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notationNotation Non omnis moriar multaque pars mei vitabit Libitinam -
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supported by UK STFC grant ST/J00037X/2 and Leverhulme Project Grant 2022.

# John Keith Stuart McKay: 1939 - 2022

Yang-Hui He<EMAIL_ADDRESS>
> “I could never have approached this subject but for the untiring interest of
> John McKay, who has often seemed to me an emissary from some advanced
> galactic civilization, sent here to speed up our evolution.”
>

These words of the topologist Morava summarise well the awe and, to a degree,
the bemusement, which the mathematics community has felt towards John McKay.
Certainly, of all mathematicians in the recent, if not entire, history of the
subject, McKay was rare: he observed the vista of mathematics from afar and
saw profound hidden connections, some of which were even mocked at first and
seen as crazy.

I had the honour and pleasure to be John’s last close collaborator and – as
his wife Trinh quotes him – “his best friend” in his old age. For these last
10 years, I have felt I was under the wing of my childhood hero. I never had a
grandfather, both of mine long being gone before my birth. Yet, rather
serendipitously, John McKay became a grandfather figure to me and my family,
intellectually and emotionally111There were three good English friends and
colleagues: John Horton Conway (1937 – 2020), Simon Phillips Norton (1952 –
2019), and John Keith Stuart McKay (1939 – 2022). Conway and Norton will
feature prominently in Section 2. Alexander Masters wrote the story of Norton
in The Genius in my Basement [2], Siobhan Roberts wrote the story of Conway in
Genius at Play [3]. I am currently writing The Genius on Skype [4], a somewhat
part three of this trilogy. . When asked by the London Mathematical Society to
write this obituary, I felt pride, and a tremendous sense of duty, as well as
pangs of sorrow. I only hope that I can, while recounting the ensuing story,
in some small way be a Dante to his Virgil, or a Boswell to his Johnson.

Figure 1: John McKay, courtesy of his wife Trinh Vo-McKay

##  1 Biography

John Keith Stuart McKay was born on 18 of November, 1939, in Kent, on the
outskirts of London, to a comfortably-off English family. His father, Ernest
Stuart McKay, ran a successful book binding business in London (the
Bookbinders of London, est. 1880) and his mother, Norah Florence (née
Culshaw), came from a well-connected background. To his English family and
friends, McKay was known as “Keith” – the “John” persona came later to his
trans-Atlantic, and global colleagues. He and his elder sister Elizabeth, who
still lives in England, grew up in a large Victorian family home, a
tranquility shaken in their early years only by the tumults of WW2, including
the time when a German V-bomb landed not too far from their dwellings. Like
many boys of his generation from English families of some affluence, John was
sent away to boarding school. From 1950 to 1958, he attended the well-known
public school, Dulwich College222I have been invited to speak at Dulwich a few
times, both independently of and specifically about, John. Taking advantage of
the friendship I have harboured with the masters, I have on many occasions
suggested that a plaque be dedicated to McKay at his alma mater., like his
father before him.

On obtaining his A-levels, McKay went to the University of Manchester to study
mathematics, where he learnt about the then-nascent subject of computing. The
university was building the Manchester Atlas Supercomputer, which was then one
of the most powerful machines in the world. This joint interest in pure
mathematics, scientific computing, as well as mathematical data would have
foundational implications for his future research. Indeed, on finishing his
undergraduate studies in 1962, McKay stayed on until 1964 as Nuffield Research
Fellow, working with the Atlas, as one of the pioneers of the use of computers
in pure mathematics.

