$
is the unique finite non-nilpotent group $G$ such that
$\sigma_{1}(G)=2+\frac{11}{|G|}$ . Then we will show that $\sigma_{1}(G)<c$
does not imply the supersolvability of $G$ for no constant $c\in(2,\infty)$.

For the proof of our results, we need the following two theorems (see Theorem
1 of [6] and Theorem 1 of [8]).

######  Theorem 1.1.

Suppose every non-normal maximal subgroup of a finite group $G$ has the same
order. Then $G$ is solvable and the nilpotent length of $G$ is at most two.

######  Theorem 1.2.

A finite group $G$ contains a cyclic maximal subgroup if and only if it is of
one of the following types:

  * a)

$G=P\times K_{1}$, where $K_{1}$ is an arbitrary finite cyclic group and $P$
is a Sylow $p$-subgroup of $G$ containing a cyclic maximal subgroup.

  * b)

$G=(K_{2}\rtimes P)\times K_{1}$, where $K_{1}$ is an arbitrary finite cyclic
group which is a Hall subgroup of $G$, $P$ is a Sylow $p$-subgroup of $G$
containing a cyclic maximal subgroup, $K_{2}\rtimes P$ is non-nilpotent, and
the centralizer $C_{P}(K_{2})$ is a maximal cyclic subgroup of $P$.

  * c)

$G=(P\rtimes K_{2})\times K_{1}$, where $K_{1}$ is an arbitrary finite cyclic
group which is a Hall subgroup of $G$ and $G_{1}=P\rtimes K_{2}$ is a non-
nilpotent group satisfying the following conditions: $P$ is a SyIow
$p$-subgroup of $G_{1}$, $C_{P}(K_{2})\supseteq\Phi(P)$, and $C_{P}(K_{2})$ is
a cyclic invariant subgroup of $G_{1}$ such that
$K_{2}C_{P}(K_{2})/C_{P}(K_{2})$ is a maximal subgroup of
$G_{1}/C_{P}(K_{2})$.

We also need two auxiliary results, taken from [11] and [7], respectively.

######  Lemma 1.3.

Let $G$ be a finite group and $[M]$ be a conjugacy class of non-normal maximal
subgroups of $G$. Then

$\sum_{H\in[M]}|H|=|G|.$

######  Lemma 1.4.

Let $G$ be a finite group. Then

$\sum_{H\leq G,\,H={\rm cyclic}}\\!\\!\\!|H|\geq|G|.$

Note that similar problems for some other functions related to the structure
of a finite group $G$, for example for the function $\psi(G)=\sum_{x\in
G}o(x)$ (where $o(x)$ denotes the order of the element $x$), have been
recently investigated by many authors (see e.g. [1, 2, 3]).

Most of our notation is standard and will not be repeated here. Basic
definitions and results on groups can be found in [5]. For subgroup lattice
concepts we refer the reader to [9].

##  2 Main results

We start with the following important lemma.

######  Lemma 2.1.

Let $G$ be a finite group with $\sigma_{1}(G)<3$. Then either $G$ is nilpotent
or a group of type c) in Theorem 1.2.

###### Proof.

Assume that $G$ is not nilpotent. If $G$ has at least two conjugacy classes of
non-normal maximal subgroups $[M_{1}]$ and $[M_{2}]$, then Lemma 1.3 leads to

$\sigma_{1}(G)>\frac{1}{|G|}\left(|G|+\sum_{i=1}^{2}\sum_{H\in[M_{i}]}|H|\right)=\frac{1}{|G|}\left(|G|+2|G|\right)=3,$

a contradiction. Thus $G$ possesses a unique conjugacy class of non-normal
maximal subgroups, say $[M]$. Moreover, $M$ is cyclic. Indeed, if this is not
true, then Lemmas 1.3 and 1.4 imply

$\sigma_{1}(G)\geq\frac{1}{|G|}\left(|G|+\\!\\!\sum_{H\in[M]}|H|+\\!\\!\\!\\!\\!\\!\\!\\!\sum_{H\leq
G,\,H={\rm
cyclic}}\\!\\!\\!\\!\\!|H|\right)\geq\frac{1}{|G|}\left(|G|+|G|+|G|\right)=3,$

contradicting again our hypothesis.

From Theorem 1.1 it follows that $G$ is solvable of nilpotent length two. More
precisely, by taking a prime $p$ dividing $[G:M]$, we infer that a Sylow
$p$-subgroup $P$ of $G$ must be normal and a $p$-complement of $G$ must be
cyclic. This shows that $G$ is a group of type c) in Theorem 1.2, completing
the proof.∎

We are now able to prove our first main result.

######  Theorem 2.2.

Let $G$ be a finite group. If $\sigma_{1}(G)<2+\frac{11}{|G|}$ , then $G$ is
supersolvable.

###### Proof.

Since all groups of order $\leq 11$ are supersolvable, we may assume that
$|G|\geq 12$. Then $\sigma_{1}(G)<3$ and therefore, by the above lemma, either
$G$ is nilpotent or $G=(P\rtimes K_{2})\times K_{1}$ with $P$, $K_{1}$ and
$K_{2}$ as in Theorem 1.2, c). In the first case we are done, while in the
second case we may assume that $G=G_{1}=P\rtimes K_{2}$ because

$\sigma_{1}(G_{1})=\sigma_{1}(G/K_{1})\leq\sigma_{1}(G)<2+\frac{11}{|G|}\leq
2+\frac{11}{|G_{1}|}\,.$ (1)

Denote by $[M]$ the unique conjugacy class of non-normal maximal subgroups of
$G$. Let $|P|=p^{r}$ and suppose that $P$ is not cyclic. Then $r\geq 2$ and
$P$ contains at least $p+1$ subgroups of order $p^{r-1}$. It follows that

$\begin{array}[]{lcl}\sigma_{1}(G)\\!\\!\\!\\!&\geq&\\!\\!\\!\\!\displaystyle\frac{1}{|G|}\left(|G|+\\!\\!\displaystyle\sum_{H\in[M]}|H|+\\!\displaystyle\sum_{H\leq
P}|H|\right)\\!\\!\geq\\!2+\displaystyle\frac{1+p+\cdots+(p+1)p^{r-1}\\!+p^{r}}{|G|}\vspace{1,5mm}\\\
&\geq&\\!\\!\\!\\!2+\displaystyle\frac{1+(p+1)p+p^{2}}{|G|}=2+\displaystyle\frac{1+p+2p^{2}}{|G|}\geq
2+\displaystyle\frac{11}{|G|}\,,\end{array}$

contradicting (1). Thus $P$ is cyclic and consequently all Sylow subgroups of
$G$ are cyclic, implying that $G$ is supersolvable. This completes the proof.∎

Next we will determine finite groups $G$ such that
$\sigma_{1}(G)=2+\frac{11}{|G|}$ . First of all, we will focus on $p$-groups.
We observe that for all primes $p$ and all positive integers $n$ we have

$\sigma_{1}(\mathbb{Z}_{p^{n}})=\displaystyle\frac{\sigma(p^{n})}{p^{n}}=\displaystyle\frac{1+p+\cdots+p^{n}}{p^{n}}=1+\displaystyle\frac{1}{p-1}-\displaystyle\frac{1}{p^{n}(p-1)}<2.$

The following lemma deals with the case of non-cyclic $p$-groups.

######  Lemma 2.3.

Let $G$ be a non-cyclic $p$-group of order $p^{n}$. Then:

  * a)

$\sigma_{1}(G)<2+\frac{11}{|G|}$ if and only if either $n=3$, $p=2$ and
$G=Q_{8}$, or $n=2$, $p\in\\{2,3,5,7\\}$ and
$G=\mathbb{Z}_{p}\times\mathbb{Z}_{p}$;

  * b)

$\sigma_{1}(G)=2+\frac{11}{|G|}$ if and only if $n=3$, $p=2$ and
$G=\mathbb{Z}_{2}\times\mathbb{Z}_{4}$.

###### Proof.

Since $G$ contains at least $p+1$ subgroups of order $p^{n-1}$, one obtains

$2+\frac{11}{p^{n}}\geq\sigma_{1}(G)\geq\displaystyle\frac{1+p+\cdots+(p+1)p^{n-1}\\!+p^{n}}{p^{n}}=2+\displaystyle\frac{1+p+\cdots+p^{n-1}}{p^{n}}\,,$

that is $1+p+\cdots+p^{n-1}\leq 11$. Thus either $n=3$ and $p=2$, or $n=2$ and
$p\in\\{2,3,5,7\\}$. The conclusion follows by checking the corresponding
groups.∎

Our second main result is stated as follows.

