# A gap theorem for the ZL-amenability constant of a finite group

Yemon Choi

(15th May 2015)

###### Abstract

It was shown in [2] that the ZL-amenability constant of a finite group is
always at least $1$, with equality if and only if the group is abelian. It was
also shown in [2] that for any finite non-abelian group this invariant is at
least $301/300$, but the proof relies crucially on a deep result of D. A.
Rider on norms of central idempotents in group algebras.

Here we show that if $G$ is finite and non-abelian then its ZL-amenability
constant is at least $7/4$, which is known to be best possible. We avoid use
of Rider’s result, by analyzing the cases where $G$ is just non-abelian, using
calculations from [1], and establishing a new estimate for groups with trivial
centre.

MSC 2010 classification: 20C15 (primary); 43A20, 43A62 (secondary).

Keywords: amenability constant, character degrees, just non-abelian groups.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Notation and terminology
    2. 2.2 Hereditary properties of the ZL-amenability constant
    3. 2.3 An explicit formula and some comments
  3. 3 Bounds on $\operatorname{ass}$ for groups with trivial centre
  4. 4 Bounds on $\operatorname{\rm AM}_{\rm Z}$ for just non-abelian groups
    1. 4.1 JNA groups: definitions and two results
    2. 4.2 The ZL-amenability constant in the JNA cases
  5. 5 A lower bound for perfect groups
  6. 6 Questions for future investigation
  7. A The proofs of Lemma 4.4 and Theorem 4.5

##  1 Introduction

Given a finite group $G$, consider its complex group algebra $\mathbb{C}G$,
and recall that the centre ${\rm Z}\mathbb{C}G$ is isomorphic to
$\bigoplus_{p}\mathbb{C}p$ where the sum is over all minimal idempotents in
${\rm Z}\mathbb{C}G$. Now consider

$\Delta_{\rm Z}(G)=\sum_{p}p\otimes p\in{\rm
Z}\mathbb{C}G\mathbin{\otimes}{\rm Z}\mathbb{C}G\,.$ (1.1)

${\rm Z}\mathbb{C}G\mathbin{\otimes}{\rm Z}\mathbb{C}G$ sits naturally inside
$\mathbb{C}G\mathbin{\otimes}\mathbb{C}G\equiv\mathbb{C}(G\times G)$, and
there is a natural $\ell^{1}$-norm $\|{\cdot}\|_{1}$ that we can put on
$\mathbb{C}(G\times G)$. We define the ZL-amenability constant of $G$, denoted
by $\operatorname{\rm AM}_{\rm Z}(G)$, to be $\|{\Delta_{\rm Z}(G)}\|_{1}$.
The original reasons for studying $\operatorname{\rm AM}_{\rm Z}(G)$ arose in
connection with Banach algebras and (non-abelian) Fourier analysis; however,
the problem which we address in this paper can be stated purely in terms of
finite groups and the behaviour of their irreducible characters.

$\operatorname{\rm AM}_{\rm Z}(G)$ seems to have first been studied explicitly
in work of A. Azimifard, E. Samei and N. Spronk, although they used different
terminology and notation. One of their observations, paraphrased into our
notation, is that $\operatorname{\rm AM}_{\rm Z}(G)\geq 1$ with equality if
and only if $G$ is abelian; this is proved using the Schur orthogonality
relations and a clever use of an “associated minorant”
$\operatorname{ass}(G)$. (See Example 2.7 and Proposition 2.9 for further
details.) The same paper also contains the following “gap” result, which lies
significantly deeper.

####  Theorem  (Azimifard–Samei–Spronk, [2]).

There exists $\delta>0$ such that $\operatorname{\rm AM}_{\rm Z}(G)\geq
1+\delta$ for every finite non-abelian group $G$.

The purpose of the present paper is to determine exactly how big $\delta$ can
be.

######  Remark 1.1.

To provide background context, let us briefly mention why the authors of [2]
wished to have this gap result. If $G$ is a compact group its convolution
algebra $L^{1}(G)$ is an example of an amenable Banach algebra; in contrast,
it is shown in [2] that for many choices of compact group $G$, the centre of
$L^{1}(G)$ is not amenable.111We shall not discuss amenability of Banach
algebras in this article, but it has proved to be a fundamental and fruitful
notion in certain areas of functional analysis; it may be viewed as a weakened
version of “homological dimension zero”, suitably interpreted. One source of
such examples is given by $G=\prod_{n=1}^{\infty}G_{n}$ where each $G_{n}$ is
a finite non-abelian group, and this works by combining Theorem 1 with a
tensor product argument to show that $\operatorname{\rm AM}_{\rm
Z}(\prod_{i=1}^{n}G_{i})\geq(1+\delta)^{n}$ for each $n$. Thus results for
finite groups are used to provide compact groups that generate interesting
examples of Banach algebras.

Now let us return to Theorem 1. Examining the original proof given in [2], one
sees that it relies crucially on the following hard result of D. A. Rider.

####  Theorem A (Rider; see [10, Lemma 5.2]).

Let $K$ be a compact non-abelian group, equipped with a choice222Strictly
speaking, Rider only states and proves this when $\lambda(K)=1$. However, the
more general version stated here follows easily from the case $\lambda(K)=1$
by a rescaling argument. of Haar measure $\lambda$. Let $f$ be a finite linear
combination of irreducible characters of $K$, regarded as an element of the
convolution algebra $L^{1}(K,\lambda)$. Let
$\|{f}\|_{1}:=\int_{K}|{f(x)}|\,d\lambda(x)$. If $f$ is an idempotent in
$L^{1}(K,\lambda)$, then either $\|{f}\|_{1}=1$ or $\|{f}\|_{1}>1+1/300$.

Theorem 1, with $\delta=1/300$, follows from Theorem 1 by taking $K=G\times
G$, $\lambda$ to be counting measure and $f=\Delta_{\rm Z}(G)\in\ell^{1}(K)$.
Nevertheless, one would hope for a more direct proof of Theorem 1, or a better
value of $\delta$. Rider notes that the value $1/300$ in Theorem 1 could be
improved, but his method does not give any indication of the likely order of
magnitude of the optimal constant.333In some private calculations the present
author has been able to get up to $1/80$, but not up to $1/10$. Furthermore,
this existing proof of Theorem 1 ignores the fact that $\operatorname{\rm
AM}_{\rm Z}(G)$ is the norm of a very special element of ${\rm
Z}\mathbb{C}(G\times G)$.

In some recent work of the present author with M. Alaghmandan and E. Samei
[1], the exact ZL-amenability constants were calculated for several families
of finite groups. In that work, the smallest value obtained was $7/4$,
achieved by the dihedral group of order $8$, and no smaller value could be
found. This demonstrated a large gap between the value of $\delta$ provided by
Theorem 1, and what one observed in practice.

In this paper we show that the inequality $\operatorname{\rm AM}_{\rm
Z}(G)\geq 1+1/300$ is a severe underestimate. Specifically, we prove the
following new bound.

######  Theorem 1.2.

Let $G$ be a finite, non-abelian group. Then $\operatorname{\rm AM}_{\rm
Z}(G)\geq 7/4$.

######  Remark 1.3.

As previously noted, the ${\rm ZL}$-amenability constant of the dihedral group
of order $8$ is exactly $7/4$. So the bound in Theorem 1.2 is best possible.
Furthermore, we will see later in Example 4.11 that there is an infinite
sequence $(G_{n})$ of finite $2$-groups, each one indecomposable as a direct
product of smaller factors, such that $\operatorname{\rm AM}_{\rm
Z}(G_{n})=7/4$ for all $n$.

We wish to immediately highlight two features of our approach to Theorem 1.2.
Firstly, we completely bypass Rider’s result; and secondly, we avoid any
difficult structure theory for finite groups. In fact, all the group theory
and character theory we need can be found in introductory texts. We need some
results from [1], but the same comments apply there.

### Structure and style of the paper

In Section 2 we give a rapid definition of $\operatorname{\rm AM}_{\rm Z}(G)$
and establish some basic properties that will be needed. These properties were
already established in [2], using a Banach-algebraic perspective; here we
shall give complete proofs that only need Schur orthogonality for finite
groups, so that the reader does not need to take results from [2] on trust.

Also, following an idea used by Azimifard, Samei and Spronk in their proof of
Theorem 1, we introduce an auxiliary invariant $\operatorname{ass}(G)$ which
is a lower bound for $\operatorname{\rm AM}_{\rm Z}(G)$ and is easier to
estimate from below. In Section 3 we obtain a new lower bound on
$\operatorname{ass}(G)$ for all groups with trivial centre (excluding the
degenerate case of the $1$-element group). This is perhaps the main
conceptually new idea in this paper that was missing from [1] and [2].
Nevertheless, to get from here to the optimal constant in Theorem 1.2 requires
another ingredient, and that is to reduce the problem to the case of finite
groups that are “just non-abelian”. The precise statement is Theorem 4.3: the
necessary definitions, and the proof of that theorem, take up all of Section
4. We also need two technical results on just non-abelian groups (Lemma 4.4
and Theorem 4.5): both can be extracted from the existing literature, but for
the reader’s convenience we provide full proofs in the appendix.

These sections, together with the Appendix, provide a complete proof of
Theorem 1.2. Figure 1 shows how the key results are assembled in this proof.

$\left.\begin{array}[]{lcc}&&\framebox{Corollary \ref{c:AMZL-of-prod-and-
quot}\ref{li:quot}}\\\ &&\\\
\left.\begin{tabular}[]{c}\framebox{Theorem~{}\ref{t:bound for trivial
centre}}\\\ \framebox{Theorem~{}\ref{t:is-dihedral}}\\\
\framebox{Lemma~{}\ref{l:baby
JN2}}\end{tabular}\right\\}&\Longrightarrow&\framebox{Theorem~{}\ref{t:bound
for JNA
cases}}\end{array}\right\\}\Longrightarrow\framebox{Theorem~{}\ref{t:mainthm}}$
Figure 1: Results needed to prove the main result

For sake of completeness and interest: in Section 5 we use similar but easier
ideas to those of Section 3 to obtain a lower bound on $\operatorname{ass}(G)$
for all perfect groups $G$; and in Section 6 we list several natural questions
that we have not yet resolved, concerning the set of possible values of
$\operatorname{\rm AM}_{\rm Z}(\cdot)$.

The style of this paper is intended to be somewhat expository. The aim, both
here and in the earlier paper [1], has been to make the proofs accessible to
researchers in abstract harmonic analysis who are interested in amenability
constants of various Banach algebras, but who may not have much exposure to
structure theory for finite groups. Therefore, the proofs are spelled out in
some detail, and we have avoided more specialized results or techniques from
finite group theory. On the other hand, this paper does not presume any
knowledge of (or interest in) abstract harmonic analysis; so finite group
theorists may find Theorem 1.2 and its proof of some interest, and perhaps
take the results obtained en route as a challenge to find sharper versions or
better estimates.

##  2 Preliminaries

###  2.1 Notation and terminology

We assume familiarity with basic character theory of finite groups over the
complex field $\mathbb{C}$, up to the Schur orthogonality relations. All of
this can be found in Isaacs’s book [6]; alternatively, see [4, Chapter 7] or
[7].

Given a finite group $G$, we denote by $\operatorname{Conj}(G)$ the set of
conjugacy classes of $G$, and we write $\operatorname{Irr}(G)$ for the set of
irreducible characters of $G$ (working over $\mathbb{C}$). The degree of a
character $\phi$ is just $\phi(e)$ (which equals the dimension of any
representation affording that character) and we say that $\phi$ is linear if
$\phi(e)=1$. If $C\in\operatorname{Conj}(G)$ and $\phi$ is a character on $G$,
we write $\phi(C)$ for the value of $\phi$ on any element of $C$; thus
$|\phi(C)|\leq\phi(e)$.

The element $\Delta_{\rm Z}(G)$ mentioned in the introduction is the diagonal
element or separability idempotent for the commutative $\mathbb{C}$-algebra
$Z(\mathbb{C}G)$. For the definition of these terms for general algebras over
a field, see e.g. [5, Section 6.7]; further explanation in the special case of
the centre of a group algebra, and discussion of the connection with
amenability for Banach algebras, can be found in [1, Section 2.1].

To keep our presentation self-contained, we will not use general properties of
separability idempotents, and instead adopt a “hands-on” definition. First
some terminology: if ${\mathbb{I}}$ is a non-empty finite indexing set, then
by the “natural” $\ell^{1}$-norm on the vector space $\mathbb{C}^{\mathbb{I}}$
we simply mean
$\|{a}\|_{1}\equiv\|{a}\|_{\ell^{1}({\mathbb{I}})}:=\sum_{i\in{\mathbb{I}}}|a_{i}|$.

######  Definition 2.1.

Let $G$ be a finite group. If $\phi\in\operatorname{Irr}(G)$, we may regard
$\phi\mathbin{\otimes}\phi$ as an element of $\mathbb{C}(G\times G)$ (since
the underlying vector space of $\mathbb{C}(G\times G)$ is $\mathbb{C}^{G\times
G}$). Let

$\Delta_{\rm
Z}(G):=\sum_{\phi\in\operatorname{Irr}(G)}\frac{\phi(e)}{|G|}\phi\mathbin{\otimes}\frac{\phi(e)}{|G|}\phi\in\mathbb{C}(G\times
G)\,.$

Then, equipping $\mathbb{C}^{G\times G}$ with its natural $\ell^{1}$-norm,
define the ZL-amenability constant of $G$ to be the $\ell^{1}$-norm of
$\Delta_{\rm Z}(G)$.