This theme of mathematics and computers continued in 1964 when McKay was
offered a PhD place at the University of Edinburgh under the joint supervision
of Sidney Michaelson, Scotland’s first professor of computer science, and W.
Douglas Munn, an algebraist specialising in representation theory of groups
and semi-groups. Appropriately, McKay’s thesis was entitled Computing with
Finite Groups, a great and first synthesis of both fields [5]. The
Acknowledgments of the dissertation read: “I thank my supervisors, Professor
W. D. Munn and Professor S. Michaelson, for much patience and many kindnesses
during the preparation of this thesis. My thanks go also to my wife Wendy who
has tirelessly typed and retyped the manuscript and organised me.” From this
we can deduce several facts about John. First, he was appreciative of his
mentors. Indeed, throughout his life, John was a kind and gentle man, to his
teachers, peers, and the younger generation. Second, he got married during his
PhD years. This was to Wendy Sue-A-Quan, a fellow student at Edinburgh. Third,
Wendy organised him. For those who knew McKay, the opinion was ubiquitous that
he was the archetype of the absent-minded professor, brilliant, soft-spoken,
and charmingly disorganised.

With a doctorate – the thesis was formally completed and submitted some years
later in 1971 – McKay became a research fellow by returning to the Atlas
Computer world, this time at Chilton, just outside of Oxford, in 1967. The
same year, he became a member of the London Mathematical Society. Chilton gave
McKay ample opportunity to collaborate with Oxford University, and in
particular Graham Higman, then the Waynflete Chair of pure mathematics, in the
explicit construction of some sporadic simple groups – specifically, the third
Janko group $J_{3}$ and the Held group $He$ – with the help of the computer.
One of John’s contemporaries at Chilton was John Leech, the originator of the
eponymous lattice [6], which is the unique even unimodular lattice in 24
dimensions and the best one for sphere packing therein. Leech tried in vain to
get the group theory community interested in this, despite his suspicion that
the enormous group of automorphisms of the lattice should be of universal
interest. Luckily, McKay got interested, then got John Horton Conway
interested, and this led to Conway’s discovery and construction of three new
sporadic simple groups, now called $Co_{1}$, $Co_{2}$, and $Co_{3}$. The group
$Co_{1}$, which is the quotient of the automorphism group of the Leech lattice
by its centre of size 2, turned out to play a crucial role in the construction
of the legendary Monster group, with which we will spend much time shortly.

One of John’s greatest powers was connecting people with ideas. Siobhan
Roberts, in her award-winning biography of Conway [3], describes it well in
her playfully entertaining way:

> “[Leech] dangled the problem under the nose of a few symmetry aficionados,
> Donald Coxeter among them, and he spread the word, telling his friend John
> McKay. McKay tempted Conway, and Conway eventually took the bait …McKay was
> forever peddling mathematical unions, people with people, people with
> ideas.”

After Chilton, McKay’s international life began. In 1969, John crossed the
Atlantic with wife Wendy, to be Visiting Assistant Professor at the California
Institute of Technology. There he continued to work on character tables, and
formed a life-long friendship and collaboration with Hershy Kisilevsky, who
would end up in Montreal with him some years later. This brief sojourn in
California was followed by McKay moving to Montreal in 1971 to take up a
permanent post as Associate Professor in the School of Computer Science at
McGill University. The reason for moving to Canada, as John once said to me,
was simple: he believed that the ratio between typical academic salary and
cost-of-living was at the time maximised in Canada. This could well be true.

McKay didn’t get along with McGill, perhaps the place was too much
establishment for him. His good friend and colleague, John Harnad, who visited
him till the very last days, wrote in the obituary for the Canadian Centre de
Recherches Mathématiques (CRM) [7]:

> “John McKay was a good friend, a distinguished colleague, and a very special
> person, with a unique character. He was remarkable in many ways, both as an
> imaginative, creative mathematician, well-known for his incredible knack of
> discovering unexpected relations between seemingly unrelated fields, and as
> a clear-minded, perceptive thinker, with great honesty and integrity. It
> followed from this that he could not let things that he perceived as wrong
> to just go by, without probing their causes. He always did this with self-
> effacing good humour, in the spirit of inquiry rather than judgement, but
> sometimes with more persistence than certain academic administrators could
> much appreciate …Besides his remarkable mathematical insights and
> contributions, it would be very incomplete to not say a bit more about John
> McKay’s human qualities, his very high ethical integrity, and his
> wonderfully wide scope of interests and knowledge, on historical,
> scientific, and cultural matters. In particular, he had a very deep sense of
> the rights and wrongs regarding how things are done in an academic
> institutional setting, and never ceased to question things that seemed to
> him as unjustified or inappropriately prioritised.