######  Theorem 2.4.

Let $G$ be a finite non-cyclic group. If $\sigma_{1}(G)=2+\frac{11}{|G|}$ ,
then either $G=\mathbb{Z}_{2}\times\mathbb{Z}_{4}$ or $G=A_{4}$.

###### Proof.

We can easily see that $G=\mathbb{Z}_{2}\times\mathbb{Z}_{4}$ is the unique
group $G$ of order $\leq 11$ satisfying $\sigma_{1}(G)=2+\frac{11}{|G|}$ .
Assume that $|G|\geq 12$. Then $\sigma_{1}(G)<3$ and therefore, by Lemma 2.1,
either $G$ is nilpotent or $G=(P\rtimes K_{2})\times K_{1}$ with $P$, $K_{1}$
and $K_{2}$ as in Theorem 1.2, c).

In the first case, let $G=G_{1}\times\cdots\times G_{k}$ be the decomposition
of $G$ as a direct product of Sylow $p$-subgroups. For $k=1$, we have no
solution $G$ of $\sigma_{1}(G)=2+\frac{11}{|G|}$ by Lemma 2.3, b). For $k\geq
2$, we get $\sigma_{1}(G_{i})<2+\frac{11}{|G_{i}|}$ , $\forall\,i=1,...,k$.
Then Lemma 2.3, a), shows that every $G_{i}$ is of one of the following types:
$Q_{8}$, $\mathbb{Z}_{p_{i}}\times\mathbb{Z}_{p_{i}}$ or
$\mathbb{Z}_{p_{i}^{n_{i}}}$ for some prime $p_{i}$ and some positive integer
$n_{i}$. Since

$\sigma_{1}(Q_{8})=\frac{23}{8}>2,\,\sigma_{1}(\mathbb{Z}_{p_{i}}\times\mathbb{Z}_{p_{i}})=2+\frac{p_{i}+1}{p_{i}^{2}}>2,\forall\,i=1,...,k$

and

$\sigma_{1}(G)=\sigma_{1}(G_{1})\cdots\sigma_{1}(G_{k}),$

we infer that at most one of $G_{i}$’s is non-cyclic. Suppose that
$G_{1}=Q_{8}$ and $G_{i}=\mathbb{Z}_{p_{i}^{n_{i}}}$, for all $i=2,...,k$.
Then our condition becomes

$\displaystyle\frac{23}{8}\,\frac{\sigma(p_{2}^{n_{2}})}{p_{2}^{n_{2}}}\cdots\frac{\sigma(p_{k}^{n_{k}})}{p_{k}^{n_{k}}}=2+\frac{11}{8p_{2}^{n_{2}}\cdots
p_{k}^{n_{k}}}\,,$

i.e.

$23\,\sigma(p_{2}^{n_{2}})\cdots\sigma(p_{k}^{n_{k}})=16p_{2}^{n_{2}}\cdots
p_{k}^{n_{k}}+11\,,$

a contradiction (left side $>$ right side). Similarly, if we suppose
$G_{1}=\mathbb{Z}_{p_{1}}\times\mathbb{Z}_{p_{1}}$ and
$G_{i}=\mathbb{Z}_{p_{i}^{n_{i}}}$, for all $i=2,...,k$, then we obtain

$\displaystyle\frac{2p_{1}^{2}+p_{1}+1}{p_{1}^{2}}\,\frac{\sigma(p_{2}^{n_{2}})}{p_{2}^{n_{2}}}\cdots\frac{\sigma(p_{k}^{n_{k}})}{p_{k}^{n_{k}}}=2+\frac{11}{p_{1}^{2}p_{2}^{n_{2}}\cdots
p_{k}^{n_{k}}}\,,$

i.e.

$(2p_{1}^{2}+p_{1}+1)\,\sigma(p_{2}^{n_{2}})\cdots\sigma(p_{k}^{n_{k}})=2p_{1}^{2}p_{2}^{n_{2}}\cdots
p_{k}^{n_{k}}+11\,,$

a contradiction (left side $>$ right side). Thus every $G_{i}$ is cyclic, and
so is $G$, contradicting our hypothesis.

In the second case, a similar reasoning shows that we have no solution
$G=(P\rtimes K_{2})\times K_{1}$ of $\sigma_{1}(G)=2+\frac{11}{|G|}$ with
$K_{1}\neq 1$. Consequently, we may assume $G=P\rtimes K_{2}$. Let $|P|=p^{r}$
and $|K_{2}|=n$. Note that $p\nmid n$ and $n\mid|{\rm Aut}G|$. As in the proof
of Theorem 2.2, we get

$2+\frac{11}{|G|}=\sigma_{1}(G)\geq
2+\displaystyle\frac{\displaystyle\sum_{H\leq P}|H|}{|G|}$

and thus

$\displaystyle\sum_{H\leq P}|H|\leq 11.$

It follows that either $r=2$, $p=2$ and
$P\in\\{\mathbb{Z}_{4},\mathbb{Z}_{2}\times\mathbb{Z}_{2}\\}$, or $r=1$,
$p\in\\{2,3,5,7\\}$ and $P=\mathbb{Z}_{p}$.

If $P=\mathbb{Z}_{4}$, then $n\mid 2$, a contradiction. If
$P=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$, then $n\mid 6$ and so $n=3$; thus
$G=A_{4}$, which is a solution of $\sigma_{1}(G)=2+\frac{11}{|G|}$ . If
$P=\mathbb{Z}_{p}$ with $p\in\\{2,3,5,7\\}$, then $n$ divides $1$, $2$, $4$ or
$6$, respectively. We infer that the unique possibilities are

$(p,n)\in\\{(3,2),(5,2),(5,4),(7,2),(7,3),(7,6)\\}.$

By checking all these semidirect products
$G=\mathbb{Z}_{p}\rtimes\mathbb{Z}_{n}$ we find no other solution of
$\sigma_{1}(G)=2+\frac{11}{|G|}$ . Hence in this case the unique solution is
$G=A_{4}$, as desired.∎

Obviously, Theorem 2.4 leads to the following characterizations of
$\mathbb{Z}_{2}\times\mathbb{Z}_{4}$ and $A_{4}$.

######  Corollary 2.5.

$\mathbb{Z}_{2}\times\mathbb{Z}_{4}$ is the unique finite non-cyclic nilpotent
group $G$ satisfying $\sigma_{1}(G)=2+\frac{11}{|G|}$ , while $A_{4}$ is the
unique finite non-nilpotent group $G$ satisfying
$\sigma_{1}(G)=2+\frac{11}{|G|}$ .

Finally, we remark that $\sigma_{1}(G)<\frac{31}{12}=\sigma_{1}(A_{4})$ does
not imply the supersolvability of $G$, a counterexample being
$SmallGroup(56,11)$ (also known as the Frobenius group $F_{8}$). In fact, we
are able to prove that there is no constant $c\in(2,\infty)$ such that
$\sigma_{1}(G)<c$ implies the supersolvability of $G$.

######  Theorem 2.6.

There are sequences of finite non-supersolvable groups
$(G_{n})_{n\in\mathbb{N}}$ such that $\sigma_{1}(G_{n})$ approaches $2$
monotonically from above as $n$ tends to infinity.

###### Proof.

Let $(q_{n})_{n\in\mathbb{N}}$ be the sequence of odd prime numbers. By
Dirichlet’s theorem we infer that for every $n\in\mathbb{N}$ there is a prime
$p_{n}$ such that $q_{n}\mid p_{n}-1$. Let $G_{n}$ be the unique non-abelian
group of order $p_{n}^{2}q_{n}$. Clearly, $G_{n}$ is not supersolvable. It
contains one subgroup of order $1$, $p_{n}+1$ subgroups of order $p_{n}$, one
subgroup of order $p_{n}^{2}$, $p_{n}^{2}$ subgroups of order $q_{n}$, and one
subgroup of order $p_{n}^{2}q_{n}$. Then

$\sigma_{1}(G_{n})=\frac{1+(p_{n}+1)p_{n}+p_{n}^{2}+p_{n}^{2}q_{n}+p_{n}^{2}q_{n}}{p_{n}^{2}q_{n}}=2+\frac{2+\displaystyle\frac{1}{p_{n}}+\displaystyle\frac{1}{p_{n}^{2}}}{q_{n}}\,,$

which approaches $2$ monotonically from above as $n$ tends to infinity, as
desired.∎

We end this paper by formulating a natural problem concerning the above study.