We could of course define $\operatorname{\rm AM}_{\rm Z}(G)$ without
explicitly naming $\Delta_{\rm Z}(G)$, but some hereditary properties of
$\operatorname{\rm AM}_{\rm Z}$ with respect to taking products and quotients
of groups are best understood in terms of corresponding properties of
$\Delta_{\rm Z}$. This will be done in the next section.

###  2.2 Hereditary properties of the ZL-amenability constant

The next two lemmas are known results, if one uses the definition of
$\Delta_{\rm Z}(G)$ that is given in [2] and [1]. We shall give character-
theoretic proofs that just use Definition 2.1.

######  Lemma 2.2.

Let $G$ and $H$ be finite groups. If we identify $\mathbb{C}(G\times H\times
G\times H)$ with $\mathbb{C}(G\times G)\mathbin{\otimes}\mathbb{C}(H\times H)$
in the natural way, then $\Delta_{\rm Z}(G\times H)$ is identified with
$\Delta_{\rm Z}(G)\mathbin{\otimes}\Delta_{\rm Z}(H)$.

###### Proof.

This is a straightforward calculation using $\operatorname{Irr}(G\times
H)=\\{\phi\mathbin{\otimes}\psi\mathbin{\colon}\phi\in\operatorname{Irr}(G),\psi\in\operatorname{Irr}(H)\\}$
and $(\phi\mathbin{\otimes}\psi)(e_{G\times H})=\phi(e_{G})\psi(e_{H})$. ∎

######  Lemma 2.3.

Let $G$ and $H$ be finite groups, and suppose $q:G\to H$ is a surjective group
homomorphism. Extend $q$ to an algebra homomorphism
$Q:\mathbb{C}G\to\mathbb{C}H$ in the natural way. Then
$(Q\mathbin{\otimes}Q)(\Delta_{\rm Z}(G))=\Delta_{\rm Z}(H)$.

###### Proof.

The map $\phi\mapsto\phi\circ q$ defines an injection
$q^{*}:\operatorname{Irr}(H)\hookrightarrow\operatorname{Irr}(G)$. Now

$Q\left(\sum_{y\in H}\sum_{x\in q^{-1}(y)}a_{x}\delta_{x}\right)=\sum_{y\in
H}\left(\sum_{x\in q^{-1}(y)}a_{x}\right)\delta_{y}\qquad(a\in\mathbb{C}G).$

In particular, for each $\phi\in\operatorname{Irr}(H)$,

$Q\left(\frac{q^{*}(\phi)(e_{G})}{|G|}q^{*}(\phi)\right)=\frac{\phi(e_{H})}{|G|}|\ker(q)|\phi=\frac{\phi(e_{H})}{|H|}\phi\;.$

Let ${\mathcal{R}}$ denote the set $\operatorname{Irr}(G)\setminus
q^{*}(\operatorname{Irr}(H))$. Then

$Q\left(\sum_{\psi\in\operatorname{Irr}(G)\setminus{\mathcal{R}}}\frac{\psi(e_{G})}{|G|}\psi\mathbin{\otimes}\frac{\psi(e_{G})}{|G|}\psi\right)=\Delta_{\rm
Z}(H)\,;$

so to complete the proof, it suffices to show that
${\mathcal{R}}\subseteq\ker(Q)$.

Our argument is indirect. Given $f\in{\rm Z}\mathbb{C}G$, for each $y\in H$ we
pick $x\in q^{-1}(y)$ and note that

$\delta_{y}*Q(f)=Q(\delta_{x}*f)=Q(f*\delta_{x})=Q(f)*\delta_{y}\,.$

Hence $Q({\rm Z}\mathbb{C}G)\subseteq{\rm Z}\mathbb{C}H$. Since
$\operatorname{Irr}(H)$ spans ${\rm Z}\mathbb{C}H$, it now suffices to prove
that

$\sum_{y\in H}Q(\theta)(y)\ \overline{\phi(y)}=0\qquad\text{for all
$\theta\in{\mathcal{R}}$ and all $\phi\in\operatorname{Irr}(H)$.}$

Equivalently, it suffices to prove that for each $\theta\in{\mathcal{R}}$ and
each $\phi\in\operatorname{Irr}(H)$,

$0=\sum_{y\in H}\left(\sum_{x\in
q^{-1}(y)}\theta(x)\right)\overline{\phi(y)}=\sum_{x\in
G}\theta(x)\overline{(\phi\circ q)(x)}\,.$

But this is immediate from Schur orthogonality for $\operatorname{Irr}(G)$. ∎

######  Corollary 2.4 (${\rm ZL}$-amenability constants of products and
quotients).

  1. (i)

Let $G_{1}$ and $G_{2}$ be finite groups. Then $\operatorname{\rm AM}_{\rm
Z}(G_{1}\times G_{2})=\operatorname{\rm AM}_{\rm Z}(G_{1})\operatorname{\rm
AM}_{\rm Z}(G_{2})$.

  2. (ii)

Let $G$ be a finite group and let $N\unlhd G$. Then $\operatorname{\rm
AM}_{\rm Z}(G/N)\leq\operatorname{\rm AM}_{\rm Z}(G)$.

###### Proof.

For any finite indexing sets ${\mathbb{I}}$ and ${\mathbb{J}}$, the
$\ell^{1}$-norm has the following property: for each
$a\in\mathbb{C}^{\mathbb{I}}$ and $b\in\mathbb{C}^{\mathbb{J}}$, we have
$\|{a\mathbin{\otimes}b}\|_{\ell^{1}({\mathbb{I}}\times{\mathbb{J}})}=\|{a}\|_{\ell^{1}({\mathbb{I}})}\|{b}\|_{\ell^{1}({\mathbb{J}})}$.
Now, using Lemma 2.2, part (i) follows.

To prove part (ii), note that $Q:\mathbb{C}G\to\mathbb{C}H$ does not increase
the $\ell^{1}$-norm of elements. The same is true for
$Q\mathbin{\otimes}Q:\mathbb{C}(G\times G)\to\mathbb{C}(H\times H)$. Therefore
$\|{(Q\mathbin{\otimes}Q)(\Delta_{\rm Z}(G))}\|_{1}\leq\|{\Delta_{\rm
Z}(G)}\|_{1}$, and applying Lemma 2.3 does the rest. ∎

######  Remark 2.5.

Both parts of Corollary 2.4 were already proved in [2, §1], using the abstract
characterization of $\Delta_{\rm Z}(G)$ as the unique “diagonal element” or
“separability idempotent” for the algebra ${\rm Z}\mathbb{C}G$. Our approach
has the benefit of just using Definition 2.1 and Schur orthogonality for
finite groups, but the approach in [2] works in a somewhat broader context.

###  2.3 An explicit formula and some comments

In [2, Theorem 1.8] the authors give the following concrete formula for the
ZL-amenability constant of a finite group:

$\operatorname{\rm AM}_{\rm
Z}(G)=\frac{1}{|G|^{2}}\sum_{C\in\operatorname{Conj}(G)}\sum_{D\in\operatorname{Conj}(G)}|C||D|\left|\sum_{\phi\in\operatorname{Irr}(G)}\phi(e)^{2}\phi(C)\overline{\phi(D)}\right|\;.$
(2.1)

We could have taken this as the definition of $\operatorname{\rm AM}_{\rm
Z}(G)$, if we wished to avoid discussing $\Delta_{\rm Z}(G)$ and
$\ell^{1}$-norms, but then the proof of Corollary 2.4 would become even more
opaque.

######  Remark 2.6.

The presence of complex conjugates in Equation (2.1) may need some
explanation. First, note that the same expression without any complex
conjugates follows immediately from Definition 2.1 and the definition of the
$\ell^{1}$-norm on $\mathbb{C}(G\times G)$. Then observe that there is a
bijection on $\operatorname{Conj}(G)$ which exchanges the conjugacy class of
$x$ with that of $x^{-1}$; and if we denote this bijection by
$D\longleftrightarrow D^{-1}$, then
$|D^{-1}|\phi(D^{-1})=|D|\overline{\phi(D)}$ for each
$D\in\operatorname{Conj}(G)$ and each character $\phi$.

######  Example 2.7 (ZL-amenability constants of finite abelian groups).

Let $G$ be a finite abelian group. It was noted in [2] that in this case
$\Delta_{\rm Z}(G)=|G|^{-1}\sum_{x\in
G}\delta_{x}\mathbin{\otimes}\delta_{x^{-1}}$, which clearly has
$\ell^{1}$-norm exactly $1$. (See the remarks in [1, Section 2.1] for further
explanation of this identity.) Here we give a more direct proof. By Equation
(2.1)

$\operatorname{\rm AM}_{\rm Z}(G)=\frac{1}{|G|^{2}}\sum_{x,y\in
G}\left|\sum_{\phi\in\widehat{G}}\phi(x)\overline{\phi(y)}\right|\,.$

By Schur (column) orthogonality, most terms in this sum vanish and summing the
remaining ones gives

$\operatorname{\rm AM}_{\rm Z}(G)=|G|^{-2}\sum_{x\in
G}\sum_{\phi\in\widehat{G}}1=1,$

as required.

In general it seems hard to get good lower bounds on the expression in (2.1),
since there could be considerable cancellation when we sum over $\phi$, as in
Example 2.7. A key idea in [2] is to introduce a minorant for
$\operatorname{\rm AM}_{\rm Z}(G)$ that is easier to estimate well from below.
The authors of [2] do not give this invariant a name, but since it is the main
object of study in Section 3 it deserves some ad hoc notation.

######  Definition 2.8 (An auxiliary minorant).

For $G$ a finite group, we define

$\operatorname{ass}(G)=\frac{1}{|G|^{2}}\sum_{C\in\operatorname{Conj}(G)}\sum_{\phi\in\operatorname{Irr}(G)}|C|^{2}|\phi(C)|^{2}\phi(e)^{2}\
.$ (2.2)

######  Proposition 2.9 (Azimifard–Samei–Spronk, [2]).

$\operatorname{\rm AM}_{\rm Z}(G)\geq\operatorname{ass}(G)\geq 1$ for every
finite group $G$. Moreover, if $G$ is non-abelian then
$\operatorname{ass}(G)>1$.

###### Proof.

This is demonstrated in the proof of Theorem 1.8 and Corollary 1.9 in [2], but
for sake of completeness we include the argument here. Note that
$\operatorname{ass}(G)$ is obtained by considering the right-hand side of
(2.1) and only summing over the terms indexed by
$\\{(C,C)\mathbin{\colon}C\in\operatorname{Conj}(G)\\}\subset\operatorname{Conj}(G)\times\operatorname{Conj}(G)$.
Thus $\operatorname{\rm AM}_{\rm Z}(G)\geq\operatorname{ass}(G)$. Moreover,
since $|C|^{2}\geq|C|$, we have

$\displaystyle\operatorname{ass}(G)$
$\displaystyle\geq|G|^{-2}\sum_{C\in\operatorname{Conj}(G)}\sum_{\phi\in\operatorname{Irr}(G)}|C||\phi(C)|^{2}\phi(e)^{2}$
$\displaystyle=\frac{1}{|G|}\sum_{\phi\in\operatorname{Irr}(G)}\phi(e)^{2}\sum_{C\in\operatorname{Conj}(G)}\frac{|C|}{|G|}|\phi(C)|^{2}$
$\displaystyle=1\,,$

where the final equality follows from the Schur orthogonality relations (row
and column). Finally, if $G$ is non-abelian, then there is at least one
$C\in\operatorname{Conj}(G)$ and $\phi\in\operatorname{Irr}(G)$ such that
$|C|^{2}|\phi(C)|^{2}>|C||\phi(C)|^{2}$, so that running the argument above we
see that the inequality is strict. ∎

######  Remark 2.10.

Combining Proposition 2.9 with Rider’s hard result (Theorem 1) one sees that
if $G$ is non-abelian then $\operatorname{\rm AM}_{\rm Z}(G)\geq 1+1/300$,
i.e. we get a proof of Theorem 1. However, it is important to note that this
does not tell us anything about

$\inf\\{\operatorname{ass}(G)\mathbin{\colon}\text{$G$ is finite and non-
abelian}\\}.$

Indeed, the present paper is unable to determine if this infimum is strictly
greater than $1$.

A closer inspection of the proof of Proposition 2.9 yields the inequality

$\operatorname{ass}(G)-1\geq\frac{s(G)^{2}-s(G)}{|G|}$

where $s(G)=\min\\{|C|\mathbin{\colon}C\in\operatorname{Conj}(G),|C|>1\\}$.
However, in cases where $Z(G)=\\{e\\}$, we can do much better. This will be
the topic of the next section.

##  3 Bounds on $\operatorname{ass}$ for groups with trivial centre

In this section we obtain a new lower bound for $\operatorname{ass}(G)$ when
$G$ has trivial centre. This is the key idea needed for a new proof of Theorem
1 with a much better constant than that of [2].