In 1974, John moved a few blocks away to Concordia University, where he
remained until his final days, being promoted to full professor of computer
science in 1979, and then to a joint full professor in mathematics and
computer science in 1990, and Distinguished Emeritus Professor when he retired
at the age of 70 in 2009. A peregrination that paralleled his move to
Concordia was when John and Wendy divorced in 1985, and while Wendy went on to
marry fellow mathematician Robert Moody, John married Trinh Vo in 1988. Trinh
had been by John’s side with great affection and devotion. She definitely
organised his life. For a decade after John and I became friends and
collaborators in 2010, we talked to each other almost every day on Skype [4].
This was greatly facilitated by the 5-hour time difference between Montreal
and London/Oxford, because his mornings coincided well with my afternoons.
Trinh would join us from time to time, especially when I had my two children,
whom they saw growing up over the video chats. In many ways, McKay became
grandpa John.

When it became too difficult for Trinh to take care of John after he had a
fall – I visited them in 2016 when I was giving a lecture at McGill and saw
the precariousness of their stairs – she arranged for John to be at Louis-Riel
Nursing Home in 2017, where he remained until his last day. Skype became more
difficult and Trinh tried whenever she could to get John onto an iPad so that
he could continue to chat to me and to his other friends. One rather poignant
tradition Louis-Riel has is that a short obituary be written for every person
at the time of admission. Trinh asked me to do this and so more than five
years ago I had the chance to rehearse the life story I’m writing now. That
brief homage to John was published by his alma mater in the Dulwich College
Alleyne Club Yearbook [9]. John died peacefully on 19 April 2022 with Trinh by
his side. Just a few days before, she managed to get me on the phone with him
because, as she put it, my voice “always cheered him up.”

McKay’s life-long contributions to mathematics and to bridging ideas and
people were duly celebrated and recognised when he was elected a Fellow of the
Royal Society of Canada in 2000. In 2003 he won the CRM-Fields Prize for
Mathematics, Canada’s highest honour for a mathematician. In 2007 a conference
was jointly organised by Concordia University and the Université de Montréal
to honour John and to marvel at what he brought to the world of mathematics.
It was appropriately named “Groups and Symmetries: From Neolithic Scots to
John McKay” [10].

##  2 Mathematical Miracles

> “John McKay was an exceptional mathematician. His broad mathematical
> interests and knowledge, together with his curiosity, intuition and
> insights, led to his discovery of important new connections between
> different areas of mathematics and the opening up of whole new areas of
> research.”
>

Einstein described the year 1905 as his annus mirabilis wherein he published
four papers, each of which revolutionised our understanding of the universe.
In recounting McKay’s contributions to mathematics it is also expedient to
emphasise his three miracles, which transpired over the course of a decade,
and in each case spurred whole new research areas that remain active to this
day. They are (i) Moonshine, for which he was most famous, (ii) McKay’s A-D-E
Correspondence, which he considered his favourite, and (iii) the McKay
conjecture on group representations, which was his first major observation.
Since John’s most extraordinary ability was to link unimaginably disparate
branches of mathematics, and since I deeply believe every graduate student
should know and appreciate these miracles, I shall introduce them with
sufficient generality and necessary context to at least attempt a conveyance
of their beauty.

(a)  (b)

Figure 2: (a) John showing the T-shirt to my then infant daughter Kitty, in a
Skype call in 2014; (b) A well-known photo of McKay at the black-board,
courtesy of John Harnad.