Open problem. Find all finite cyclic groups $G$ satisfying
$\sigma_{1}(G)=2+\frac{11}{|G|}$ .

Note that this is equivalent to finding all positive integers
$n=p_{1}^{n_{1}}\cdots p_{k}^{n_{k}}$ such that

$\left(1+p_{1}+\cdots+p_{1}^{n_{1}}\right)\cdots\left(1+p_{k}+\cdots+p_{k}^{n_{k}}\right)=2p_{1}^{n_{1}}\cdots
p_{k}^{n_{k}}+11.$

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Marius Tărnăuceanu

Faculty of Mathematics

“Al.I. Cuza” University

Iaşi, Romania

e-mail<EMAIL_ADDRESS>


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##  Appendix A $SU(N)$ generators and identities

$SU(N)$ is the special unitary group of degree $N$ whose elements $U$ are
$N\times N$ unitary matrices, $U^{\dagger}U=\mathds{1}$, satisfying
$\det(U)=1$. It is a Lie group, with its elements being continuously generated
by real parameters $\theta^{a}\in\mathds{R}$. Each element can be written as

$U=e^{i\theta^{a}t^{a}}$ (A.1)

where $t^{a}$ are the $N^{2}-1$ group generators, corresponding to each
parameter $\theta^{a}$. The generators are hermitian and traceless matrices

$\displaystyle(t^{a})^{\dagger}=t^{a},$ $\displaystyle\tr(t^{a})=0,$ (A.2)

that span a vector space underlying the corresponding Lie algebra,
$\mathfrak{su}(N)$. The generators obey the commutation relation

$[t^{a},t^{b}]=if^{abc}t^{c}$ (A.3)

where $f^{abc}$ are the antisymmetric structure constants, specific for each
group and non-zero for a non-abelian group. A fundamental property of Lie
groups is the Jacobi identity

$[t^{a},[t^{b},t^{c}]]+[t^{b},[t^{c},t^{a}]]+[t^{c},[t^{a},t^{b}]]=0$ (A.4)

implying

$f^{ade}f^{bcd}+f^{bde}f^{cad}+f^{cde}f^{abd}=0.$ (A.5)

There are two main irreducible representations of the groups $SU(N)$. The
fundamental representation consists of $N$-dimensional complex vectors, with
the group as well as the algebra elements being $N\times N$ matrices. For QCD,
$N=3$, this corresponds to the representation of the 3-spinor quark field. The
usual choice of the normalization of the generators is

$f^{acd}f^{bcd}=N\delta^{ab}$ (A.6)

from which we can derive for the fundamental representation,

$\Tr\left(t^{a}t^{b}\right)=\frac{\delta^{ab}}{2}.$ (A.7)

The structure constants may be written as

$f^{abc}=-2i\tr([t^{a},t^{b}]t^{c})$ (A.8)

and the product of two generators has the general form,

$t^{a}t^{b}=\frac{\delta^{ab}}{2N}+\frac{1}{2}d^{abc}t^{c}+\frac{1}{2}if^{abc}t^{c}$
(A.9)

where the totally symmetric object is defined as
$d^{abc}=2\Tr\left(t^{a}\\{t^{b},t^{c}\\}\right)$, making use of the anti-
commutator defined as

$\\{t^{a},t^{b}\\}=\frac{\delta^{ab}}{N}+d^{abc}t^{c}.$ (A.10)

Additional identities may be obtained

$\displaystyle\Tr\left(t^{a}t^{b}t^{c}\right)=\frac{1}{4}(d^{abc}+if^{abc})$
(A.11) $\displaystyle f^{abc}f^{abc}=N(N^{2}-1)$ (A.12) $\displaystyle
f^{abm}f^{cdm}=\frac{2}{N}\left(\delta^{ac}\delta^{bd}-\delta^{ad}\delta^{bc}\right)+d^{acm}d^{dbm}-d^{adm}d^{bcm}$
(A.13) $\displaystyle f^{abm}d^{cdm}+f^{acm}d^{dbm}+f^{adm}d^{bcm}=0$ (A.14)

with a further relation for $N=3$,

$\delta^{ab}\delta^{cd}+\delta^{ac}\delta^{bd}+\delta^{ad}\delta^{bc}=3\left(d^{abm}d^{cdm}+d^{acm}d^{dbm}+d^{adm}d^{bcm}\right).$
(A.15)

The other important representation is the adjoint representation to which the
generators belong and acts on the vector space spanned by the generators
themselves – it is an $N^{2}-1$ dimensional representation. In QCD, the 8
gluon fields live on the adjoint representation of the group $SU(3)$ and
transform accordingly. The representation matrices of the generators are given
by the structure constants

$(t^{b})_{ac}=if^{abc}.$ (A.16)

A useful relation is the trace of four generators in the adjoint
representation

$\Tr(t^{a}t^{b}t^{c}t^{d})=\delta^{ad}\delta^{bc}+\frac{1}{2}\left(\delta^{ab}\delta^{cd}+\delta^{ac}\delta^{bd}\right)+\frac{N}{4}\left(f^{adm}f^{bcm}+d^{adm}d^{bcm}\right).$
(A.17)

In this representation, the covariant derivative

$D_{\mu}\eta(x)=(\partial_{\mu}-igA_{\mu}^{a}t^{a})\eta(x)$ (A.18)

takes the component form

$\displaystyle(D_{\mu}\eta(x))_{a}$
$\displaystyle=\partial_{\mu}\eta_{a}(x)-igA_{\mu}^{b}(t^{b})_{ac}\eta_{c}(x)$
(A.19)
$\displaystyle=\partial_{\mu}\eta_{a}(x)+gf^{abc}A_{\mu}^{b}\eta_{c}(x).$
(A.20)

##  Appendix B Lattice tensors

###  B.1 Construction of the lattice basis

####  B.1.1 Momentum polynomial under a transposition

We consider a brief proof of the transformation of a polynomial of a vector
$p$ under a transposition is given. A transposition is defined by an exchange
of two components of a vector, $\sigma\leftrightarrow\rho$, under the
operation $T^{\sigma\rho}$. A matrix form for this operator is

$\begin{cases}T^{(\sigma\rho)}_{\mu\nu}=\delta_{\mu\nu},\leavevmode\nobreak\
\mu\neq\sigma,\rho\\\ T^{(\sigma\rho)}_{\sigma\nu}=\delta_{\rho\nu}\\\
T^{(\sigma\rho)}_{\rho\nu}=\delta_{\sigma\nu}\end{cases}$ (B.1)

which reproduces the correct transformation on the vector $p$:

$\begin{cases}p^{\prime}_{\nu}=p_{\nu},\leavevmode\nobreak\
\nu\neq\sigma,\rho\\\ p^{\prime}_{\sigma}=p_{\rho},\\\
p^{\prime}_{\rho}=p_{\sigma}.\end{cases}$ (B.2)

Considering the transformation for an arbitrary order of $p$

$(p^{\prime}_{\mu})^{n}=p^{\prime}_{\mu}...p^{\prime}_{\mu}=T^{(\sigma\rho)}_{\mu\nu_{1}}p_{\nu_{1}}...T^{(\sigma\rho)}_{\mu\nu_{n}}p_{\nu_{n}}$
(B.3)

and considering the case $\mu\neq\sigma,\rho$ the correct transformation is
immediate since all components are left unchanged,

$(p^{\prime}_{\mu})^{n}=p_{\mu}...p_{\mu}=(p_{\mu})^{n}=T^{(\sigma\rho)}_{\mu\nu}(p_{\nu}).$
(B.4)

For $\mu=\sigma,\rho$, the transformation is

$\displaystyle(p^{\prime}_{\sigma})^{n}$
$\displaystyle=T^{(\sigma\rho)}_{\sigma\nu_{1}}p_{\nu_{1}}...T^{(\sigma\rho)}_{\sigma\nu_{n}}p_{\nu_{n}}$
$\displaystyle=\delta_{\sigma\nu_{1}}p_{\nu_{1}}...\delta_{\sigma\nu_{n}}p_{\nu_{n}}$
$\displaystyle=T^{(\sigma\rho)}_{\sigma\nu}(p_{\nu})^{n}=(p_{\rho})^{n}.$
(B.5)

This is the same transformation as for the vector $p$, and thus the polynomial
transforms accordingly.