######  Theorem 3.1 (Lower bound on $\operatorname{ass}(G)$ when
$Z(G)=\\{e\\}$).

Let $G$ be a finite group with trivial centre, and let $s$ be the smallest
size of its non-trivial conjugacy classes. Then

$\operatorname{ass}(G)-1\geq\frac{s-1}{s+1}\cdot\frac{2s}{s+1}\geq\frac{4}{9}\
.$

######  Remark 3.2.

The author has not found any groups with trivial centre for which
$\operatorname{ass}(G)=13/9$. Probably one can improve on this lower bound by
investigating the examples with $s=2$ more closely.

The proof of Theorem 3.1 will occupy the rest of this section. To streamline
some of the formulas which follow, we introduce the matrix
$A:\operatorname{Irr}(G)\times\operatorname{Conj}(G)\to[0,1]$ defined by

$A_{\phi,C}=|G|^{-1}|C|\
|\phi(C)|^{2}\qquad(\phi\in\operatorname{Irr}(G),C\in\operatorname{Conj}(G)).$
(3.1)

Note that rows and columns of $A$ add up to $1$. We define

$\mu:=\max_{\phi\in\operatorname{Irr}(G)}A_{\phi,e}=\max_{\phi\in\operatorname{Irr}(G)}|G|^{-1}\phi(e)^{2}\in(0,1).$

######  Proposition 3.3.

Let $G$ be a finite group with trivial centre, and let $s=s(G)$ be as defined
in the statement of Theorem 3.1. Then

$\frac{\operatorname{ass}(G)-1}{s-1}\geq\sum_{\phi\in\operatorname{Irr}(G)}A_{\phi,e}(1-A_{\phi,e})\geq\max(2\mu,1)(1-\mu).$
(3.2)

###### Proof.

We have

$\displaystyle\operatorname{ass}(G)$
$\displaystyle=\sum_{\phi\in\operatorname{Irr}(G)}\sum_{C\in\operatorname{Conj}(G)}|C|A_{\phi,C}A_{\phi,e}$
$\displaystyle\geq\sum_{\phi\in\operatorname{Irr}(G)}\left(A_{\phi,e}^{2}+\sum_{C\in\operatorname{Conj}(G)\setminus\\{e\\}}sA_{\phi,C}A_{\phi,e}\right)$
(definition of $s$)
$\displaystyle=(1-s)\sum_{\phi\in\operatorname{Irr}(G)}A_{\phi,e}^{2}+s\sum_{\phi\in\operatorname{Irr}(G)}\sum_{C\in\operatorname{Conj}(G)}A_{\phi,C}A_{\phi,e}$
$\displaystyle=(1-s)\sum_{\phi\in\operatorname{Irr}(G)}A_{\phi,e}^{2}+s\sum_{\phi\in\operatorname{Irr}(G)}A_{\phi,e}$
(rows of $A$ sum to $1$)
$\displaystyle=(s-1)\sum_{\phi\in\operatorname{Irr}(G)}A_{\phi,e}\left(1-A_{\phi,e}\right)+\sum_{\phi\in\operatorname{Irr}(G)}A_{\phi,e}\,.$

Since columns of $A$ sum to $1$, we obtain the first inequality in (3.2).

To get the second inequality: fix $\psi\in\operatorname{Irr}(G)$ with
$A_{\psi,e}=\mu=\max_{\phi\in\operatorname{Irr}(G)}A_{\phi,e}$. For each
$\phi\in\operatorname{Irr}(G)\setminus\\{\psi\\}$ we have $1-A_{\phi,e}\geq
1-\mu$; also, since columns of $A$ sum to $1$,

$1-A_{\phi,e}=\sum_{\gamma\in\operatorname{Irr}(G)\setminus\\{\phi\\}}A_{\gamma,e}\geq
A_{\psi,e}=\mu\,.$

Hence

$\displaystyle\sum_{\phi\in\operatorname{Irr}(G)}A_{\phi,e}\left(1-A_{\phi,e}\right)$
$\displaystyle=\mu(1-\mu)+\sum_{\phi\in\operatorname{Irr}(G)\setminus\\{\psi\\}}A_{\phi,e}\left(1-A_{\phi,e}\right)$
$\displaystyle\geq\mu(1-\mu)+\sum_{\phi\in\operatorname{Irr}(G)\setminus\\{\psi\\}}A_{\phi,e}\max(\mu,1-\mu)$
$\displaystyle=\mu(1-\mu)+\max(\mu,1-\mu)(1-\mu)$ (rows of $A$ sum to $1$)
$\displaystyle=(1-\mu)\max(2\mu,1),$

as required. ∎

######  Example 3.4.

Let $q$ be a prime power $\geq 3$ and consider
$G=\operatorname{Aff}({\mathbf{F}}_{q})$, the affine group of the finite field
with $q$ elements. The character theory of this group is well known, and gives
$\mu=|G|^{-1}(q-1)^{2}=1-q^{-1}$. For this group $s=q-1$, so that

$\operatorname{ass}(\operatorname{Aff}({\mathbf{F}}_{q}))-1\geq(s-1)(1-\mu)\max(2\mu,1)=(q-2)\cdot\frac{1}{q}\cdot
2\left(1-\frac{1}{q}\right);$ (3.3)

and the right-hand side of (3.3), being a monotone increasing function of $q$,
is minimized when $q=3$ (with minimum value $4/3$). In fact an explicit but
tedious calculation shows that
$\operatorname{ass}(\operatorname{Aff}({\mathbf{F}}_{q}))=3-4q^{-1}$, so that
our lower bound is a slight underestimate.

Example 3.4 shows that even within the class of groups with trivial centre,
$1-\mu$ can be arbitrarily small. However, note that in this example, when
$1-\mu$ is small $s$ is large. The following result shows that this happens
more generally.

######  Lemma 3.5.

Let $G$ be a finite group. Then $1\leq(1+s)(1-\mu)$.

###### Proof.

Choose $\psi$ with $A_{\psi,e}=\mu$ and choose $C$ with $|C|=s$. Then
$A_{\psi,C}\leq 1-\mu$ (since rows of $A$ sum to $1$), while

$A_{\phi,C}=\frac{s}{|G|}|\phi(C)|^{2}\leq\frac{s}{|G|}\phi(e)^{2}=sA_{\phi,e}\quad\text{for
all $\phi\in\operatorname{Irr}(G)$.}$

Since the columns of $A$ sum to $1$, this gives

$1=A_{\psi,C}+\sum_{\phi\neq\psi}A_{\phi,C}\leq(1-\mu)+s\sum_{\phi\neq\psi}A_{\phi,e}=(1-\mu)+s(1-\mu)$

as required. ∎

######  Remark 3.6.

  1. (i)

The idea behind the proof of Lemma 3.5 comes from considering cases such as
Example 3.4, where the ratio $\mu$ is large in the sense of being close to
$1$. Picking $\phi\in\operatorname{Irr}(G)$ with $A_{\phi,e}=\mu$ and
$C\in\operatorname{Conj}(G)$ with $|C|=s$, one reasons as follows:

     * •

since the column labelled by $e$ adds up to $1$, $A_{\phi,e}$ must be small
for every $\phi\in\operatorname{Irr}(G)\setminus\\{\psi\\}$;

     * •

since the row labelled by $\psi$ adds up to $1$, $A_{\psi,C}$ has to be small;

     * •

in turn, this forces $A_{\phi,C}$ to be large for every
$\phi\in\operatorname{Irr}(G)\setminus\\{\psi\\}$.

  2. (ii)

There are groups for which equality is achieved in Lemma 3.5. For instance, if
$G$ is the dihedral group of order $6$, we have $s=2$, $\mu=2^{2}/6=2/3$,
$(s+1)(1-\mu)=1$.

  3. (iii)

Lemma 3.5 implies that if a finite group has an irreducible character with
“relatively high degree”, not only must the group have trivial centre, but all
conjugacy classes other than the identity have to be “quite large”. Such
observations are probably not new, but we are unaware of any references in the
literature which provide such inequalities.

###### Proof of Theorem 3.1.

Let $h:[0,1]\to[0,\infty)$ be the function $h(t)=(1-t)\max(2t,1)$. This is a
continuous, decreasing function. By Proposition 3.3 and Lemma 3.5,

$\operatorname{ass}(A)-1\geq(s-1)h(\mu)\geq(s-1)h\left(\frac{s}{s+1}\right)\
.$

Now since $s/(s+1)\geq 2/3>1/2$,

$(s-1)h\left(\frac{s}{s+1}\right)=\frac{s-1}{s+1}\cdot\frac{2s}{s+1}\,;$

and the right hand side is minimized when we take $s-2$, with value $4/9$. ∎

######  Remark 3.7.

We could have tried the same approach for a non-abelian finite group $G$ with
non-trivial centre. The proof of Proposition 3.3 goes through much as
before444We need to know that $|\phi(x)|=\phi(e)$ for all $x\in Z(G)$ and all
$\phi\in\operatorname{Irr}(G)$. This is an easy consequence of Schur’s lemma,
see e.g. [6, Lemma 2.25]., but this time we merely obtain

$\operatorname{ass}(G)-1\geq(s-1)\sum_{\phi\in\operatorname{Irr}(G)}\frac{\phi(e)^{2}}{|G|}\left(1-\frac{|Z(G)|\phi(e)^{2}}{|G|}\right).$

Now if we define $\mu=\max_{\phi\in\operatorname{Irr}(G)}|G|^{-1}\phi(e)^{2}$
as before, then $|Z(G)|\mu\leq 1$, but this time we can have equality, which
means that some of the terms in the sum above will be zero. It is not clear
how to obtain good lower bounds on the remaining contributions to the sum.

##  4 Bounds on $\operatorname{\rm AM}_{\rm Z}$ for just non-abelian groups

###  4.1 JNA groups: definitions and two results

Throughout, we denote the derived subgroup (or commutator subgroup) of $G$ by
$[G,G]$ rather than $G^{\prime}$.

######  Definition 4.1.

We say that a quotient of a group $G$ is proper if it is not equal to $G$
itself. A group $G$ is just non-abelian (JNA for short) if it is non-abelian
yet all its proper quotients are abelian.

######  Remark 4.2.

Since every homomorphism from $G$ to an abelian group factors through the
quotient homomorphism $G\to G/[G,G]$, a group $G$ is JNA if and only if all
its non-trivial normal subgroups contain $[G,G]$. In alternative terminology:
if $G$ is JNA then $[G,G]$ is the minimal non-trivial normal subgroup of $G$.

Recall that by Corollary 2.4(ii), if $H$ is a quotient of $G$ then
$\operatorname{\rm AM}_{\rm Z}(G)\geq\operatorname{\rm AM}_{\rm Z}(H)$. Now if
$G$ is a finite, non-abelian group, there exists a quotient of $G$ which is
JNA. (The proof is a routine induction which we omit.) Hence

$\inf\\{\operatorname{\rm AM}_{\rm Z}(G)\mathbin{\colon}\text{$G$ is finite
and non-abelian}\\}=\inf\\{\operatorname{\rm AM}_{\rm
Z}(G)\mathbin{\colon}\text{$G$ is finite and JNA}\\}.$

Therefore Theorem 1.2 will follow from the following special case.

######  Theorem 4.3.

Let $G$ be a finite JNA group. Then $\operatorname{\rm AM}_{\rm Z}(G)\geq
7/4$.

Our proof of Theorem 4.3 works by looking at various cases. The main point is
that if $G$ is finite JNA and has non-trivial centre, then it can be handled
by the main result of [1]. For this we use the following lemma, which is a
simpler (but weaker) version of [6, Lemma 12.3].

######  Lemma 4.4.

Let $G$ be a finite JNA group with non-trivial centre. Then every non-linear
$\phi\in\operatorname{Irr}(G)$ either has degree $1$ (i.e. is linear) or has
degree $|G:Z(G)|^{1/2}$. Moreover, there is a prime $p$ such that $|G|$ is a
power of $p$; $|[G,G]|=p$; and every conjugacy class has size $1$ or size $p$.

We will also need the following result, which the author was unable to find
stated explicitly in the literature.

######  Theorem 4.5 (Folklore?).

Let $G$ be a finite JNA group which has trivial centre and has a conjugacy
class of size $2$. Then $G$ is isomorphic to the dihedral group of order $2p$,
for some odd prime $p$.

To keep this article as self-contained as possible, the Appendix includes
elementary proofs of Lemma 4.4 and Theorem 4.5. In the rest of this subsection
we offer some remarks on just non-abelian groups to provide extra context for
the reader, and to acknowledge the sources for the ideas behind Lemma 4.4 and
Theorem 4.5. The reader who just wants to get on with the proof of Theorem 4.3
can skip to Section 4.2.

######  Remark 4.6 (JN2 groups and JM groups).

If $G$ is a JNA group with non-trivial centre, then
$\\{e\\}\subsetneq[G,G]\subseteq Z(G)$, so that $G$ is $2$-step nilpotent.
Finite nilpotent JNA groups were classified by M. F. Newman [9], who
christened them finite JN2 groups; notable examples are the extra-special
$p$-groups for any given prime $p$. His earlier paper [8] gives a
classification of finite solvable JNA groups with trivial centre, calling them
finite JM groups; examples include the affine groups of finite fields (cf.
Example 3.4). In general, every finite JM group is necessarily of the form
${\mathbf{F}}\rtimes{\bf D}$ for some finite field ${\mathbf{F}}$ and some
subgroup ${\bf D}\leq{\mathbf{F}}^{\times}$. This is a necessary condition and
not a sufficient one: see [8, Theorem 6.4] for the full classification.