###  2.1 “This is Moonshine!”

McKay’s most famous equation is

$196884=196883+1\ .$ (1)

The Fields Institute makes T-shirts with this printed on them and indeed next
to John’s ashes in Cimetière Notre-Dame-des-Neiges is the photo of John
pointing to this equation (see Figure 2). What on earth does this equation
mean?! Let’s explain the left and right in turn.

####  2.1.1 The Modular Invariant

Consider one of the most important concepts in mathematics, the modular group
$SL(2;\mathbb{Z}):=\left\\{\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right):a,b,c,d\in\mathbb{Z};ad-bc=1\right\\}$, the group of
$2\times 2$ integer matrices with unit determinant333 Sometimes, the modular
group is defined projectively as $PSL(2;\mathbb{Z})$ where the determinant
$ad-bc=\pm 1$. Much of our discussions carry through the $\pm$ sign. . This
group is ubiquitous, manifesting in combinatorics, geometry, number theory,
and so on. For one thing, the infinite number of parallelograms formed by the
columns of its elements all have the same unit area as that of the identity
element $\mathbb{I}_{2\times 2}$, which corresponds to the unit square. For
instance, the fact that the extremely skew one whose sides are generated by
vectors $\\{489,100\\}$ and $\\{2000,409\\}$ has area 1 is highly non-trivial.

Is there a function invariant under its action? To address this, we complexify
the plane and recast the $SL(2;\mathbb{Z})$ action as the integer linear
fractional transformation $\tau\mapsto\frac{a\tau+b}{c\tau+d}$ for
$\tau\in\mathbb{C}$, and seek an analytic function invariant under this
transformation. Such a function would generalise well-known ones that are
invariants under important subgroups of transformations, such as the
trignometric function $\sin(2\pi\tau)$ being invariant under
$\tau\mapsto\tau+1$. The functions invariant up to a weight factor are the
famous modular forms. The Eisenstein series prescribes them: for
$k\in\mathbb{Z}_{>1}$, we have

$E_{2k}(\tau)=(2\zeta(2k))^{-1}\sum\limits_{(0,0)\neq(m,n)\in\mathbb{Z}^{2}}(m+n\tau)^{-2k}\quad\Rightarrow\quad
E_{2k}\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{2k}E_{2k}(\tau)\ ,$ (2)

with the weight-factor $(c\tau+d)^{2k}$. The series is absolutely convergent
for $\tau\in\mathcal{H}$, the upper-half plane, where $\Im(\tau)>0$. The
numerical factor $2\zeta(2k)$ – which involves the Riemann zeta-function
evaluated at even positive integers, and thus can be expressed in terms of
Bernoulli numbers – has deep number-theoretical meaning which will not concern
us presently.

Now, at the cost of introducing a pole, we can get rid of the weight factors,
by defining

$j(\tau):=1728\frac{E_{4}(\tau)^{3}}{E_{4}(\tau)^{3}-27E_{6}(\tau)^{2}}\quad\Rightarrow\quad
j\left(\frac{a\tau+b}{c\tau+d}\right)=j(\tau)\ .$ (3)

Incidentally, Ramanujan’s familiarity with the $1728=12^{3}$ pre-factor of
this function is surely the fundamental reason behind the Taxicab legend (the
Lambert series of $E_{4}$ having the coefficient $n^{3}$). Thus defined, it is
clear that the properties of $E_{2k}$ under modular transformations from (2)
renders $j(\tau)$ invariant under $SL(2;\mathbb{Z})$. However, it is not
referred to as a “modular form of weight zero” because it is not holomorphic:
unlike the modular forms $E_{2k}(\tau)$, $j(\tau)$ has a simple pole at
$\tau=i\infty$.