####  B.1.2 Second order tensors under $H(4)$ symmetry

Here we show that there is no mixing among the diagonal and off-diagonal
elements under a general $H(4)$ transformation, using the fact that these
transformations can be formed by products of transpositions and inversions.

The transposition operator for the exchange of components
$\sigma\leftrightarrow\rho$ was defined in B.1. For the inversion of the
component $\rho$, we define the operator as

$\displaystyle P^{\rho}_{\mu\nu}=\delta_{\mu\nu},\leavevmode\nobreak\
\mu\neq\rho$ (B.6) $\displaystyle P^{\rho}_{\rho\nu}=-\delta_{\rho\nu}.$ (B.7)

The transformation for a second order tensor under transpositions and
inversions is

$\displaystyle
D^{\prime}_{\mu\nu}=T^{(\sigma\rho)}_{\mu\tau}T^{(\sigma\rho)}_{\mu\varepsilon}D_{\tau\varepsilon},$
(B.8) $\displaystyle
D^{\prime}_{\mu\nu}=P^{(\rho)}_{\mu\tau}P^{(\rho)}_{\mu\varepsilon}D_{\tau\varepsilon}.$
(B.9)

Now we consider the transformation of diagonal elements $\mu=\nu$. For
transpositions there are three distinct situations,

$\begin{cases}D^{\prime}_{\sigma\sigma}=\delta_{\rho\tau}\delta_{\rho\varepsilon}D_{\tau\varepsilon}=D_{\rho\rho}\\\
D^{\prime}_{\rho\rho}=D^{\prime}_{\sigma\sigma}\\\
D^{\prime}_{\mu\mu}=D_{\mu\mu},\leavevmode\nobreak\
\mu\neq\rho,\sigma\end{cases}$ (B.10)

and we see that no off-diagonal terms appear.

A similar analysis can be considered for the inversions using B.9

$\begin{cases}D^{\prime}_{\rho\rho}=(-\delta_{\rho\tau})(-\delta_{\rho\varepsilon})D_{\tau\varepsilon}=D_{\rho\rho}\\\
D^{\prime}_{\mu\mu}=D_{\mu\mu},\leavevmode\nobreak\ \mu\neq\rho\end{cases}$
(B.11)

and again for this transformation, no off-diagonal terms appear for the
diagonal transformation.

We now consider the off-diagonal transformation, $\mu\neq\nu$. For the
transpositions there are again three distinct cases

$\begin{cases}D^{\prime}_{\sigma\nu}=\sum_{\tau,\varepsilon}\delta_{\rho\tau}\delta_{\nu\varepsilon}=D_{\rho\nu}\\\
D^{\prime}_{\rho\nu}=D^{\prime}_{\sigma\nu}\\\
D^{\prime}_{\rho\sigma}=D_{\sigma\rho}\end{cases}$ (B.12)

and no diagonal terms are involved. On the other hand for inversions there are
two cases

$\begin{cases}D^{\prime}_{\mu\nu}=-D_{\mu\nu},\leavevmode\nobreak\
&\mu=\rho\wedge\nu=\rho\\\ D^{\prime}_{\mu\mu}=D_{\mu\mu},\leavevmode\nobreak\
&\mu\neq\rho\wedge\nu\neq\rho.\end{cases}$ (B.13)

We conclude that a general $H(4)$ transformation does not mix the diagonal and
off-diagonal elements for second order tensors.

###  B.2 General construction for projectors

The projectors $\mathcal{P}^{k}$ are necessary to extract form factors
corresponding to each basis element. Here we describe the general form of
constructing projectors, for an arbitrary vector space.

Given a general tensor $\Gamma$, this object will be described by a basis of
$N$ tensor elements $\tau^{j}$,

$\Gamma=\sum_{j=1}^{N}\gamma^{j}\tau^{j}$ (B.14)

where $\gamma^{j}$ are the corresponding dressing functions. Suppose we want
to extract one of the form factors $\gamma^{k}$ by acting on $\Gamma$ with an
operator $\mathcal{P}^{k}$ (this operation involves the necessary index
contractions to build a scalar). The operation is of the form,

$\mathcal{P}^{k}\Gamma=\mathcal{P}^{k}\left(\sum_{j=1}^{N}\gamma^{j}\tau^{j}\right)=\gamma^{k}.$
(B.15)

From this we may extract the relation

$\mathcal{P}^{k}\tau^{j}=\delta^{kj}.$ (B.16)

using the completeness of the basis, and the linearity of the operator.
Considering the most general form of the projector $\mathcal{P}^{k}$,
constructed from basis elements

$\mathcal{P}^{k}=\sum_{i=1}^{N}A_{ki}\tau^{i}$ (B.17)

and substitute this into eq. B.16, to obtain

$\sum_{i=1}^{N}A_{ki}\tau^{i}\tau^{j}=\delta^{kj}\Leftrightarrow
A_{ki}=(\tau^{k}\tau^{i})^{-1}.$ (B.18)

This reduces the extraction of the form factors to a matrix inversion problem.
We need only to build the matrix with elements $A_{ki}^{-1}=\tau^{k}\tau^{i}$,
where the contraction of indices referred before is assumed, and obtain its
inverse $A$

$\mathcal{P}^{k}=\sum_{i=0}^{N}(\tau^{k}\tau^{i})^{-1}\tau^{i}.$ (B.19)

With this mechanism, it is straightforward to understand why it is impossible
to build well defined projectors when there are redundant basis elements that
can be written as a linear combination of the remaining elements. In this
case, not all rows will be linearly independent, and it is know from linear
algebra that matrices with this property are singular, i.e. non-invertible,
and the projectors cannot be defined.

####  B.2.1 Projectors for the lattice bases

We use the previous mechanism to build the projectors for the tensor bases
considered throughout the work. We begin with the general form for second
order tensors in the continuum

$D_{\mu\nu}(p)=A(p)\delta_{\mu\nu}+B(p)p_{\mu}p_{\nu}$ (B.20)

with the elements $\tau^{1}=\delta_{\mu\nu}$ and $\tau^{2}=p_{\mu}p_{\nu}$.
The matrix $A^{-1}$ for a $N_{d}$ dimensional space is

$A^{-1}=\matrixquantity(N_{d}^{2}&p^{2}\\\ p^{2}&p^{4}),$ (B.21)

and its inverse

$A=\frac{1}{p^{4}(N_{d}-1)}\matrixquantity(p^{4}&-p^{2}\\\ -p^{2}&N_{d}).$
(B.22)

The projectors are built with eq. B.17

$\displaystyle\mathcal{P}^{1}_{\mu\nu}=\frac{1}{N_{d}-1}\left(\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right)$
(B.23)
$\displaystyle\mathcal{P}^{2}_{\mu\nu}=\frac{1}{N_{d}-1}\left(-\frac{\delta_{\mu\nu}}{p^{2}}+N_{d}\frac{p_{\mu}p_{\nu}}{p^{4}}\right),$
(B.24)

and the extraction of the respective form factors follows immediately

$\displaystyle
A(p)=\frac{1}{N_{d}-1}\left(\sum_{\mu}D_{\mu\mu}(p)-\frac{1}{p^{2}}\sum_{\mu\nu}p_{\mu}p_{\nu}D_{\mu\nu}(p)\right)$
(B.25) $\displaystyle
B(p)=\frac{1}{N_{d}-1}\left(-\frac{1}{p^{2}}\sum_{\mu}D_{\mu\mu}(p)+\frac{N_{d}}{p^{2}}\sum_{\mu\nu}p_{\mu}p_{\nu}D_{\mu\nu}(p)\right).$
(B.26)

This procedure can be simplified when considering the tensor form

$D_{\mu\nu}(p)=D(p^{2})\left(\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right),$
(B.27)

with the form factor extracted with

$D(p^{2})=\frac{1}{N_{d}-1}\sum_{\mu}D_{\mu\mu}(p).$ (B.28)