######  Remark 4.7 (Looking for a proof of Theorem 4.5).

If we knew the hypotheses of Theorem 4.5 force $G$ to be solvable, we could
extract this result from Newman’s classification of JM groups. However, doing
so is somewhat cumbersome: one has to analyze the conjugacy classes in ${\bf
F}\rtimes{\bf D}$ to show that $|{\bf D}|=2$, explain why $|{\bf F}|$ is an
odd prime, and so on. Alternatively, once we know $G$ is solvable, we could
appeal to the results of Isaacs and Passman that were mentioned earlier; but
this seems to require some form of Frobenius’s theorem on malnormal subgroups
([6, Theorem 7.2] or [4, Chapter 7, Theorem 8.6]). In both cases, of course,
we would still need a separate argument to explain why the conditions in
Theorem 4.5 imply that the group is solvable! This is one reason why we chose
to give a self-contained proof in the Appendix.

######  Remark 4.8 (Groups with two character degrees).

The present paper also owes an indirect debt to work of Isaacs and Passman on
“groups with two character degrees”. Rather than try to pick out those parts
of the original papers which are relevant to the present article, the reader
can find a good exposition in Chapter 12 of [6]. In particular, the dichotomy
theorem contained in [6, Lemma 12.3] implies that every finite JM group is
Frobenius with abelian complement and kernel, so that its ZL-amenability
constant can be calculated explicitly using the methods of [1, §3.2]. In fact,
the idea of looking at ZL-amenability constants of JNA groups arose from
reading about this work in [6, Chapter 12], while seeking to extend the
results of [1].

###  4.2 The ZL-amenability constant in the JNA cases

To fully exploit Lemma 4.4 and Theorem 4.5, we will apply the main theorem of
[1], stated below for clarity.

######  Theorem 4.9 ([1, Theorem 2.4]).

Let $m$ be an integer greater than or equal to $2$, and suppose $G$ is a
finite non-abelian group such that $\phi(e)\in\\{1,m\\}$ for all
$\phi\in\operatorname{Irr}(G)$. Then

$\operatorname{\rm AM}_{\rm
Z}(G)-1=2(m^{2}-1)\sum_{C\in\operatorname{Conj}(G)}\frac{|C|}{|G|}\left(1-\frac{|C|}{|[G,G]|}\right).$

The following calculation was done in [1, Example 3.13] for extraspecial
$p$-groups, but works in greater generality. For convenience we give the
proof.

######  Proposition 4.10.

Let $G$ be a finite JNA group with non-trivial centre. Then there is a prime
$p$ and an integer $k\geq 1$ such that

$\operatorname{\rm AM}_{\rm
Z}(G)-1=2\left(1-\frac{1}{p^{2k}}\right)\left(1-\frac{1}{p}\right).$ (4.1)

In particular, $\operatorname{\rm AM}_{\rm Z}(G)\geq 7/4$.

###### Proof.

Let $p$ be the prime from Lemma 4.4 and let $k$ satisfy
$|G:Z(G)|^{1/2}=p^{k}$. By the lemma every $\phi\in\operatorname{Irr}(G)$ has
degree $1$ or $p^{k}$, so $k$ is a positive integer (strictly positive since
$G$ is non-abelian). Moreover, Lemma 4.4 also implies

$\frac{|C|}{|[G,G]|}=\begin{cases}p^{-1}&\quad\text{if $C\in Z(G)$}\\\
1&\quad\text{if $C\in\operatorname{Conj}(G)\setminus Z(G)$.}\end{cases}$

Applying Theorem 4.9 yields

$\displaystyle\operatorname{\rm AM}_{\rm Z}(G)-1$
$\displaystyle=2(p^{2k}-1)\sum_{C\in\operatorname{Conj}(G)}\frac{|C|}{|G|}\left(1-\frac{|C|}{|[G,G]|}\right)$
$\displaystyle=2(p^{2k}-1)|Z(G)|\cdot\frac{1}{|G|}\left(1-\frac{1}{p}\right)$
$\displaystyle=2\frac{(p^{2k}-1)}{|G:Z(G)|}\left(1-\frac{1}{p}\right)$
$\displaystyle=2\left(1-\frac{1}{p^{2k}}\right)\left(1-\frac{1}{p}\right).$

To finish: note that the right-hand side of Equation (4.1) is an increasing
function of $k$ and of $p$. Taking $k=1$ and $p=2$ gives the minimum value
$3/4$, as required. ∎

######  Example 4.11 (JNA groups achieving the minimal ZL-amenability
constant).

The calculations in [1, Example 3.13] show that the only extraspecial
$p$-groups with ZL-amenability constant equal to $7/4$ are the dihedral and
quaternion groups of order $8$. However, for any $n\in\mathbb{N}$ the group
denoted in [9] by $M(2^{n})$ is a JNA group of order $2^{n+2}$ whose centre
has order $2^{n}$. The proof of Proposition 4.10 then yields
$\operatorname{\rm AM}_{\rm Z}(M(2^{n}))=7/4$.

We now turn to JNA groups with trivial centre. If $G$ is such a group and it
has a conjugacy class of size $2$, then (Theorem 4.5) $G$ is isomorphic to the
dihedral group $C_{p}\rtimes C_{2}$ for some odd prime $p$. As shown in [1,
Section 3.1], Theorem 4.9 applies to $C_{p}\rtimes C_{2}$ and straightforward
calculations yield

$\operatorname{\rm AM}_{\rm Z}(C_{p}\rtimes
C_{2})=1+3\left(1-\frac{1}{p}\right)^{2}\geq
1+3\left(1-\frac{1}{3}\right)^{2}=\frac{7}{3}.$ (4.2)

It remains to deal with JNA groups where every conjugacy class, except that of
$e$, has size $\geq 3$. (Recall that this class includes all finite simple
non-abelian groups!) Here, it is no longer clear that we can exploit the
results of [1], but instead we can turn to the results of Section 3.

######  Proposition 4.12.

Let $G$ be a finite group with at least two elements and trivial centre.
Suppose that $G$ has no conjugacy classes of size $2$. Then $\operatorname{\rm
AM}_{\rm Z}(G)\geq\operatorname{ass}(G)\geq 7/4$.

###### Proof.

Let
$s=\min\\{|C|\mathbin{\colon}C\in\operatorname{Conj}(G)\setminus\\{e\\}\\}$.
The conditions put on our group ensure that $s\geq 3$. Then, by Theorem 3.1,

$\displaystyle\operatorname{ass}(G)-1$
$\displaystyle\geq\frac{s-1}{s+1}\frac{2s}{s+1}$
$\displaystyle=2\left(1-\frac{2}{s+1}\right)\left(1-\frac{1}{s+1}\right)$
$\displaystyle\geq 2\left(1-\frac{2}{4}\right)\left(1-\frac{1}{4}\right)$
$\displaystyle=\frac{3}{4}\,,$

as claimed. ∎

###### Proof of Theorem 4.3.

Combine Proposition 4.10, Proposition 4.12 and the inequality (4.2). ∎

######  Remark 4.13.

When $G$ has trivial centre and has a conjugacy class of size $2$, the direct
application of Theorem 3.1 merely gives $\operatorname{\rm AM}_{\rm
Z}(G)\geq\operatorname{ass}(G)\geq 13/9$, which is not good enough. This is
why we had to deal with such cases separately (and also use the JNA condition,
which is not used in Proposition 4.12).

##  5 A lower bound for perfect groups

Since every non-abelian simple group is JNA, and since we proved Theorem 1.2
by passing from the given non-abelian finite group to some JNA quotient, it is
natural to wonder what can be said about the ZL-amenability constant of a
finite simple group. In this section we obtain a lower bound that is far
better than what we obtain from Theorem 3.1, although it is probably far short
of the true infimum. In fact, our bound holds for all finite perfect groups,
and our approach does not require any structure theory.

######  Proposition 5.1 (An alternative bound for $\operatorname{ass}$).

Let $G$ be a finite group, and let
$m=\min\\{\phi(e)\mathbin{\colon}\phi\in\operatorname{Irr}(G)\text{ and
}\phi(e)>1\\}$. Then

$\operatorname{ass}(G)-1\geq(m^{2}-1)\sum_{C\in\operatorname{Conj}(G)}\frac{|C|}{|G|}\left(1-\frac{|C|}{|[G,G]|}\right).$

Equality holds if every non-linear irreducible character of $G$ has degree
$m$.

###### Proof.

This is like the proof of [1, Theorem 2.4], only simpler; in some sense it is
dual to the argument used in the proof of Proposition 3.3.

As in Equation 3.1, let $A_{\phi,C}:=|G|^{-1}|C||\phi(C)|^{2}$. Let
$\varepsilon$ denote the trivial character of $G$, sometimes called the
augmentation character, which takes the value $1$ on every conjugacy class.
Then

$\operatorname{ass}(G)=\sum_{C\in\operatorname{Conj}(G)}A_{\varepsilon,C}\sum_{\phi\in\operatorname{Irr}(G)}\phi(e)^{2}A_{\phi,C}\,.$

Let
${\mathcal{L}}=\\{\phi\in\operatorname{Irr}(G)\mathbin{\colon}\phi(e)=1\\}$.
Then for each $C\in\operatorname{Conj}(G)$,

$\displaystyle\sum_{\phi\in\operatorname{Irr}(G)}\phi(e)^{2}A_{\phi,C}$
$\displaystyle\geq\sum_{\phi\in{\mathcal{L}}}A_{\phi,C}+m^{2}\sum_{\phi\in\operatorname{Irr}(G)\setminus{\mathcal{L}}}A_{\phi,C}$
(definition of $m$)
$\displaystyle=(1-m^{2})\sum_{\phi\in{\mathcal{L}}}A_{\phi,C}+m^{2}\sum_{\phi\in\operatorname{Irr}(G)}A_{\phi,C}$
$\displaystyle=(1-m^{2})\sum_{\phi\in{\mathcal{L}}}A_{\phi,C}+m^{2}$ (columns
of $A$ sum to $1$)
$\displaystyle=(1-m^{2})|{\mathcal{L}}|A_{\varepsilon,C}+m^{2}$ (definition of
${\mathcal{L}}$).

Multiplying this inequality by $A_{\varepsilon,C}$ and summing over all $C$,

$\operatorname{ass}(G)\geq\sum_{C\in\operatorname{Conj}(G)}A_{\varepsilon,C}\;+\;(m^{2}-1)\sum_{C\in\operatorname{Conj}(G)}A_{\varepsilon,C}(1-|{\mathcal{L}}|A_{\varepsilon,C}).$

Since rows of $A$ sum to $1$, and $|{\mathcal{L}}|$ equals the order of the
quotient group $G/[G,G]$, the result follows. ∎

The following result is surely folklore, but we include a proof since it is
straightforward.

######  Lemma 5.2.

Let $G$ be a finite group. Suppose it has a conjugacy class $C$ of size
$|G|/2$. Then $K=G\setminus C$ is an index $2$ subgroup of $G$. In particular,
$G$ is not perfect.

###### Proof.

We divide the proof into several steps.

#### Step 1.

Let $t\in C$. Then its centralizer $Z_{G}(t)$ has order $|G|/|C|=2$. Since
$\langle t\rangle\subseteq Z_{G}(T)$, $t$ is an involution and
$Z_{G}(t)=\\{1,t\\}$.

#### Step 2.

If $x\in G$ is an involution, and $y\in C$ and $xy$ is an involution, then
$x=e$ or $x=y$.

[Proof: we have $xy=(xy)^{-1}=y^{-1}x^{-1}=yx$, so $x\in Z_{G}(y)$; now use
Step 1.]

#### Step 3.

Let $x,y\in C$. Then $xy\in K$.

[Proof: if $x=y$ then $xy=e$; while if $x$ and $y$ are distinct elements of
$C$, then by Step 2, $xy$ cannot be an involution, so in particular $xy\notin
C$.]

#### Step 4.

Let $t\in C$. Then the functions $L_{t},R_{t}:G\to G$ defined by $L_{t}(x)=tx$
and $R_{t}(x)=xt$ are involutive bijections $C\leftrightarrow K$.

[Proof: by Step 3, $L_{t}(C)\subseteq K$; and since $t$ is an involution,
$L_{t}$ is injective. So $|L_{t}(C)|\leq K$. But $|K|=|G\setminus C|=|C|$,
hence $L_{t}:C\to K$ is surjective. The argument for $R_{t}$ is similar.]

#### Step 5.

$K$ is closed under multiplication.

[Proof: let $a,b\in K$. Pick $t\in C$; by Step 4, $at,tb\in C$. Since $t$ is
an involution, $ab=(at)(tb)\in C\cdot C\subseteq K$ by Step 3.]