This function $j$ is usually attributed to Klein [11] and is mostly called the
Klein $j$-invariant. However, McKay had always told me that it would have been
known to at least Gauss and Eisenstein, so he insisted on it being simply
called the modular invariant function, which emphasises the fact that it is
unique, in the sense that it is the only function on the upper-half-plane
invariant under $SL(2;\mathbb{Z})$, which is holomorphic away from the simple
pole at $i\infty$ and normalised so that

$j(\exp(2\pi i/3))=0\ ,\quad j(i)=1728\ .$ (4)

Any other modular invariant (with different values at these points) is a
rational function in $j$.

There are countless beautiful properties of $j(\tau)$, ranging from geometry
(e.g., it being a detector of isomorphism classes of elliptic curves) to
number theory (e.g., its connection to class field theory), but the one most
pertinent to us is its series development. Defining $q:=\exp(2\pi i\tau)$, the
Laurent series in $q$ (and equivalently, the Fourier series in $\tau$) is

$j(q)=q^{-1}+744+196884q+21493760q^{2}+864299970q^{3}+20245856256q^{4}+\cdots$
(5)

We can see that the leading term of $q^{-1}$ signifies the simple pole at
$q=0$, or $\tau=i\infty$. All coefficients are integers, as can be seen from
the integer Fourier expansion of the Eisenstein $E_{2k}$, but the meaning of
these seemingly meaningless integers had to wait almost a century until McKay
came along.

####  2.1.2 Finite Simple Groups

Now, for something completely different. Parallel to analysis, geometry and
number theory, another great development of the 19th century was the theory of
groups, with its incipience as the tragic scribbles of Galois the night before
his fateful duel. Ever since Galois’ idea of organising the permutations
amongst the roots of polynomials – in what we now call the Galois group – the
classification of finite groups had been a major theme in abstract algebra.

First, the problem boils down to the classification of finite simple groups,
i.e., those which do not have a non-trivial proper normal subgroup. We remind
ourselves that a normal subgroup $N$ of a group $G$ is a subgroup closed under
conjugation: $gng^{-1}\in N$ for all $n\in N$ and all $g\in G$; this normality
is denoted as $N\triangleleft G$. Normality permits one to form the quotient
$G/N$. Every finite group $G$ has a composition series which in a way encodes
its structure: $\mathbb{I}=H_{0}\triangleleft
H_{1}\triangleleft\ldots\triangleleft H_{n}=G$ where each $H_{i}$ is a maximal
proper normal subgroup of $H_{i+1}$ and each quotient $H_{i+1}/H_{i}$ is
simple. The Jordan–Hölder theorem dictates that, up to permutations and
isomorphisms, the composition series is unique. The composition series for a
finite group is like prime factorisation for integers, and thus simple groups
to finite groups are like prime numbers to integers.

The classification of finite simple groups is itself a saga. Galois noticed
that the alternating group $\mathcal{A}_{n}$ on $n\geq 5$ elements was simple.
This, together with what we now call Galois Theory of field extensions, was
enough to resolve the long-standing problem of why generic polynomials of
degree 5 and above have no solutions in radicals. In this sense, the
classification problem began in 1832. It is a remarkable journey spanning more
than a century when the classification was officially completed. A significant
portion of the proof involved computer algebra, for which McKay was a pioneer.
The definitive volumes, consisting of almost ten thousand pages detailing the
steps, are still an ongoing project [12]. For those uninitiated to pore over
these tomes, there is a leisurely but technical account in [13] which is very
readable and even comes with a song at the end to help you remember the
punchline:

###### Theorem 2.1

The finite simple groups are

  1. 1.

Cyclic groups $\mathbb{Z}/p\mathbb{Z}$ of size $p$ prime;

  2. 2.

The Alternating group $\mathcal{A}_{n}$ of even permutations on $n\geq 5$
elements;

  3. 3.

The Lie groups defined over finite fields;

  4. 4.