We consider now the lattice basis 3.16. As referred in the construction of the
basis, the diagonal elements do not mix with off-diagonal, which allow us to
analyse them independently. The reducibility of the group representation
splits the five dimensional matrix into two square matrices of size two and
three. It is thus important to use two different index contractions, one
considering only diagonal terms,
$\sum_{\mu}\tau_{\mu\mu}^{i}\tau_{\mu\mu}^{j}$, and the second considering
only off-diagonal elements
$\sum_{\mu\neq\nu}\tau_{\mu\nu}^{i}\tau_{\mu\nu}^{j}$. Starting with the
diagonal elements $\tau^{1}=\delta_{\mu\mu}$, $\tau^{2}=p_{\mu}^{2}$ and
$\tau^{3}=p_{\mu}^{4}$. The contraction matrix $A^{-1}$ is

$A^{-1}=\matrixquantity(N_{d}&p^{2}&p^{[4]}\\\ p^{2}&p^{[4]}&p^{[6]}\\\
p^{[4]}&p^{[6]}&p^{[8]}).$ (B.29)

Hence, the diagonal form factors are

$\displaystyle
E(p)=\frac{1}{\Delta_{1}}\bigg{[}\sum_{\mu}D_{\mu\mu}(p^{[4]}p^{[8]}-(p^{[6]})^{2})$
$\displaystyle+\sum_{\mu}p_{\mu}^{2}D_{\mu\mu}(p^{[4]}p^{[6]}-p^{2}p^{[8]})$
$\displaystyle+\sum_{\mu}p_{\mu}^{4}D_{\mu\mu}(p^{2}p^{[6]}-(p^{[4]})^{2})\bigg{]}$
(B.30) $\displaystyle
F(p)=\frac{1}{\Delta_{1}}\bigg{[}\sum_{\mu}D_{\mu\mu}(p^{[4]}p^{[6]}-p^{2}p^{[8]})$
$\displaystyle+\sum_{\mu}p_{\mu}^{2}D_{\mu\mu}(N_{d}p^{[6]}-(p^{[4]})^{2})$
$\displaystyle+\sum_{\mu}p_{\mu}^{4}D_{\mu\mu}(p^{2}p^{[4]}-N_{d}p^{[6]})\bigg{]}$
(B.31) $\displaystyle
G(p)=\frac{1}{\Delta_{1}}\bigg{[}\sum_{\mu}D_{\mu\mu}(p^{2}p^{[6]}-(p^{[8]})^{2})$
$\displaystyle+\sum_{\mu}p_{\mu}^{2}D_{\mu\mu}(p^{2}p^{[4]}-N_{d}p^{[6]})$
$\displaystyle+\sum_{\mu}p_{\mu}^{4}D_{\mu\mu}(p^{2}p^{[4]}-N_{d}p^{[6]})\bigg{]}$
(B.32)

with

$\Delta_{1}=N_{d}\left(p^{[4]}p^{[8]}-(p^{[6]})^{2}\right)+p^{2}\left(p^{[4]}p^{[6]}-p^{2}p^{[8]}\right)+p^{[4]}\left(p^{2}p^{[6]}-(p^{[4]})^{2}\right).$
(B.33)

Similarly we can repeat the procedure for the two dimensional, off-diagonal
case, obtaining both form factors,

$\displaystyle
H(p)=\frac{2}{\Delta_{2}}\bigg{[}\sum_{\mu\neq\nu}p_{\mu}p_{\nu}D_{\mu\nu}(p^{[4]}p^{[6]}-p^{[10]})-\sum_{\mu\neq\nu}p_{\mu}^{3}p_{\nu}^{3}D_{\mu\nu}(p^{2}p^{[4]}-p^{[6]})\bigg{]}$
(B.34) $\displaystyle
I(p)=\frac{1}{\Delta_{1}}\bigg{[}\sum_{\mu\neq\nu}p_{\mu}p_{\nu}D_{\mu\nu}(p^{[8]}-(p^{[4]})^{2})+\sum_{\mu\neq\nu}p_{\mu}^{3}p_{\nu}^{3}D_{\mu\nu}(p^{4}-p^{[4]})\bigg{]}$
(B.35)

with

$\Delta_{2}=2\left(p^{2}p^{[4]}-p^{[6]}\right)\left(p^{[8]}-(p^{[4]})^{2}\right)+2\left(p^{4}-p^{[4]}\right)\left(p^{[4]}p^{[6]}-p^{[10]}\right).$
(B.36)

Having all projectors for the lattice basis, we need to consider the case of
the generalized diagonal kinematics where these projectors are not possible to
obtain. This analysis is done for each individual configuration. Starting with
the diagonal, $(n,n,n,n)$, the gluon propagator is

$\displaystyle D_{\mu\mu}(p)=(E(p)+n^{2}F(p)+n^{4}G(p))\delta_{\mu\mu}$
$\displaystyle D_{\mu\nu}(p)=n^{2}H(p)+2n^{4}I(p),\leavevmode\nobreak\
\mu\neq\nu$ (B.37)

and in this case we can only extract two form factors, for the diagonal and
off-diagonal terms. These are extracted with

$\displaystyle
E(p)+n^{2}F(p)+n^{4}G(p)=\frac{1}{N_{d}}\sum_{\mu}D_{\mu\mu}(p),$ (B.38)
$\displaystyle
n^{2}H(p)+2n^{4}I(p)=\frac{1}{N_{d}(N_{d}-1)}\sum_{\mu\neq\nu}D_{\mu\nu}(p).$
(B.39)

The mixed configurations, $(n,n,0,0)$ and $(n,n,n,0)$ have non-diagonal terms
and the gluon propagator reads

$\displaystyle D_{\mu\mu}(p)=E(p)\delta_{\mu\mu}+(F(p)+n^{2}G(p))p_{\mu}^{2}$
$\displaystyle
D_{\mu\nu}(p)=(H(p)+2I(p)n^{2})p_{\mu}p_{\nu},\leavevmode\nobreak\
\mu\neq\nu.$ (B.40)

For these configurations we consider the parameter $k$ representing the number
of non-vanishing components. The contractions of tensor basis elements are
summarized by

(B.41)

with corresponding inverses

(B.42)

With this, the form factors follow easily

$\displaystyle
E(p^{2})=\frac{1}{kn^{4}(N_{d}-k)}\sum_{\mu}D_{\mu\mu}(p)\left(kn^{4}\delta_{\mu\mu}-kn^{2}p_{\mu}^{2}\right)$
(B.43) $\displaystyle
F(p^{2})+n^{2}G(p^{2})=\frac{1}{kn^{4}(N_{d}-k)}\sum_{\mu}D_{\mu\mu}(p)\left(-kn^{2}\delta_{\mu\mu}+N_{d}p_{\mu}^{2}\right)$
(B.44) $\displaystyle
H(p^{2})+2n^{2}I(p^{2})=\frac{1}{k(k-1)n^{4}}\sum_{\mu\neq\nu}D_{\mu\nu}(p)p_{\mu}p_{\nu}.$
(B.45)

Lastly, for on-axis momenta, $(n,0,0,0)$, only diagonal terms survive

$D_{\mu\mu}(p)=E(p)+(F(p)+n^{2}G(p))p_{\mu}^{2},$ (B.46)

and the form factors are extracted with

$\displaystyle E(p)=\frac{1}{3}\sum_{\mu\neq 1}D_{\mu\mu}(p),$ (B.47)
$\displaystyle n^{2}F(p)+n^{4}G(p)=D_{11}(p)-E(p).$ (B.48)

##  Appendix C Results – Additional figures

###  C.1 Gluon propagator

####  C.1.1 Continuum relations – mixed diagonal configurations

In this section the continuum relations for the momentum configurations
$(n,n,n,0)$ and $(n,n,0,0)$ are computed. The procedure follows similarly as
the other two diagonal kinematics. For both cases the lattice gluon propagator
reads

$\displaystyle
D_{\mu\mu}=E(p^{2})\delta_{\mu\mu}+(F(p^{2})+n^{2}G(p^{2}))p_{\mu}^{2}$
$\displaystyle
D_{\mu\nu}=(H(p^{2})+2n^{2}I(p^{2}))p_{\mu}p_{\nu},\leavevmode\nobreak\
\mu\neq\nu.$ (C.1)