#### Final step.

Since $C$ is the conjugacy class of an involution, it is closed under
inversion. Hence $K=G\setminus C$ is also closed under inversion. Combined
with Step 5, this shows $K$ is a subgroup of $G$; and since $K$ is the union
of conjugacy classes, it is normal. Finally, since $G/K$ is abelian of order
$2$, $G$ is not perfect. ∎

######  Remark 5.3.

In Step 1 of the proof we showed, in effect, that $C$ consists of self-
centralizing involutions. So if $t\in C$ then $\langle t\rangle$ is malnormal
in $G$, hence by Frobenius’s theorem ([6, Theorem 7.2] or [4, Chapter 7,
Theorem 8.6]) it admits a complement, and Lemma 5.2 follows. However,
Frobenius’s theorem remains quite a deep result and seems excessive in this
instance.

######  Theorem 5.4.

Let $G$ be a finite perfect group. Then $\operatorname{ass}(G)\geq 3$.

###### Proof.

If $C\in\operatorname{Conj}(G)$ then $|C|\leq|G|/3$ (since $|C|$ is a proper
factor of $|G|$, and it cannot be $|G|/2$ by Lemma 5.2). Applying Proposition
5.1 and using $[G,G]=G$ yields

$\operatorname{ass}(G)-1\geq(m^{2}-1)\sum_{C\in\operatorname{Conj}(G)}\frac{|C|}{|G|}\left(1-\frac{1}{3}\right)=\frac{2(m^{2}-1)}{3}\,.$

Since $m\geq 2$ the result follows. ∎

######  Remark 5.5.

If $G$ is simple one can improve this lower bound. A finite simple group has
no irreducible $2$-dimensional representations over $\mathbb{C}$ (see e.g. [7,
Corollary 22.13]); hence $m\geq 3$ when $m$ is as defined in Proposition 5.1.
Running the argument in the proof of Theorem 5.4 then yields
$\operatorname{ass}(G)\geq 19/3$.

Moreover: since quotients of perfect groups are perfect, every finite perfect
group $G$ has a quotient group $K$ which is non-abelian and simple. It follows
that if $G$ is a finite perfect group,

$\operatorname{\rm AM}_{\rm Z}(G)\geq\operatorname{\rm AM}_{\rm
Z}(K)\geq\operatorname{ass}(K)\geq\frac{19}{3}\,.$

While this argument does not imply $\operatorname{ass}(G)\geq 19/3$ for all
finite perfect groups, we know of no finite perfect group whose associated
minorant is less than $19/3$.

##  6 Questions for future investigation

Our first two questions concern the associated minorant
$\operatorname{ass}(G)$. Although it is technically easier to deal with than
$\operatorname{\rm AM}_{\rm Z}(G)$ for particular groups, we do not know if it
has the same good hereditary properties.

#### Question 0.

Is there a finite group $G$ and $N\lhd G$ such that
$\operatorname{ass}(G)<\operatorname{ass}(G/N)$?

If the answer is negative, then we can obtain lower bounds on
$\operatorname{ass}(G)$ by passing to JNA quotients, and hence the next
question would have a positive answer.

#### Question 1.

Is $\inf\\{\operatorname{ass}(G)\colon\text{$G$ non-abelian and finite}\\}$
strictly greater than $1$?

To motivate the next two questions, we present a small consequence of our
earlier work on the JN2 cases.

######  Theorem 6.1.

Let $G$ be a finite nilpotent group. If $\operatorname{\rm AM}_{\rm
Z}(G)<59/27$ then $G$ is the product of a $2$-group with an abelian group.

###### Proof.

Since $G$ is nilpotent it factorizes as a direct product of finite $p$-groups,
say $G=H_{1}\times\dots\times H_{m}$. (More precisely, it is the direct
product of its Sylow $p$-subgroups; see e.g. [3, Section 9.8, Theorem 5].)

Suppose at least one of these groups, say $H_{1}$, is a non-abelian $p$-group
where $p\geq 3$. Then $H_{k}$ admits a JNA quotient $K$, and $K$ must also be
a $p$-group; the proof of Proposition 4.10 then implies that

$\operatorname{\rm AM}_{\rm Z}(K)-1\geq 2(1-p^{-2})(1-p^{-1})\geq
2\left(1-\frac{1}{9}\right)\left(1-\frac{1}{3}\right)=\frac{32}{27}\,.$

But since $\operatorname{\rm AM}_{\rm Z}(G)\geq\operatorname{\rm AM}_{\rm
Z}(H_{1})\geq\operatorname{\rm AM}_{\rm Z}(K)$, this contradicts the
assumption on $\operatorname{\rm AM}_{\rm Z}(G)$. Thus each of the groups
$H_{1},\dots,H_{m}$ is either abelian or a $2$-group, and the result follows.
∎

#### Question 2.

Does Theorem 6.1 remain true if one drops the word “nilpotent”?

Note that a positive answer to Question 6 would yield a positive answer to the
next question.

#### Question 3.

Is $\inf\\{\operatorname{\rm AM}_{\rm Z}(G)\colon\text{$G$ a finite, non-
nilpotent group}\\}$ strictly greater than $7/4$? Can we at least show that a
group with ZL-amenability constant $7/4$ has to be nilpotent?

######  Remark 6.2.

An explicit calculation shows that $\operatorname{ass}(C_{3}\rtimes
C_{2})=5/3$. Therefore, with our current methods we have no hope of proving
results of the form “if $G$ has trivial centre then $\operatorname{\rm
AM}_{\rm Z}(G)\geq 2$”, because without the JNA condition the best we can do
for groups with trivial centre is to use the lower bound provided by
$\operatorname{ass}(G)$.

Moving away from questions of lower bounds, we could look at the structure of
the set

$\mathcal{ZL}=\\{\operatorname{\rm AM}_{\rm Z}(G)\colon\text{$G$ a finite
group}\\}.$

of possible values for ZL-amenability constants.

#### Question 4.

Is $\mathcal{ZL}$ closed? If not, does it contain any non-isolated points?

The results summarized in [1, Section 3.4] show that there exist strictly
increasing sequences in $\mathcal{ZL}$ which converge. The same results
suggest that these limiting values are limit points of $\mathcal{ZL}$, and
hence that $\mathcal{ZL}$ is not closed, but they are not conclusive.

#### Question 5.

Does $\mathcal{ZL}$ contain any strictly decreasing sequences? That is, does
there exist a sequence $(G_{n})$ of finite groups such that $\operatorname{\rm
AM}_{\rm Z}(G_{n+1})<\operatorname{\rm AM}_{\rm Z}(G_{n})$ for all $n$?

Our final question connects the study of $\operatorname{\rm AM}_{\rm Z}(G)$
back to the world of infinite-dimensional Banach algebras.

#### Question 6.

Consider an inverse system of finite groups

$\dots G_{3}\to G_{2}\to G_{1}\to G_{0}$

in which all connecting maps are surjective group homomorphisms. If
$\sup_{n}\operatorname{\rm AM}_{\rm Z}(G_{n})<\infty$, is the inverse limit
$\varprojlim_{n}G_{n}$ always virtually abelian?

If the answer is positive, then as noted in [2], we could prove the following:
if $G$ is a profinite group which is not virtually abelian, then the centre of
$L^{1}(G)$ is non-amenable.

### Acknowledgements

The main ideas needed for the proof of Theorem 1.2 were originally worked out
in 2013, while the author worked at the University of Saskatchewan. He thanks
M. Alaghmandan and E. Samei for their interest and encouragement.

Other important parts, such as an early version of Theorem 4.5, were obtained
while attending the Conference in Abstract Harmonic Analysis held in Granada,
May 2013. The author thanks the conference organizers for an engaging
conference and a stimulating atmosphere. During this time the author was
supported by NSERC Discovery Grant 402153-2011.

The results of Section 5 were obtained during the author’s present position at
Lancaster University, which is also where this paper took shape as various
proofs were streamlined, rearranged and improved. Thanks are due to the
referee for an attentive reading and constructive feedback, in particular for
drawing his attention to the reference cited in Remark 5.5.

##  Appendix A The proofs of Lemma 4.4 and Theorem 4.5

These proofs are included for the convenience of readers who are more familiar
with harmonic analysis and Banach algebras than finite group theory. To
clarify the proof of Lemma 4.4, it is helpful to divide the statement of the
lemma into smaller pieces.

#### Restatement of Lemma 4.4.

Let $G$ be a finite JNA group which has non-trivial centre. Then there is a
prime $p$ such that:

  1. (i)

$|G|$ is a power of $p$;

  2. (ii)

$|[G,G]|=p$;

  3. (iii)

every conjugacy class has size $1$ or size $p$;

  4. (iv)

every non-linear $\phi\in\operatorname{Irr}(G)$ has degree $|G:Z(G)|^{1/2}$.

#### Proof of (i).

Since $Z(G)$ is non-trivial, $G/Z(G)$ is abelian. So $G$ is $2$-step
nilpotent. Hence there exist subgroups $H_{1},\dots,H_{s}$ which each have
prime power order, such that $G\cong H_{1}\times\dots\times H_{s}$. (More
precisely, $G$ factorizes as the direct product of Sylow subgroups, see e.g.
[3, Section 9.8, Theorem 5].) At least one of these subgroups, say $H_{1}$,
must be non-abelian; but then since $G$ quotients onto $H_{1}$, the JNA
condition forces $G=H_{1}$.

#### Proof of (ii).

By (i), $G$ has order $p^{N}$ for some prime $p$ and some $N\geq 2$. So
$|Z(G)|=p^{m}$ for some $1\leq m\leq N-1$. By the JNA condition $[G,G]$ is
contained in every non-trivial subgroup of $Z(G)$. Hence $[G,G]$ is an abelian
$p$-group with no non-trivial proper subgroups, so it must be cyclic of order
$p$.

#### Proof of (iii).

Let $C\in\operatorname{Conj}(G)\setminus Z(G)$. Since $|C|$ divides $|G|$ it
is a power of $p$. On the other hand, there is an injection $C\to[G,G]$ (fix
$x\in C$ and consider the map $C\to[G,G]$, $y\mapsto yx^{-1}$). By part (ii),
$|[G,G]|=p$, and so $|C|=1$ or $|C|=p$.

#### Proof of (iv).

Let ${\mathcal{L}}$ denote the set of linear characters on $G$, and let
$C\in\operatorname{Conj}(G)\setminus Z(G)$. By Schur orthogonality (column
sums)

$\frac{|G|}{|C|}-\sum_{\phi\in\operatorname{Irr}(G)\setminus{\mathcal{L}}}|\phi(C)|^{2}=\sum_{\psi\in{\mathcal{L}}}|\psi(C)|^{2}=|{\mathcal{L}}|\,.$

On the other hand: since any finite abelian group has the same cardinality as
its dual group,

$|{\mathcal{L}}|=|(G/[G,G])^{\wedge}|=|G/[G,G]|=\frac{|G|}{|[G,G]|}.$

By (ii) and (iii), $|C|=p=|[G,G]|$. Combining this with the previous two
identities forces

$\sum_{\phi\in\operatorname{Irr}(G)\setminus{\mathcal{L}}}|\phi(C)|^{2}=0,$

which in turn forces $\phi(C)=0$ for all
$\phi\in\operatorname{Irr}(G)\setminus{\mathcal{L}}$.

Now let $\phi\in\operatorname{Irr}(G)\setminus{\mathcal{L}}$. By the previous
paragraph, $\phi(C)=0$ for all $C\in\operatorname{Conj}(G)\setminus Z(G)$.
Hence, by Schur orthogonality (row sums),

$|G|=\sum_{x\in Z(G)}|\phi(x)|^{2}=|Z(G)|\phi(e)^{2}\,.$

Rearranging yields the desired identity, and the proof of the lemma is
complete. $\Box$

Now we turn to Theorem 4.5, which we paraphrase as follows.

#### Theorem 4.5, paraphrased.

Let $G$ be a finite JNA group which has trivial centre and has a conjugacy
class of size $2$. Then $G$ contains an involution $t$ and an element $r$ of
odd prime order $p$, such that $|G|=2p$ and $rt=tr^{-1}$.

Originally the present author re-invented a proof of this result, but the
argument which follows is streamlined slightly using some ideas from Newman’s
paper [8]. To illustrate where we use the various hypotheses on $G$, the proof
is broken up into several lemmas.

######  Lemma A.1.

Let $G$ be a finite group with a conjugacy class $\\{a,\overline{a}\\}$ of
size $2$. Then for each $g\in G$, conjugation by $g$ either fixes both $a$ and
$\overline{a}$ or it swaps them. In particular, $a$ and $\overline{a}$
commute.

###### Proof.

When a group acts on a set of size two, each element either acts trivially or
swaps the two points. This gives us the first claim in the lemma. The second
claim follows by considering the conjugation action of $a$ on
$\\{a,\overline{a}\\}$. ∎

######  Lemma A.2.

Let $G$ be a finite JNA group with a conjugacy class $\\{a,\overline{a}\\}$ of
size $2$. Put $r=\overline{a}a^{-1}\neq\\{e\\}$. Then $[G,G]=\langle
r\rangle$, and the order of $r$ is prime. Moreover, $r$ has order $2$ if and
only if it is central.

###### Proof.

Let $g\in G$. By Lemma A.1, either $grg^{-1}=r$ or
$grg^{-1}=a(\overline{a})^{-1}=(\overline{a})^{-1}a^{-1}=r^{-1}$. Thus the
conjugation map $x\mapsto gxg^{-1}$ sends $r$ to $r^{\pm 1}$, and since
$r^{-1}\in\langle r\rangle$ this shows $\langle r\rangle\lhd G$. So by the JNA
condition $[G,G]\subseteq\langle r\rangle$. On the other hand, let $g\in G$ be
such that $gag^{-1}=\overline{a}$; then $r=gag^{-1}a^{-1}\in[G,G]$. Thus
$[G,G]=\langle r\rangle$.