26 Sporadic groups.

Note that I am using “size” instead of “group order” and avoided using
“degree” for the alternating groups. This is a nod to John, who always thought
the words “order” and “degree” have been far too abused in mathematics. In the
above, Types (i-iii) are somewhat ‘classical’ objects and come in infinite
families. Type (i) is clearly simple because they don’t have any non-trivial
proper subgroups. Type (ii) is Galois’ original observation. Type (iii) is
interesting in that Lie groups are usually associated with continuous
symmetries, but defined over finite fields $\mathbb{F}_{p^{r}}$ they become
discrete and finite; to Lie groups we will shortly return in the next section.
Type (iv), the sporadic groups444 There is the Tits group, which is sometimes
called the 27th sporadic, and sometimes grouped together with the Lie-type
family.  are the most mysterious: they are all outliers and have rather large
size – the smallest of these, Mathieu’s group $M_{11}$ discovered in 1861, is
of size 7920. The existence and construction of these 26 occupied a good part
of the century-long story of the classification. Aside from much active
research they continue to engender, their very being remains puzzling. It is
as if I told you that there are families of prime numbers like the Mersenne
primes or Fermat primes, but then there are 26 specific primes of considerable
size which are simply out there and are special and there are no more!

####  2.1.3 Monstrous Moonshine

The largest of the sporadics is of size

$\begin{array}[]{c}808017424794512875886459904961710757005754368000000000=\\\
2^{46}\cdot 3^{20}\cdot 5^{9}\cdot 7^{6}\cdot 11^{2}\cdot 13^{3}\cdot 17\cdot
19\cdot 23\cdot 29\cdot 31\cdot 41\cdot 47\cdot 59\cdot 71\ .\end{array}$ (6)

The enormity of this number gave rise to the name Monster. Its existence was
first predicted independently by Bernd Fischer and Robert Griess [14] in the
early 1970s. In 1982, Griess published the first construction of the Monster,
and a simplified construction was later published by Conway. John Thompson
proved uniqueness, subject to the existence of a faithful complex
representation of dimension 196883, and Simon Norton showed that such a
representation must exist, subject to the structure of the prime-element
centralisers. A complete uniqueness proof for the Monster was finally
published in 1989 by Griess, Meierfrankenfeld, and Segev [15].

This faithful complex representation of dimension $196883=47\cdot 59\cdot 71$
is fascinating. It is the smallest one. In other words, the smallest matrices
which generate the Monster – in fact only two such matrices are needed, as all
finite simple groups are generated by two matrices! – are of size
$196883\times 196883$. With today’s technology, writing down these two
matrices is not terrible, it requires about 1 Tb of disk-space. Writing down
all the elements of the Monster is impossible – it has more elements than the
number of atoms of earth. Conway summarises the discovery of the Monster well:
“There’s never been any kind of explanation of why it’s there, and it’s
obviously not there just by coincidence. It’s got too many intriguing
properties for it all to be just an accident.”

The legend goes as follows. In 1978, John Conway555 Remember that Conway and
McKay knew each other from the late 1960s when McKay dangled Leech’s lattice
to Conway. It turns out that the Monster is intimately related to the
automorphism group of the Leech lattice.  showed the number 196883 to John
McKay666At the time, this 196883-dimensional representation was still
conjectural. Based on this conjectured degree, Fischer, Livingstone, and
Thorne, in an amazing piece of work at the University of Birmingham,
constructed the entire character table of the Monster. Thus the data were
available for Thompson, Conway, and Norton to check that McKay’s ideas were
not so crazy! When McKay saw the number, his reply was Eq. (1). Conway replied
that this is ridiculous, “it’s moonshine!” Moonshine is English slang for (1)
illegal home-brew and (2) something outrageously crazy. Since both Johns were
English, they would get the reference.