Using the extraction for the form factors built in section B.2 and also the
continuum parametrization

$D_{\mu\nu}^{c}(p)=D(p^{2})\left(\delta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right)$

the proof of the continuum relations is follows simply,

$\displaystyle E(p^{2})$
$\displaystyle=\frac{D(p^{2})}{kn^{4}(N_{d}-k)}\sum_{\mu}\left(\delta_{\mu\mu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right)\left(kn^{4}\delta_{\mu\mu}-kn^{2}p_{\mu}^{2}\right)$
$\displaystyle=\frac{D(p^{2})}{kn^{4}(N_{d}-k)}\sum_{\mu}\left(kn^{4}-kn^{2}p_{\mu}^{2}-\frac{kn^{4}}{p^{2}}p_{\mu}^{2}+\frac{kn^{2}}{p^{2}}p_{\mu}^{4}\right)$
$\displaystyle=D(p^{2})$ $\displaystyle F(p^{2})+n^{2}G(p^{2})$
$\displaystyle=\frac{D(p^{2})}{kn^{4}(N_{d}-k)}\sum_{\mu}\left(\delta_{\mu\mu}-\frac{p_{\mu}p_{\nu}}{p^{2}}\right)\left(-kn^{2}\delta_{\mu\mu}+N_{d}p_{\mu}^{2}\right)$
$\displaystyle=\frac{D(p^{2})}{kn^{4}(N_{d}-k)}\sum_{\mu}\left(-kn^{2}+N_{d}p_{\mu}^{2}-\frac{kn^{2}}{p^{2}}p_{\mu}^{2}+\frac{N_{d}}{p^{2}}p_{\mu}^{2}\right)$
$\displaystyle=-\frac{D(p^{2})}{p^{2}}$ $\displaystyle
H(p^{2})+2n^{2}I(p^{2})$
$\displaystyle=\frac{D(p^{2})}{k(k-1)n^{4}}\sum_{\mu\neq\nu}\frac{-p_{\mu}p_{\nu}}{p^{2}}p_{\mu}p_{\nu}$
$\displaystyle=-\frac{D(p^{2})}{p^{2}}.$

Notice that $p^{2}=kn^{2}$ with the parameter $k$ defined in section B.2 and
$N_{d}=4$ the dimensionality of the lattice. In addition, this result is
independent of the use o lattice or improved momentum.

$0$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$4$$0$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$4$$\scriptscriptstyle(n,n,n,0)$$0$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$4$$0$$0.5$$1$$1.5$$2$$2.5$$\scriptscriptstyle(n,n,0,0)$

$\scriptstyle p^{2}\Gamma(p^{2})$

$\hat{p}\leavevmode\nobreak\ ($\mathrm{G}\mathrm{e}\mathrm{V}$)$$\scriptstyle
d(\hat{p}^{2})$$\scriptstyle\hat{p}^{2}E(\hat{p}^{2})$$\scriptstyle\hat{p}^{2}(n^{2}F(\hat{p}^{2})+n^{4}G(\hat{p}^{2}))$$\scriptstyle-\hat{p}^{4}H(\hat{p}^{2})-p^{6}I(\hat{p}^{2})$$\hat{p}\leavevmode\nobreak\
($\mathrm{G}\mathrm{e}\mathrm{V}$)$$\scriptstyle
d(\hat{p}^{2})$$\scriptstyle\hat{p}^{2}E(\hat{p}^{2})$$\scriptstyle\hat{p}^{2}(n^{2}F(\hat{p}^{2})+n^{4}G(\hat{p}^{2}))$$\scriptstyle-\hat{p}^{4}H(\hat{p}^{2})-p^{6}I(\hat{p}^{2})$
Figure C.1: Form factors from the lattice basis for the mixed configurations
$p=(n,n,n,0)$ (left) and for $p=(n,n,0,0)$ (right) both as a function of
improved momentum. Shown for comparison is the benchmark result
$d(\hat{p}^{2})$.

The analysis of the continuum relations for these two configurations is seen
in fig. C.1. The continuum relations are exactly satisfied among all three
form factors for both configurations. The benchmark result was shown for
comparison, and it is noticeable that the further from the diagonal, the worse
the correspondence becomes. The configuration $(n,n,0,0)$ deviates from the
gluon propagator dressing function for higher momentum, while the result for
$(n,n,n,0)$ remains compatible through all range of momenta similarly to the
full diagonal momenta.


# M-MELD: A Multilingual Multi-Party Dataset for Emotion Recognition in
Conversations

###### Abstract

Expression of emotions is a crucial part of daily human communication. Emotion
recognition in conversations (ERC) is an emerging field of study, where the
primary task is to identify the emotion behind each utterance in a
conversation. Though a lot of work has been done on ERC in the past, these
works only focus on ERC in the English language, thereby ignoring any other
languages. In this paper, we present Multilingual MELD (M-MELD), where we
extend the Multimodal EmotionLines Dataset (MELD) [1] to 4 other languages
beyond English, namely Greek, Polish, French, and Spanish. Beyond just
establishing strong baselines for all of these 4 languages, we also propose a
novel architecture, DiscLSTM, that uses both sequential and conversational
discourse context in a conversational dialogue for ERC. Our proposed approach
is computationally efficient, can transfer across languages using just a
cross-lingual encoder, and achieves better performance than most uni-modal
text approaches in the literature on both MELD and M-MELD. We make our data
and code publicly on GitHub 111https://github.com/Sreyan88/M-MELD.

Index Terms—  emotion recognition, multilingual, discourse learning, graph
learning

##  1 Introduction

In the past few years, ERC has become increasingly popular as an emerging
research topic in Natural Language Processing (NLP), with potential
applications in areas of empathetic dialog systems [2], improved human-
computer interaction [3], and social media opinion mining [4]. Researchers
have proposed several methodologies [5, 6, 7] to solve this task and have
primarily evaluated their systems on 4 popular benchmark datasets, namely MELD
[8], IEMOCAP [9], DailyDialog [10], and EmoryNLP [11]. ERC, particularly
emotion recognition in multiparty conversation (ERMC), is more challenging
than tasks like sentiment classification due to the presence of emotional
dynamics in conversations [8].

Though a lot of datasets and systems have been proposed for ERC, these
datasets are in English, thereby ignoring any other languages. Different
languages function in different ways. Building uni-modal text ERC systems for
languages beyond English is especially challenging owing to the fact that
recent literature shows how multi-modal cues [12] and external knowledge [7,
13] is important for ERC. Multi-modal systems, though outperform text-only
systems, may not be applicable in all real-world scenarios like chatbots,
where only textual information is available. For text-only ERC, external cues
like commonsense [13] or psychological knowledge [7] depend on external
sources of knowledge like knowledge graphs, which restricts their use to ERC
models trained on English due to the lack of these sources in any other
language.

Main Contributions: In this paper we attempt to bridge this gap between
English and non-English languages for ERC by first proposing a new dataset,
M-MELD. M-MELD consists of over 4504 human-annotated text dialogues and 45871
individual utterances in 4 different languages, namely Greek, Polish, French,
and Spanish, spanning 7 distinct emotions and balanced equally across 2 high-
resource (French and Spanish) and 2 low-resource (Greek and Polish) languages.
We describe the annotation procedure in Section 3, where we also describe why
in-context human-translated dialogues are required for training better ERC
systems and training over machine translated (MT) dialogues results in worse
performance. Second, we also present a new model for ERC namely DiscLSTM, or
Discourse-aware LSTM, which incorporates discourse-aware structured graph
information into sequential recurrence-based learning. Precisely, we first
train a discourse parsing model using cross-lingual embeddings from XLM-
RoBERTa [14] on a popular human-annotated dataset annotated for discourse
relations between human utterances [15]. Next, we infer discourse relations
for dialogues in multiple languages and use these cues to incorporate a long-
distance conversational background in DiscLSTM. More information on DiscLSTM
can be found in Section 4. To sum up, our main contributions are as follows:

  * •

We propose M-MELD, the first dataset for ERC in languages beyond English.
M-MELD has over 45871 utterances in 4504 dialogues with human-translated
utterances for ERC and is balanced across languages. Our proposed dataset is
also useful in multilingual sentiment classification and as a human-annotated
parallel corpus for learning MT systems. Additionally, we establish strong
baselines for ERC with M-MELD.