Let $p$ be a prime which divides the order of $r$. Then $\langle
r^{p}\rangle<\langle r\rangle=[G,G]$. Moreover $\langle r^{p}\rangle\lhd G$,
since the only conjugates of $r^{p}$ are $r^{p}$ itself and $r^{-p}$, which
both belong to $\langle r^{p}\rangle$. So by the JNA condition, $r^{p}=e$.
Thus $r$ has order $p$.

Finally: since the conjugacy class of $r$ is $\\{r^{\pm 1}\\}$, $r$ is central
if and only if $r=r^{-1}$, i.e. if and only if $r^{2}=e$. ∎

Choose some $t\in G$ such that $\overline{a}=tat^{-1}$. Let
$H=Z_{G}(a)=Z_{G}(\overline{a})$: this has index $2$ in $G$, so is normal, and
$G=H\sqcup Ht$. Note that $\langle r\rangle\subseteq H$, by another
application of Lemma A.1.

######  Lemma A.3.

Let $G$ be as above, and suppose in addition that $Z(G)=\\{e\\}$. Then
$H=[G,G]$.

###### Proof.

By our previous remarks, $[G,G]=\langle r\rangle\subseteq H$. So it suffices
to prove $|H|\leq|[G,G]|$, and we can do this by exhibiting an injection
$H\to[G,G]$. (Note that this injection will not be a left inverse to the
inclusion of sets $[G,G]\hookrightarrow H$.)

Consider the function $\theta:H\to[G,G]$ defined by
$\theta(h)=tht^{-1}h^{-1}$. Since $[G,G]\subseteq Z_{G}(H)$,

$t(hk)t^{-1}=(tht^{-1})(tkt^{-1})=\theta(h)h\theta(k)k=\theta(h)\theta(k)hk\quad\text{for
all $h,k\in H$.}$

Hence $\theta(hk)=\theta(h)\theta(k)$ for all $h,k\in H$, i.e. $\theta$ is a
homomorphism. Now consider the subgroup

$K=\ker(\theta)=\\{h\in H\colon ht=th\\}=H\cap Z_{G}(t).$

Since $K$ is the kernel of a homomorphism $H\to G$, $K\unlhd H$, and so $K\lhd
G$ since $G=H\cup Ht$ and $K\subseteq Z_{G}(t)$. Suppose $K\neq\\{e\\}$. Then
$K\supseteq[G,G]$, by the JNA condition, and so $r\in K$ by Lemma A.2. But
this would imply $rt=tr$, and since $r\in Z_{G}(H)$ it would follow from
$G=H\cup Ht$ that $r\in Z(G)$, contradicting the assumption that
$Z(G)=\\{e\\}$. Thus $K=\\{e\\}$, so $\theta$ is injective, as required. ∎

###### Finishing the proof of Theorem 4.5.

Let $G$ be finite JNA with trivial centre and a conjugacy class of size $2$.
Combining the previous lemmas, we obtain $r,t\in G$ such that:
$trt^{-1}=r^{-1}$; $r$ has odd prime order; and $G=\langle r,t\rangle$ (this
follows from Lemma A.3 and the fact $H=\langle r\rangle$). It only remains to
show that $t^{2}=e$. Since $trt^{-1}=r^{-1}$, $t^{2}$ commutes with $r$, and
hence with everything in $G=\langle r,t\rangle$. So $t^{2}\in Z(G)=\\{e\\}$,
and the proof of Theorem 4.5 is complete. ∎

## References

  * [1] M. Alaghmandan, Y. Choi, and E. Samei, ZL-amenability constants of finite groups with two character degrees, Canad. Math. Bull., 57 (2014), pp. 449–462.
  * [2] A. Azimifard, E. Samei, and N. Spronk, Amenability properties of the centres of group algebras, J. Funct. Anal., 256 (2009), pp. 1544–1564.
  * [3] P. M. Cohn, Algebra. Vol. 1, John Wiley & Sons, Ltd., Chichester, second ed., 1982.
  * [4] , Algebra. Vol. 2, John Wiley & Sons, Ltd., Chichester, second ed., 1989.
  * [5] , Algebra. Vol. 3, John Wiley & Sons, Ltd., Chichester, second ed., 1991.
  * [6] I. M. Isaacs, Character theory of finite groups, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1976.  Pure and Applied Mathematics, No. 69.
  * [7] G. James and M. Liebeck, Representations and characters of groups, Cambridge University Press, New York, second ed., 2001.
  * [8] M. F. Newman, On a class of metabelian groups, Proc. London Math. Soc. (3), 10 (1960), pp. 354–364.
  * [9] , On a class of nilpotent groups, Proc. London Math. Soc. (3), 10 (1960), pp. 365–375.
  * [10] D. Rider, Central idempotent measures on compact groups, Trans. Amer. Math. Soc., 186 (1973), pp. 459–479.

Yemon Choi

Department of Mathematics and Statistics
Fylde College, Lancaster University
Lancaster, United Kingdom LA1 4YF

Email<EMAIL_ADDRESS>


# LyricJam: A system for generating lyrics for live instrumental music

Olga Vechtomova, Gaurav Sahu, Dhruv Kumar
University of Waterloo
{ovechtom, gsahu<EMAIL_ADDRESS>
###### Abstract

> We describe a real-time system that receives a live audio stream from a jam
> session and generates lyric lines that are congruent with the live music
> being played. Two novel approaches are proposed to align the learned latent
> spaces of audio and text representations that allow the system to generate
> novel lyric lines matching live instrumental music. One approach is based on
> adversarial alignment of latent representations of audio and lyrics, while
> the other approach learns to transfer the topology from the music latent
> space to the lyric latent space. A user study with music artists using the
> system showed that the system was useful not only in lyric composition, but
> also encouraged the artists to improvise and find new musical expressions.
> Another user study demonstrated that users preferred the lines generated
> using the proposed methods to the lines generated by a baseline model.

## Introduction

Music artists have different approaches to writing song lyrics. Some write
lyrics first, and then compose music, others start with the music and let the
mood and the emotions that emerge from the music guide their choice of lyric
themes, words and phrases. The latter type of songwriting is often
experimental in nature, and is commonly found in genres such as rock, jazz and
electronic music, whereby one or more artists jam and in the process converge
on the desired musical expressions and sounds. In this work, we describe a
system we designed to support this type of songwriting process, where artists
play musical instruments, while the system listens to the audio stream and
generates lyric lines that are congruent with the music being played in real
time. The artists see the lines as they are generated in real time,
potentially entering a feedback loop, where the lines suggest emotions and
themes that artists can use to guide their musical expressions and
instrumentation as they jam. The generated lines are not intended to be the
complete song lyrics, instead acting as snippets of ideas and expressions
inspiring creativity. After the jam session is over, the lines are saved, and
the artist can use them as inspiration to write the lyrics for their song.

The goal of our system is to generate lyric lines that are congruent with
emotions and moods evoked by live instrumental music. Past research in
musicology has found a correlation between some aspects of music and emotions.
In one large-scale study, researchers found evidence that certain harmonies
have strong associations with specific emotions, for example, the diminished
seventh is associated with the feeling of despair (?). In another study,
researchers found correlation between major chords and lyric words with
positive valence (?). In addition to chords and harmonies, various sound
textures and effects also contribute to the emotional intent of the music.
Therefore, in this work we use raw audio input that captures all aspects of
music. Our model learns associations between raw audio clips and their
corresponding lyrical content from the training dataset, and at inference
time, relies on these learned associations to generate new lines based on the
live audio stream.

Our approach is based on training a variational autoencoder for learning the
representations of mel-spectrograms of audio clips (spec-VAE), and a
conditional variational autoencoder for learning the representations of lyric
lines (text-CVAE). The advantage of using variational autoencoders as
generative models is their ability to learn a continuous latent space that can
then be sampled to generate novel lines, which is an important requirement for
creative applications.

At inference time, the model must be able to generate new lyric lines given an
audio clip being recorded from live jam session. In order to do that we need a
way to align the latent representations learned by the spec-VAE and the latent
representations learned by the text-VAE. We propose two novel approaches to
achieve this alignment.

The first approach (Figure 1) is based on training a separate Generative
Adversarial Network (GAN) model that takes the spectrogram embedding from
spec-VAE, and learns to predict the lyric line embedding in the text-CVAE. The
GAN-predicted embedding is then sent to the text-CVAE decoder to generate
text.

Figure 1: GAN-based alignment of music and lyrics representations (Approach
1).

The second approach (Figure 2) learns to transfer the latent space topology of
the spec-VAE to the text-CVAE latent space. To achieve this we use the learned
posteriors from the spec-VAE as priors in text-CVAE during its training. The
text-CVAE learns to encode lyric lines corresponding to a given audio clip in
the region of the latent space corresponding to that clip. Also, since similar
sounding audio clips are encoded in neighboring regions, the text-CVAE
correspondingly learns to encode lines for these clips in neighboring regions.
For example, ambient music clips would be encoded in proximal regions of spec-
VAE, and so would be the lines corresponding to these clips. The intuition is
that lines corresponding to similar sounding audio clips (e.g. ambient) would
have similar emotional intent, as opposed to, for example, aggressive sounding
music. At inference time, when an artist plays an ambient music piece, the
system would feed its spectrogram to the spec-VAE encoder to get the
parameters of its posterior distribution. Since the spec-VAE posterior
distributions are also prior distributions in the text-CVAE, the system
samples latent codes from the corresponding prior of the text-CVAE, generating
new lines reflecting ambient music.

To summarize the contributions of this paper are as follows:

  1. 1.

To our knowledge this is the first real-time system that receives a live audio
stream from a jam session and generates lyric lines that are congruent with
the live music being played.

  2. 2.

We propose two novel approaches to align the latent spaces of audio and text
representations that allow the system to generate novel lyric lines matching
the live music audio clip.

  3. 3.

We discuss our findings based on observations and interviews of musicians
using the system.

In the next section, we describe the GAN-based approach of aligning the spec-
VAE and text-VAE latent embeddings. Next, we will describe our second approach
of aligning the topologies of the two VAEs. We will then follow up with the
implementation details, experiments and user studies.

## Approach 1: GAN-based alignment of music and lyrics representations

The model consists of three neural network models that are trained
consecutively in three stages (Figure 1).

Figure 2: Latent space topology transfer from spec-VAE to text-CVAE (Approach
2).

### Training stage 1: spec-VAE

In this stage, we train the spectrogram variational autoencoder (VAE) model to
learn the latent representations of audio clips.

First we convert the raw waveform audio files into MEL-spectrogram images
using the same method as used in ? (?). These spectrograms are then used as
input for the spec-VAE.

The variational autoencoder (?) is a stochastic neural generative model that
consists of an encoder-decoder architecture. The encoder transforms the input
image $\bm{x}$ into the approximate posterior distribution $q_{\phi}(z|x)$
learned by optimizing parameters $\phi$ of the encoder. The decoder
reconstructs $\bm{x}$ from the latent variable $\bm{z}$, sampled from
$q_{\phi}(z|x)$. In our implementation, we use convolutional layers as the
encoder and a deconvolutional layers as the decoder. Standard normal
distribution was used as the prior distribution $p(z)$. The VAE is trained on
the loss function that combines the reconstruction loss (Mean Squared Error)
and KL divergence loss that regularizes the latent space by pulling the
posterior distribution to be close to the prior distribution.

### Training stage 2: text-CVAE

Unlike the vanilla VAE used for encoding spectrograms, we use conditional VAE
(CVAE) for encoding lyrics.

The CVAE learns a posterior distribution that is conditioned not only on the
input data, but also on a class c: $q_{\phi}(z|x,c)$. Here, we define the
class as the spectrogram corresponding to a given line. Every conditional
posterior distribution is pulled towards the same prior, here the standard
normal distribution.

During training, every input data point consists of a lyric line and its
corresponding spectrogram. We first pass the spectrogram through the spec-VAE
encoder to get the parameters of the posterior distribution (a vector of means
and a vector of standard deviations). We then sample from this posterior to
get a vector $z^{(s)}$ that is then concatenated with the input of the encoder
and the decoder during training. The reason why we used sampling as opposed to
the mean $z^{(s)}$ vector is to induce the text-VAE model to learn
conditioning on continuous data, as opposed to discrete classes. This prepares
it to better handle conditioning on unseen new spectrograms at inference.

Both the encoder and the decoder in the text-CVAE are Long Short Memory
Networks (LSTMs). The sampled $z^{(s)}$ is concatenated with the word
embedding input to every step of the encoder and the decoder.

#### Training stage 3: GAN

In this phase, we train a generative adversarial network (GAN), which learns
to align audio and text latent codes. The GAN architecture has a generator
$\mathit{G}$ and a discriminator $\mathit{D}$. For training the GAN on a given
spectrogram-text pair $\\{x^{(s)},x^{(t)}\\}$111Superscripts (s) and (t) in
our notation refer to spectrogram and text, respectively, we follow these
steps:

  1. 1.