Conway’s outburst is well-justified. The left hand-side of Eq. (1) lives in
the world of number theory and analytic functions; the right hand-side lives
in the world of finite groups and representation theory. The two worlds are so
distant in the landscape of mathematics that it would indeed be crazy for them
to be inter-related. To make sure this is not just a numerical coincidence,
they checked the next few (huge) coefficients

$\begin{array}[]{rcl}1&=&r_{1}\\\ 196884&=&r_{1}+r_{2}\\\
21493760&=&r_{1}+r_{2}+r_{3}\\\ 864299970&=&2r_{1}+2r_{2}+r_{3}+r_{4}\\\
20245856256&=&3r_{1}+3r_{2}+r_{3}+2r_{4}+r_{5}=2r_{1}+3r_{2}+2r_{3}+r_{4}+r_{6}\\\
\ldots\end{array}$ (7)

where on the left are the Fourier coefficients of $j(q)$ in Eq. (5) and
$r_{n=1,\ldots,194}=\\{$ 1, 196883, 21296876, 842609326, 18538750076,
19360062527, …, 258823477531055064045234375 } are the dimensions of the
irreducible representations of the Monster - there is a finite number, a total
of 194, of them because it is a finite group. Thus all the infinite number of
integer coefficients of the modular invariant are encoded as simple linear
combinations of the dimensions of the irreducible representations of the
Monster sporadic simple group. This can no longer be just a coincidence,
especially because it involves such astronomical integers.

Thompson, who got the Fields Medal for his work on the simple groups and who
discovered a sporadic that bears his name, suggested that there might exist a
graded vector-space, now called the moonshine module,
$V^{\natural}=\oplus_{n}V_{n}$ such that $j(q)-744=\sum_{n}{\rm
Tr}(\mathbb{I}|_{V_{n}})q^{n}$, because the trace of the identity $\mathbb{I}$
on a vector-space is its dimension777 The constant term 744 does not show up
in Moonshine and needed to be subtracted away. However, it is a remarkable
fact that $744/3=248$ and has to do with the fact that $j(q)^{1/3}$ encodes
the irreducible representations of the Lie group $E_{8}$ which we will
encounter shortly [21]. . This would then generalise to other elements $g$ \-
or conjugacy classes, since trace is conjugacy invariant - of the Monster as

$j_{g}(q)=\sum_{n}{\rm Tr}(g|_{V_{n}})q^{n}\ .$ (8)

This expansion is now called the McKay–Thompson series. After Conway took the
bait, he and Norton formalised his initial exclamation of incredulity [16]
into:

######  Conjecture 2.2.

(The Moonshine Conjecture) The McKay–Thompson series $j_{g}(q)$ is the unique
modular invariant of a group lying between $\Gamma_{0}(N)$ and its normaliser
$\Gamma_{0}(N)^{+}$ in the modular group $SL(2;\mathbb{Z})$.

Here, $\Gamma_{0}(N)$ is the level $N$ congruence subgroup of the modular
group consisting of the usual elements $\left(\begin{array}[]{cc}a&b\\\
c&d\end{array}\right)$, $ad-bc=1$, but with the further congruence condition
$c\bmod N\equiv 0$. When $N=1$ and $g=\mathbb{I}$, $j_{g}(q)$ is the modular
invariant $j(q)-744$ for the entire modular group, associated to McKay’s
original observation. There is, of course, a lot more to the story. The
conjecture had many more precise details, and entire books have been devoted
thereto, e.g., [17, 18, 19] contain excellent accounts888For a short,
enjoyable, but still technical, account, the reader is referred to [20], by my
former student Tatitscheff, who worked with John and me.. My intention here is
merely to entice the reader with some highlights. After immediate response
from the community [22, 23] in constructing the Monster module $V^{\natural}$,
using ideas from conformal field theory and string theory, in particular
vertex operator algebras, Borcherds proved the Moonshine Conjectures for the
Monster, for which he was awarded the Fields Medal in 1998.

This surprising connection between modularity, finite groups and lattices,
representation theory as well as theoretical physics, prompted by McKay’s
penetrating intuition, has become a field in and of itself, under the name of
Moonshine. Generalising beyond the Monster group to other sporadics, lattices
beyond the Leech lattice, and $j(q)$ to partition functions, remains an active
field of research: since the 1990s, there have been some 300 papers on this
subject on the ArXiv.