  * •

We propose DiscLSTM, a simple yet powerful model for ERC. Unlike most state-
of-the-art systems in the literature, DiscLSTM can be easily adapted across
languages and does not require external knowledge, is more resource friendly,
and outperforms our text-based uni-modal baselines taken from literature.

##  2 Related Work

One of the first works in this space was DialogRNN [6], which proposed
modeling dialog dynamics with stacked RNNs. Following DialogRNN, the same
authors proposed DialogueGCN [5], which treats each dialogue as a graph, with
vertices as individual utterances, and edges connecting a vertex with its past
and future turns. DAG-ERC [16] uses a Directed Acyclic Graph, which combines
the benefits of graph and recurrence models with its structural properties.
Very recently, MMGCN [12] proposed fusing information from multiple modalities
by the use of spectral domain GCN to encode the multimodal contextual
information. The work closest to DiscLSTM is [17], where the authors use
discourse relations between utterances to build a conversational graph and
show that ER in both multi-party and two-party conversations benefit from
conversational discourse structures. Another popular system is DialogXL which
uses dialog-aware self-attention. Finally, CoMPM [18] combines context
embedding and pre-trained speaker memory to reflect the dialogue context and
EmotionFlow [19] learns user-specific features to model the spread impact of
emotion in a conversation.

##  3 Multilingual MELD (M-MELD)

In this section, we describe in detail the annotation procedure for M-MELD.
MELD is a dataset for ERMC. In addition to the text transcript of the
utterance, MELD also consists of acoustic and visual cues for each utterance.
MELD has more than 1400 dialogues and 13000 utterances from the TV series
Friends. Each utterance in dialogue has been labeled one of these seven
emotions:- Anger, Disgust, Sadness, Joy, Neutral, Surprise, and Fear. In this
paper, and for M-MELD, we are only concerned with the text modality for each
utterance.

The most trivial procedure to curate M-MELD would be to use state-of-the-art
Machine Translation (MT) models to translate MELD utterances into any desired
language. However, MT systems largely rely on the assumption that sentences
can be translated in isolation. Indeed, we hypothesize that translation of
conversations or dialogues requires translating sentences coherently with
respect to conversational flow in order for all aspects of the information
exchange, including speaker intent, attitude, and style, to be correctly
communicated. Thus, as a first step, we curate human-annotated translations
for M-MELD from foreign language subtitles
222https://www.tvsubtitles.net/tvshow-65-1.html. The next step involves
finding the exact utterances in the subtitles that match the utterances in
MELD. Though MELD has timestamps for each utterance relative to each episode,
it should be noted that these timestamps are prone to changing when in another
foreign language, as utterance durations might shorten or lengthen. Thus, we
hire 4 professional annotators (1 for each language) to manually match each
utterance in MELD dialogues to their actual utterances in the subtitles. To
reduce the annotation burden, we show them English utterances that occur at a
maximum of +-10s from the foreign utterance to be annotated, as the original
sentence was guaranteed to be found within that window. Statistics for our
M-MELD dataset can be found in Table 1.

Table 1: Dataset Statistics for M-MELD Language | # Dialogues | # Utterances
---|---|---
Train | Dev | Test | Train | Dev | Test
French | 633 | 97 | 224 | 6537 | 964 | 2198
Greek | 870 | 103 | 240 | 9003 | 1062 | 2366
Spanish | 769 | 111 | 268 | 7890 | 1064 | 2546
Polish | 858 | 96 | 235 | 8928 | 989 | 2324

##  4 Discourse-aware LSTM

###  4.1 Problem Formulation

The general problem of ERC can be formulated as follows. Suppose there are $m$
participants
{$p\textsubscript{1},p\textsubscript{2},p\textsubscript{2},\cdots,p\textsubscript{m}$}
in a conversation or dialogue with $n$ number of utterances
{$e\textsubscript{1},\cdots,e\textsubscript{i},\cdots,e\textsubscript{n}$},
where utterance $e\textsubscript{i}$ is uttered by $p(e\textsubscript{i})$,
and $p(.)$ denotes the mapping between an utterance and it’s speaker. The
primary objective of ERC is to predict the emotion label $y\textsubscript{i}$
for utterance $e\textsubscript{i}$ based on the context of the dialogue to
which $e\textsubscript{i}$ belongs. We denote a dialogue as
$U\textsubscript{j}$, where utterance $e\textsubscript{i}\in
U\textsubscript{j}$ and
$D=\\{U\textsubscript{1},\cdots,U\textsubscript{j},\cdots,U\textsubscript{T}$},
where dataset $D$ has a total of $T$ dialogues.

###  4.2 Utterance-level Feature Extraction

We formulate each conversation $U_{j}$ as a graph and treat each utterance
embedding $u\textsubscript{i}\in U_{j}$ as a node in the graph. In line with
prior methods [5, 16], we use the XLM-RoBERTalarge transformer model, fine-
tuned on the task-specific ERC dataset to extract sentence-level features
$u\textsubscript{i}\in\mathbb{R}^{1024}$ for each individual utterance $e_{i}$
in the dialogue. More precisely, similar to [5], we add a _[CLS]_ token at the
beginning of each tokenized utterance we feed into our XLM-RoBERTalarge model;
the output embedding of the _[CLS]_ acts as our pooled utterance
representation. Conversation $U_{j}$ can now be denoted by $U_{j}$ =
{$u\textsubscript{1},\cdots u\textsubscript{i},\cdots,u\textsubscript{n}$} or
a set of utterance representations from XLM-RoBERTalarge.

###  4.3 Dialogue Discourse Parsing

For dialogue discourse parsing, we first train a state-of-the-art model [20]
on a human-annotated multi-party dialogue corpus STAC [15]. We view each
utterance as an EDU (Elementary Discourse Unit) and use the discourse relation
types defined in STAC. [20] employs a transformer backbone. One simple trick
that enables us to obtain discourse relations in dialogues across languages is
using cross-lingual embeddings from XLM-RoBERTa to train our discourse parsing
model. Thus, even though STAC is in English, the model supports a wide range
of languages during inference time.

###  4.4 Disc-LSTM Architecture

####  4.4.1 Conversation Graph Construction

With discourse relations obtained from the previous step, we construct a
discourse graph $G$ = ($V$,$E$) for each conversation, where $V$ = {$v$$1$,
$v$$2$, $v$$3$, $\cdots$, $v$$n$} are the vertices or nodal representations of
the utterances {$u$$1$, $u$$2$, $u$$3$, $\cdots$, $u$$n$} and
$E\in\mathbb{R}^{n\times n}$ is the adjacency matrix denoting edge relation,
where $E[i][j]$ = 1 if there is a discourse relation between utterances $i$
and $j$.

####  4.4.2 Temporal Information Flow in Graph Layers

To encode discourse relations in a conversation, we use a Graph Attention
Network (GAT) [21] with the information flow through layers inspired by [22].
To feed our contextualized RoBERTa-based utterance embedding $u_{i}$ $\in$
$\mathbb{R}^{1024}$ to our graph network, we first down-project $e_{i}$ to
$g_{i}^{1}$, where a $g_{i}^{1}$ $\in$ $\mathbb{R}^{300}$ via a full-connected
layer $f(.)$ as follows:
$G^{1}=f(U)=\\{g_{1}^{1},g_{2}^{1},g_{3}^{1},\cdots,g_{n}^{1}\\}$, where
$G^{1}$ is the graph-encoded representation of our utterance output by the
first layer in the graph. For each utterance embedding $g$$i$, the attention
weights between $g$$i$ and its predecessors are calculated by using $g$$i$’s
hidden state at the $(l-1)$-th layer and the nodes $j\in\mathcal{N}_{i}$ in
the current $(l)$-th. Formally, we find the attention weights of utterance
$u$i’s hidden value with the above-mentioned nodes in the following manner
using a GAT layer:

$\alpha_{ij}^{l}=\text{softmax}_{j\in\mathcal{N}_{i}}(W_{\alpha}^{l}[g^{l}_{j}||g^{l-1}_{i}])$
(1)

where $W_{\alpha}^{l}$ are the learnable parameters and $||$ represents a
concatenation operation. We finally gather or accumulate the information using
the weights calculated above and get the subsequent layer information by:
$g^{l}_{i}=\sum_{j\in\mathcal{N}\textsubscript{i}}\alpha_{ij}g_{j}^{l}+g_{i}^{l-1}$.
We use the final graph layer embedding after multiple information propagation
steps in the graph network and obtain
$G^{l}=\\{g_{1}^{l},g_{2}^{l},g_{3}^{l},\cdots,g_{n}^{l}\\}$ and the
contextualized embeddings from the RoBERTa for our next step.