First, we pass the spectrogram $x^{(s)}$ through spec-VAE to obtain
$z^{(s)}=\mu^{(s)}+\tau(\epsilon\cdot\sigma^{(s)})$, the latent code sampled
from the posterior distribution. Here, $\mu^{(s)}$ and $\sigma^{(s)}$ denote
the mean and standard deviation predicted by the spec-VAE,
$\epsilon\sim\mathcal{N}(0,1)$ is a random normal noise, and $\tau$ is the
sampling temperature. Simultaneously, we obtain
$z^{(t)}=\mu^{(t)}+\tau(\epsilon\cdot\sigma^{(t)})$ by passing the
corresponding lyric line $x^{(t)}$ through the text-VAE.

  2. 2.

After obtaining $z^{(s)}$ and $z^{(t)}$, we proceed with the GAN training. We
pass $z^{(s)}$ through the generator network, which outputs a predicted text
latent code $\hat{z}^{(t)}$.

  3. 3.

We then pass $\hat{z}=[\hat{z}^{(t)};z^{(s)}]$ and $z=[z^{(t)};z^{(s)}]$
through the discriminator network, where ; denotes the concatenation
operation. We treat $\hat{z}$ as the negative sample, and ${z}$ as the
positive sample. The discriminator $\mathit{D}$ then tries to distinguish
between the two types of inputs. This adversarial training regime, in turn,
incentivizes $\mathit{G}$ to match $\hat{z}^{(t)}$ as closely as possible to
$z^{(t)}$.

The adversarial loss is formulated as follows:

$\begin{split}\min_{G}\max_{D}V(D,G)&=\mathbb{E}_{x\sim\mathcal{D}_{\text{train}}}[\textrm{log}D({z})+\textrm{log}(1-D(\hat{z})]\end{split}$
(1)

where $\mathcal{D}_{\text{train}}$ is the training data, and each sample
$x=\\{x^{(s)},x^{(t)}\\}$. We also add an auxiliary MSE loss to the objective
function as it is found to stabilize GAN training (?). The overall loss for
the GAN is:

$J_{GAN}=\min_{G}\max_{D}V(D,G)+\lambda_{MSE}||\hat{z}^{(t)}-{z}^{(t)}||$ (2)

At inference time, the encoder of the text-CVAE is no longer needed. A
spectrogram is input to the spec-CVAE encoder to obtain the spectrogram latent
code $z^{(s)}$, which is then fed to the generator of the GAN, which generates
the lyric latent code $z^{(t)}$. The inference process is also stochastic, as
$z^{(s)}$ is sampled from the posterior distribution for $s$. Sampling allows
us to generate diverse lines for the same spectrogram. The GAN-predicted text
latent code is then concatenated with the spectrogram latent code and input to
the text-CVAE decoder, which generates a lyric line.

## Approach 2: Latent space topology transfer from spec-VAE to text-CVAE

The intuition for this approach is to induce the text-CVAE to learn the same
latent space topology as the spec-VAE. This would mean that data points that
are close in the spec-VAE latent space are expected to be close in the text-
VAE latent space. More concretely, if two audio clips are encoded in the
neighboring regions of the spec-VAE latent space, their corresponding lyric
lines should be encoded in the neighboring regions in the text-CVAE latent
space.

The training is a two-stage process (Figure 2), where the first stage, spec-
VAE, is the same as in the GAN-based approach. For the second stage, we train
a different formulation of text-CVAE. Instead of using one prior (standard
normal) to regularize every posterior distribution, we use the posterior of
the spec-VAE as the prior for any given data point. More formally, let the
spectrogram be $x^{(s)}$ and its corresponding lyric line be $x^{(t)}$. The
posterior distribution for the spectrogram in the spec-VAE is
$q^{(s)}_{\phi}(z^{(s)}|x^{(s)})$, and the posterior distribution for the
lyric line in the text-CVAE is $q^{(t)}_{\phi}(z^{(t)}|x^{(t)},z^{(s)})$. The
KL term of the text-CVAE loss is computed between the posterior for the lyric
line and the prior which is set to be the posterior of its corresponding
spectrogram in spec-VAE:

$\begin{split}\operatorname{KL}^{(t)}=\operatorname{KL}(q^{(t)}_{\phi}(z^{(t)}|x^{(t)},z^{(s)})||q^{(s)}_{\phi}(z^{(s)}|x^{(s)}))\end{split}$
(3)

The cross-entropy reconstruction loss for text-CVAE is:

$\begin{split}J_{\text{rec}}(\phi,\theta,{z}^{(s)},x^{(t)})=-\sum_{i=1}^{n}\log
p({x}^{(t)}|\bm{z}^{(t)},\\\
{z}^{(s)},{x_{1}}^{(t)}\cdots{x_{i-1}}^{(t)})\end{split}$ (4)

The final text-CVAE loss is the combination of the reconstruction loss and the
KL term:

$\begin{split}J_{\text{CVAE}}(\phi,\theta,{z}^{(s)},x^{(t)})=J_{\text{rec}}+\lambda\operatorname{KL}^{(t)}\end{split}$
(5)

To avoid the KL term collapse problem, we used two techniques first proposed
by ? (?): KL term annealing by gradually increasing $\lambda$ and word dropout
from the decoder input.

## Ranking of generated lines with BERT

At inference time, the model generates a batch of 100 lines conditioned on a
short audio clip sampled every 10 seconds from the user’s audio source. Since
the system shows one or two lines to the user for each clip, and since not all
generated lines are equally fluent or interesting, we need a method to rank
them so that the system shows a small number of high-quality lines to the
user. For this purpose, we used a pre-trained BERT model (?), which we fine-
tuned on our custom dataset. The dataset consisted of 3600 high-quality and
low-quality lines that were manually selected by one of the authors from a
large number of lines output by a VAE trained on the same dataset as used in
our experiments. The criteria used for determining quality of lines included
the following: originality, creativity, poetic quality and language fluency.
While BERT is trained as a binary classifier, we use logits from the final
layer for ranking the lines.

## Live lyric generation system

We developed a React/NodeJS web application222The application can be accessed
at: https://lyricjam.ai that listens to the user’s audio source, which can be
either a microphone or line level input from the user’s audio-to-digital
converter, receiving input from the user’s instruments, such as guitar or
keyboard. The application samples clips from the audio stream every 10 seconds
and saves them as uncompressed PCM WAV files at 44.1kHz sampling rate. On the
server, WAV files are converted to MEL-spectrograms and sent to the spec-VAE
to obtain the latent code, which is then used in lyric generation by the text-
CVAE. The lines generated by text-CVAE are passed through BERT classifier for
ranking. The top-ranked lines are then displayed to the user on their screen.
The lines slowly float for a few seconds, gradually fading away as newly
generated lines appear. The user can view the history of all generated lines
with time stamps during the jam session in a collapsible side drawer (Figure
3).

Figure 3: LyricJam screenshot.

## Evaluation

We trained the system on the aligned lyric-music dataset created by ? (?). The
dataset consists of 18,000 WAV audio clips of original songs and their
corresponding lyrics by seven music artists.

The goals of our evaluation are two-fold:

  1. 1.

Determine whether the two proposed methods generate lines that match the mood
created by the music more accurately.

  2. 2.

Understand how the application can benefit music artists as they compose new
songs by playing musical instruments.

To answer these evaluation questions, we designed two user studies, described
in detail in the subsequent sections.

### Study 1

For this study, we developed an interactive web application, which plays
instrumental songs and every 10 seconds shows three lines to the users in
random order, two generated by the proposed models and one by the baseline
model. The user was asked to click on the line that best matches the mood of
the music currently being played. Participants for this study did not have to
be songwriters or music artists, but they were required to have a general
appreciation of music of any genre.

For the baseline, we used CVAE with standard normal prior. This is a strong
baseline, which generates lines conditioned on the music being played.
Comparison to this baseline specifically lets us evaluate the effect of the
two proposed methods: GAN-based alignment (GAN-CVAE) and topology transfer
(CVAE-spec). At inference time, the lines generated by each system were ranked
by BERT, and the top-ranked line from each system was selected for the
experiment.

We selected five instrumental songs by two artists for this study with a
variety of tempos and instrumentation, and evoking different moods. The lines
for each 10-second song clip were generated in advance, so that each user was
shown exactly the same three lines per clip. The order of line presentation
was random each time. Examples of generated lyrics can be seen in Figure 4.

Song description | CVAE | CVAE-spec | GAN-CVAE
---|---|---|---
Synthesizer, bass guitar, banjo, piano / forlorn, tense, slow | 162 | 152 | 195
Electric guitar, synthesizer / harsh, aggressive, high-tempo | 61 | 94 | 89
Modular synthesizer, keyboard / melancholy, low-tempo | 64 | 48 | 76
Piano / sombre, tense, low-tempo | 43 | 91 | 66
Modular synthesizer, keyboard / mellow, uplifting, ambient | 53 | 72 | 53
Total | 383 | 457 | 479
Table 1: Number of lines per system and instrumental song selected by the
users in Study 1

In total, 15 users participated in this study. As can be seen from Table 1,
users preferred the lines from two experimental systems over the baseline
regardless of the type of song played. The differences are statistically
significant (ANOVA, p$<$0.01).

Figure 4: Examples of generated lines for different instrumental music clips.

### Study 2

For this study we asked five musicians to use the system as they played
musical instruments. The participants were free to choose any musical
instruments they liked and play any genre of music. They were asked to observe
the generated lines as they played music, and answer the following questions
after the completion of the study:

  1. 1.

What were your thoughts after using the application?

  2. 2.

What did you like about it?

  3. 3.

What would you change about this system?

  4. 4.

Would you consider using this system again?

  5. 5.

Did the lines reflect the mood that your music conveys to you?

The instruments and genres played by the participants are as follows:

  * •

Participant A: electric guitar / rock

  * •

Participant B: keyboard / rock

  * •

Participant C: guitar and electric drums / blues, rockabilly, punk, folk,
rock.

  * •

Participant D: piano / new age, contemporary classical, Bollywood, pop;

  * •

Participant E: electric guitar / metal

All users found the experience enjoyable and would use the system again. Three
users mentioned that they liked its simplicity and minimalist design.
Interestingly, while our initial goal was to design a system to assist
musicians in lyric writing, two unanticipated new uses emerged from this user
study: (a) the system encouraged improvisation and trying new sounds and (b)
the system was perceived as a helpful jam partner. This suggests that the
system could be useful not only for lyric composition, but also for music
playing in general. Below, we elaborate in more detail on the themes that
emerged from the feedback.

#### Improvisation and experimenting with new sounds.

Users commented that the system encouraged them to try new sounds and
experiment with new musical expressions. User A: “[The system] inspires me to
try things out to see what this sound does to the machine”. User C: “there
were a few times where the system started suggesting lyrics that got me
thinking about structuring chords a bit differently and taking a song in
different directions than originally intended”.

User A also commented that the system encouraged them to be more experimental
in their music: “You become much less critical of making unmusical sounds and
you can explore the musical palette of the instrument to see what lines it
provokes.” This user also observed a game-like element of the system: “I am
trying to make it say something crazy by playing crazy sounds” and “I am
playing my guitar out of tune and try to find music in it as the machine is
trying to find meaning in it.” The user also noted that the generated lines
sometimes encouraged them to stay in a certain music mode, such as minor
chords, when they liked the lines that it generated. This suggests that the
system could act as a responsive listener for musicians, to help them bring
their music to the desired emotional expression. For example, if the musician
intends their music to be sombre, and the emotions of the lines match the
mood, they would know they are on track, and if the emotions in the lines
start shifting as they try new expressions, it could be a signal that their
music may not have the intended emotional effect.

#### System as a jam partner.

One user commented that the system acts like a jamming partner as they play:
“I don’t feel like I am alone. I feel like jamming with someone.” “It’s like
having an uncritical jam partner. It always responds to whatever you are
trying.” “The machine is trying to work with you even if you make a mistake.”
This suggests that a musician can benefit from using the system and be
encouraged to play their instrument, even if they are not actively trying to
write lyrics.

#### System as a source of lyrics.

Users commented that the system would be useful as a source of new lyrical
ideas. User C: “I could see it being very useful if I’m having a bit of
writer’s block and just noodling around to get a starting point.” User B: “I
like to jam with my band. We jam for 30 minutes or so. It would be nice to
have it running. It would do the work for you as you are jamming. If at the
end of it you have a pile of lyrics that would be great. That could be the
starting material we could use for a song.” The user emphasized that for them
it would be important to see the lines after the jam session as they may miss
some lines while playing their instruments: “I look at my hands when playing,
occasionally glancing at the screen to see the lyrics. I would like to see the
lines after the jam session.”

#### Generated lines and music mood.

Users noticed differences in lines, in particular with the changes in tempo
and instruments used. One user played on their keyboard in the piano and
Hammond B3 (organ) modes. The user played minor piano chords on the piano,
noticing that the lines were darker in theme. Switching to organ with a more
upbeat percussive sound and using major chords more often led to more upbeat
lines.

Another user noticed that the system generated interesting lines when they
played unusual musical shapes: “The system seemed to produce more interesting
lines when it was fed slightly more novel audio, e.g. a tune that had a mode
change (switching from major to minor)”.