Fig. 1: Illustration of our proposed DiscLSTM. The utterance embedding $u$ is
passed through a Graph Encoder $G$ post which it is passed to the DiscLSTM
cell together with the output of the graph encoder $g^{l}$.

####  4.4.3 Bi-Directional DiscLSTM Cell

In order to better integrate both the sequential and discourse context and to
dynamically learn the relative importance of each graph-encoded utterance
representation for modeling a conversation, we propose DiscLSTM. DiscLSTM
builds on the basic LSTM cell and takes as inputs previous cell state
$c_{t-1}$, previous hidden state $h_{t-1}$, current cell input $u_{t}$, and an
additional graph-encoded utterance representation $g_{t}$. The cell outputs
the current cell state $c_{t}$ and the current hidden state $h_{t}$. The cell
representation can be seen in Fig. 1. The following represents the propagation
of information inside the DiscLSTM cell:

$\begin{aligned}
\mathbf{f}_{t}&=\sigma\left(W_{(f)}\mathbf{u}_{t}+U_{(f)}\mathbf{h}_{t-1}+Q_{(f)}\mathbf{g}_{t}+\mathbf{b}_{(f)})\right.\\\
\mathbf{o}_{t}&=\sigma\left(W_{(o)}\mathbf{u}_{t}+U_{(o)}\mathbf{h}_{t-1}+Q_{(o)}\mathbf{g}_{t}+\mathbf{b}_{(o)}\right)\\\
\end{aligned}$

$\begin{aligned}
\mathbf{i}_{t}&=\sigma\left(W_{(i)}\mathbf{u}_{t}+U_{(i)}\mathbf{h}_{t-1}+\mathbf{b}_{(i)}\right)\\\
\mathbf{p}_{t}&=\sigma\left(W_{(p)}\mathbf{u}_{t}+Q_{(p)}\mathbf{g}_{t}+\mathbf{b}_{(p)}\right)\\\
\tilde{\mathbf{c}}_{t}&=\tanh\left(W_{(u)}\mathbf{u}_{t}+U_{(u)}\mathbf{h}_{t-1}+\mathbf{b}_{(u)}\right)\\\
\tilde{\mathbf{s}}_{t}&=\tanh\left(W_{(n)}\mathbf{u}_{t}+Q_{(n)}\mathbf{g}_{t}+\mathbf{b}_{(n)}\right)\\\
\mathbf{c}_{t}&=\mathbf{f}_{t}\odot\mathbf{c}_{t-1}+\mathbf{i}_{t}\odot\tilde{\mathbf{c}}_{t}+\mathbf{p}_{t}\odot\tilde{\mathbf{s}}_{t}\\\
\mathbf{h}_{t}&=\mathbf{o}_{t}\odot\tanh\left(\mathbf{c}_{t}\right)\end{aligned}$

where $u_{t}$ ($u_{i}$ w.r.t. utterance in a dialogue) is the XLM-RoBERTalarge
utterance representation and $g_{t}$ ($g_{i}^{l}$ w.r.t. utterance in a
dialogue) is the graph-encoded utterance representation for the $t^{th}$ time-
step in the sequential processing by the bi-directional DiscLSTM cell. The
forward and backward DiscLSTM enable the model to integrate both the
sequential and structured-discourse information from both directions in the
sequence. Finally, we concatenate the hidden state $\overrightarrow{h_{if}}$
and hidden state $\overleftarrow{h_{ib}}$ from the forward and backward states
respectively to get the final hidden state representation of $i$th utterance
$h_{i}=[\overrightarrow{h_{if}};\overleftarrow{h_{ib}}]$. This final hidden
state representation $\textbf{H}=\\{h_{1},h_{2},h_{3},\cdots,h_{n}\\}$ is then
fed to a fully-connected layer which outputs a vector representation $p_{i}$
$\in$ $\mathbb{R}^{d}$ for each utterance $e_{i}$ where $d$ equals the number
of emotion classes in the ERC dataset.

##  5 Experiments

Baselines: For evaluating systems from prior-art across languages in M-MELD
and also to compare DiscLSTM, we choose systems that do not require an
external source of knowledge, e.g., knowledge graphs, and can be easily re-
implemented across languages. To the best of our knowledge, external sources
of knowledge commonly used in recent literature attributing to their success
[7, 13] are not readily available in foreign languages beyond English. Adding
to this, systems like DialogXL depend on large-scale language-specific pre-
trained models and their foreign language counterparts are neither available
nor resource-friendly to pre-train from scratch. On the other hand, systems
like MMGCN [12] leverage multiple modalities, which is not always available in
a real-world setting.

Our first baseline is an XLM-RoBERTa transformer trained on a sequence
classification task for ER. We adopted DialogueRNN, DialogueGCN and DAG-ERC
described in Section 2 to work with utterance-level features extracted using
the methodology in Section 4.2. Additionally, we fine-tune EmotionFlow and
CoMPM end-to-end with XLM-RoBERTa backbone for each language. To prove the
need for human-translated annotations, we also evaluate performance of all our
baselines on synthetic MT data.

Experimental Setup: We use the pre-trained XLM-RoBERTa from the Huggingface
library. For training and evaluation of all our systems, including baselines
and DiscLSTM, we use a batch size of 16 and train our networks for 50 epochs
using Adam optimizer with a learning rate of $1e^{-5}$. For training XLM-
RoBERTalarge we find an optimal learning rate of $1e^{-4}$. Optimal
hyperparameters for all other baselines were obtained from their respective
papers.

##  6 Results

Following much of prior-art, we evaluate the performance of all our baselines
and our proposed DiscLSTM on the _weighted $F_{1}$_ score. Results are
presented in Table 2. As we clearly see, DiscLSTM outperforms most of our
baselines on all 4 languages in M-MELD and is close to DialogueGCN and DAS-ERC
in French and Spanish. However, one must note that unlike DialogueGCN and DAG-
ERC, DiscLSTM does not use future utterance information for ERC. In addition
to M-MELD, we also repeat our baselines for the original MELD and the results
are 62.10 / 63.16 / 63.5 / 52.81 / 50.84 / 63.75 respectively in same order as
in Table 2. DiscLSTM is also more efficient than all our graph-based baselines
as it uses much lesser edges on average than our other baselines which extend
an edge from each utterance to all future and past utterances. Additionally,
we notice an drop of 2.1% average across methods when trained on MT data.
Detailed results can be found on GitHub.

Table 2: Results on the various languages in M-MELD Model | French | Spanish | Greek | Polish
---|---|---|---|---
XLM-RoBERTa | 49.14 | 52.30 | 52.70 | 31.0
DialogueRNN [6] | 50.19 | 52.50 | 53.10 | 31.41
DialogueGCN [5] | 51.20 | 52.81 | 53.46 | 31.90
CoMPM [18] | 35.41 | 49.00 | 42.76 | 37.03
EmotionFlow [19] | 35.69 | 41.65 | 43.13 | 42.20
DAG-ERC [16] | 49.10 | 52.9 | 53.40 | 42.23
DiscLSTM (ours) | 49.43 | 53.24 | 53.46 | 43.21

##  7 Conclusion

In this paper, we present M-MELD, the first publicly available multilingual
dataset for ERC in languages beyond English, namely French, Spanish, Greek,
and Polish. We establish strong baselines from literature for all 4 languages
in M-MELD and discuss how most modern systems achieving state-of-the-art in
MELD get restricted to their usage just in English. Our dataset opens new
challenges to the research community in designing ERC systems that can be
easily transferred across languages. Additionally, we also propose a new and
efficient system for ERC, DiscLSTM, which outperforms all our unimodal text
baselines across MELD and M-MELD. A limitation of DiscLSTM is the two-step
approach where error does not propagate across stages. As part of future work
we would devise better end-to-end language-independent ERC systems.

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