The genre of music did not appear to have as strong effect as tempo or the
choice of instruments. One user who played blues, rock, folk and rockabilly
genres did not observe a strong effect of genre. The likely reason is that the
training dataset only contains rock music, and therefore the model has never
seen music of other genres. It would be interesting to see how the lines are
affected by genre, if the model is trained on a larger dataset, which is more
representative of different genres.

#### Themes in generated lyrics

Users noted presence of recurring themes as they played. They commented on
often seeing a subset of words in the lines shown to them. But these words
appeared to be different for each user, which suggests that the system
differentiated between their musical styles, and narrowed down on specific
themes. One user commented that when they experimented by playing songs with
overtly obvious emotional valence (sad and happy), the system generated lines
with markedly different words: “pain”, “falling”, “leaving” for a sad song,
and “time”, “dreams”, “believe” for a happy song.

Training the model on a larger dataset would likely lead to more lexical
diversity. Furthermore, the system always showed the users top two generated
lines ranked by BERT, which was fine-tuned on a small dataset as well. When we
started sampling from among the top-ranked lines, the diversity improved.

## Related Work

Automated creative text generation has long been getting attention from
researchers at the intersection of computational creativity and artificial
intelligence. Creative text generation includes tasks such as story (?; ?; ?)
and poetry (?; ?; ?) generation. Recently automated lyric generation has also
begun receiving attention from the research community.

? (?) approached lyrics generation from an information retrieval perspective.
They selected the suitable next line from a dataset containing rap lyrics to
produce a meaningful and interesting storyline. Similarly, ? (?) developed a
technique for cross-modal learning and used it to retrieve lyrics conditioned
on given audio. However, our vision with this work is not to generate complete
song lyrics but individual lines that act as snippets of ideas for the
artists.

With the recent advances in end-to-end text generation, lyrics generation is
often considered a conditional text generation task. ? (?) generated author-
stylized lyrics using audio-derived embeddings. ? (?) generate rap verses
conditioned on the content on any given text. ? (?) created a system for a
real-time human and robot rap battle. They generated lyrics conditioned on the
keywords present in the lyrics of the human rapper. In contrast, our model
conditions the lyrics on an audio stream.

While the above systems generate lyrics conditionally, there is no focus on
capturing the mood of the song while generating lyrics. This is important
since there are associations between chords used in the songs and the
sentiments expressed in the lyrics. For example, major chords are associated
with positive sentiment and minor chords with negative sentiment (?; ?). ? (?)
use a language model conditioned on an input melody for lyrics generation. ?
(?) created a co-creative song writing system and applied it in a therapeutic
context. The lyrics could either be fed by the user or generated by the system
conditioned on an audio track or input topics selected by the user. ? (?) also
developed a system to generate lyric lines conditioned on music audio. They
show that their system generates lyric lines that are consistent with the
emotional effect of a given music audio clip. In this work, we propose new
approaches that align the learned latent spaces of audio and text
representations, thus building upon and improving their approach. Moreover,
our system generates lyrics in real-time.

## Implementation Details

#### Text-CVAE.

We use the Tensorflow framework (?) to implement the text-CVAE. The encoder is
a single-layer bi-LSTM and a decoder is an LSTM. The hidden state dimensions
were set to 300 and the latent space to 128. The CVAE models in our
experiments were trained for 500 epochs.

#### Spec-VAE.

We use PyTorch (?) to implement the spec-VAE. The encoder has 4 Conv2d layers
interleaved with ReLU activation function (?). The decoder is a mirror image
of the encoder, with 4 ConvTranspose2d layers, interleaved with 3 ReLU
activations and 1 Sigmoid activation in the end. We use 128 latent dimensions
for the mean and sigma vectors. During training, we use a batch size of 32,
learning rate of 1e-4, and Adam optimizer (?). The sampling temperature is 1.0
for both training and inference.

#### GAN.

We use the AllenNLP library (?) to implement the GAN. The generator and
discriminator networks are 3-layered feed-forward neural networks, interleaved
with ReLU activation function. During training, we use a batch size of 32,
learning rate of 1e-3, and Adam optimizer for both the generator and the
discriminator. We set $\lambda_{MSE}=1.0$ to ensure diverse lyric lines. The
sampling temperature is 1.0 during both training and inference. Also, we train
the GAN alignment network for 6 epochs.

#### BERT.

We use the Transformers library (?) to fine-tune a bert-base model for
sequence classification on our custom dataset. The model is trained for 15
epochs using a learning rate warmup scheduler for the first 500 steps of
training with a weight decay of 0.01. We use a batch size of 16 for training
and 64 for inference.

## Conclusions

We developed a system that generates lyrics congruent with the emotions
conveyed in live instrumental music. Two novel approaches to align the learned
representations of music and lyrics have been proposed: GAN-CVAE, which
adversarially learns to predict the lyric representation from the music
representation, and CVAE-spec, which transfers the topology of the music
(spectrogram) latent space learned by the spectrogram VAE to the lyric latent
space learned by the text CVAE. Our user study showed that users selected the
lines generated by the proposed two methods significantly more often than the
lines generated by a baseline for different types of instrumental songs. For
another user study we recruited musicians performing live music. Their
feedback suggested that the system could be useful not only for lyric writing,
but also (a) to encourage musicians to improvise and try new musical
expressions, and (b) act as a non-critical jam partner. In future work, we
would like to generate lines that are not only congruent with the emotions of
the music, but whose meter and phonetic structure match the beat and the
composition of the song.

## Acknowledgments

This research was partially supported by the Natural Sciences and Engineering
Research Council of Canada (NSERC) under grant No. RGPIN-2019-04897 and
Compute Canada (www.computecanada.ca).

## References

  * [Abadi et al. 2016] Abadi, M.; Barham, P.; Chen, J.; Chen, Z.; Davis, A.; Dean, J.; Devin, M.; Ghemawat, S.; Irving, G.; Isard, M.; et al.  2016\.  Tensorflow: A system for large-scale machine learning.  In 12th $\\{$USENIX$\\}$ Symposium on Operating Systems Design and Implementation ($\\{$OSDI$\\}$ 16), 265–283.
  * [Bowman et al. 2016] Bowman, S. R.; Vilnis, L.; Vinyals, O.; Dai, A. M.; Józefowicz, R.; and Bengio, S.  2016\.  In Proceedings of The 20th SIGNLL Conference on Computational Natural Language Learning, 10–21.
  * [Brahman, Petrusca, and Chaturvedi 2020] Brahman, F.; Petrusca, A.; and Chaturvedi, S.  2020\.  Cue me in: Content-inducing approaches to interactive story generation.  In Proceedings of the 1st Conference of the Asia-Pacific Chapter of the Association for Computational Linguistics and the 10th International Joint Conference on Natural Language Processing, 588–597.  Suzhou, China: Association for Computational Linguistics.
  * [Cheatley et al. 2020] Cheatley, L.; Ackerman, M.; Pease, A.; and Moncur, W.  2020\.  Co-creative songwriting for bereavement support.  In Eleventh International Conference on Computational Creativity: ICCC’20, 33–41.  Association for Computational Creativity.
  * [Devlin et al. 2019] Devlin, J.; Chang, M.-W.; Lee, K.; and Toutanova, K.  2019\.  BERT: Pre-training of deep bidirectional transformers for language understanding.  In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), 4171–4186.  Minneapolis, Minnesota: Association for Computational Linguistics.
  * [Gardner et al. 2018] Gardner, M.; Grus, J.; Neumann, M.; Tafjord, O.; Dasigi, P.; Liu, N.; Peters, M.; Schmitz, M.; and Zettlemoyer, L.  2018\.  Allennlp: A deep semantic natural language processing platform.  arXiv preprint arXiv:1803.07640.
  * [Khan et al. 2020] Khan, K.; Sahu, G.; Balasubramanian, V.; Mou, L.; and Vechtomova, O.  2020\.  Adversarial learning on the latent space for diverse dialog generation.  In Proceedings of the 28th International Conference on Computational Linguistics, 5026–5034.  Barcelona, Spain (Online): International Committee on Computational Linguistics.
  * [Kingma and Ba 2015] Kingma, D. P., and Ba, J.  2015\.  Adam: A method for stochastic optimization.  In Bengio, Y., and LeCun, Y., eds., 3rd International Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings.
  * [Kingma and Welling 2014] Kingma, D. P., and Welling, M.  2014\.  Auto-Encoding Variational Bayes.  In 2nd International Conference on Learning Representations, ICLR 2014, Banff, AB, Canada, April 14-16, 2014, Conference Track Proceedings.
  * [Kolchinsky et al. 2017] Kolchinsky, A.; Dhande, N.; Park, K.; and Ahn, Y.-Y.  2017\.  The minor fall, the major lift: inferring emotional valence of musical chords through lyrics.  Royal Society open science 4(11):170952.
  * [Malmi et al. 2016] Malmi, E.; Takala, P.; Toivonen, H.; Raiko, T.; and Gionis, A.  2016\.  Dopelearning: A computational approach to rap lyrics generation.  In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 195–204.
  * [Manurung 2004] Manurung, H.  2004\.  An evolutionary algorithm approach to poetry generation.
  * [Nair and Hinton 2010] Nair, V., and Hinton, G. E.  2010\.  Rectified linear units improve restricted boltzmann machines.  In Icml.
  * [Nikolov et al. 2020] Nikolov, N. I.; Malmi, E.; Northcutt, C.; and Parisi, L.  2020\.  Rapformer: Conditional rap lyrics generation with denoising autoencoders.  In Proceedings of the 13th International Conference on Natural Language Generation, 360–373.
  * [Paszke et al. 2019] Paszke, A.; Gross, S.; Massa, F.; Lerer, A.; Bradbury, J.; Chanan, G.; Killeen, T.; Lin, Z.; Gimelshein, N.; Antiga, L.; Desmaison, A.; Kopf, A.; Yang, E.; DeVito, Z.; Raison, M.; Tejani, A.; Chilamkurthy, S.; Steiner, B.; Fang, L.; Bai, J.; and Chintala, S.  2019\.  Pytorch: An imperative style, high-performance deep learning library.  In Advances in Neural Information Processing Systems 32. Curran Associates, Inc.  8024–8035.
  * [Riedl and Young 2010] Riedl, M. O., and Young, R. M.  2010\.  Narrative planning: Balancing plot and character.  Journal of Artificial Intelligence Research 39:217–268.
  * [Roemmele 2016] Roemmele, M.  2016\.  Writing stories with help from recurrent neural networks.  In Proceedings of the AAAI Conference on Artificial Intelligence, volume 30.
  * [Savery, Zahray, and Weinberg 2020] Savery, R.; Zahray, L.; and Weinberg, G.  2020\.  Shimon the rapper: A real-time system for human-robot interactive rap battles.  In Eleventh International Conference on Computational Creativity: ICCC’20.
  * [Vechtomova et al. 2018] Vechtomova, O.; Bahuleyan, H.; Ghabussi, A.; and John, V.  2018\.  Generating lyrics with variational autoencoder and multi-modal artist embeddings.  arXiv preprint arXiv:1812.08318.
  * [Vechtomova, Sahu, and Kumar 2020] Vechtomova, O.; Sahu, G.; and Kumar, D.  2020\.  Generation of lyrics lines conditioned on music audio clips.  In Proceedings of the 1st Workshop on NLP for Music and Audio (NLP4MusA), 33–37.  Online: Association for Computational Linguistics.
  * [Watanabe et al. 2018] Watanabe, K.; Matsubayashi, Y.; Fukayama, S.; Goto, M.; Inui, K.; and Nakano, T.  2018\.  A melody-conditioned lyrics language model.  In Proceedings of the 2018 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long Papers), 163–172.
  * [Willimek and Willimek 2014] Willimek, D., and Willimek, B.  2014\.  Why do minor chords sound sad? the theory of musical equilibration and the emotions of chords.  Journal of Psychology & Psychotherapy 4:139.
  * [Wolf et al. 2020] Wolf, T.; Debut, L.; Sanh, V.; Chaumond, J.; Delangue, C.; Moi, A.; Cistac, P.; Rault, T.; Louf, R.; Funtowicz, M.; Davison, J.; Shleifer, S.; von Platen, P.; Ma, C.; Jernite, Y.; Plu, J.; Xu, C.; Scao, T. L.; Gugger, S.; Drame, M.; Lhoest, Q.; and Rush, A. M.  2020\.  Transformers: State-of-the-art natural language processing.  In Proceedings of the 2020 Conference on Empirical Methods in Natural Language Processing: System Demonstrations, 38–45.  Online: Association for Computational Linguistics.
  * [Yang et al. 2017] Yang, X.; Lin, X.; Suo, S.; and Li, M.  2017\.  Generating thematic chinese poetry using conditional variational autoencoders with hybrid decoders.  In Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence, IJCAI-18.
  * [Yu et al. 2019] Yu, Y.; Tang, S.; Raposo, F.; and Chen, L.  2019\.  Deep cross-modal correlation learning for audio and lyrics in music retrieval.  ACM Transactions on Multimedia Computing, Communications, and Applications (TOMM) 15(1):1–16.
  * [Zhang and Lapata 2014] Zhang, X., and Lapata, M.  2014\.  Chinese poetry generation with recurrent neural networks.  In Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing (EMNLP), 670–680.