When Conway turned 80, there was a conference and volume dedicated to him
[25]. McKay and I wrote the opening chapter [109], a one-page mathematical
(half) joke entitled “Moonshine and the Meaning of Life” where we summed the
squares of the first 24 coefficients of $j(q)$, as well as for the related
Ramanujan tau-function, and reduced modulo 70 (both 24 and 70 being key to the
Leech lattice) and obtained 42. Conway was very amused. I am honoured that the
very last published work I have with John was my editing and tidying up a set
of notes of his final in-person lecture on his thoughts and new ideas about
Moonshine [114].

Let me conclude this section with the words of Norton [2]:

> “I can explain what Monstrous Moonshine is in one sentence, it is the voice
> of God.”

###  2.2 A-D-E

Our second mathematical miracle is what John always told me to be his
favourite, even more than Moonshine. Again, this is the bridging of unrelated
fields: finite groups, graphs, geometric singularities and Lie theory. It is
now called the McKay Correspondence and is widely believed to be but the tip
of the iceberg for many more deep connections in mathematics yet to be
discovered.

####  2.2.1 Lie Groups

Sophus Lie’s original ambitious programme was to classify continuous
symmetries by considering the group of transformations on the variables of a
multi-variate analytic function. This led to the concept which we now call Lie
groups, i.e., continuous groups which are also manifolds. With many standard
examples we are familiar. For example, the circle $S^{1}$ is a group by
identifying its elements as unimodular complex numbers $\exp(i\theta)$, under
usual multiplication; this is the Lie group $U(1)$. Another example is
$SU(2)$, the group of $2\times 2$ complex matrices $M$ such that
$M^{H}M=\mathbb{I}_{2\times 2}$ and $\det(M)=1$, where $H$ is the conjugate
transpose. In other words, $M$ are the special unitary $2\times 2$ matrices,
forming the Lie group which geometrically is the 3-sphere $S^{3}$. Yet another
example, of which we will shortly make use, is $SO(3)$, the group of $3\times
3$ real matrices $M$ such that $M^{T}M=\mathbb{I}_{3\times 3}$ and
$\det(M)=1$, i.e., special orthogonal matrices; this Lie group is
topologically the real projective space $\mathbb{R}\mathbb{P}^{3}$. The above-
mentioned are examples of classical Lie groups, which are more or less
familiar to the 19th century mathematician999 To reconnect with the previous
section, these continuous Lie groups, when defined not over the reals or the
complex numbers, but over finite fields, become discrete, and furnish the Lie
family of finite simple groups..

Lie’s insight was that the Lie group, being a manifold, has a tangent space at
the group identity. This tangent space is a vector space endowed with a binary
operation inherited from the group law that renders it an algebra; the algebra
is now dubbed the Lie algebra and the binary operation, the Lie bracket. This
group-algebra correspondence transforms the study of a Lie group to that of
the Lie algebra whose vector space structure renders it more manageable.

The brilliant works of Killing, Cartan, Levi et al. subsequently reduced the
classification of Lie groups to that of Lie algebras as complex vector spaces.
In analogy to the finite groups, Lie algebras can be built up from the simple
Lie algebras, which are the ones that have no nonzero proper ideals. The Levi
structure theorem dictates that any Lie algebra, when quotiented by its
radical (largest solvable ideal), is a direct sum of simple ones. Thus again,
simple Lie algebras are the “primes” and the complete list is, rather
prosaically dubbed, as follows:

######  Theorem 2.3.

The simple Lie algebras over $\mathbb{C}$ are

  1. 1.

Type $A_{n}$: $\mathfrak{sl}_{n+1}(\mathbb{C})$;

  2. 2.

Type $B_{n}$: $\mathfrak{so}_{2n+1}(\mathbb{C})$;

  3. 3.

Type $C_{n}$: $\mathfrak{sp}_{2n}(\mathbb{C})$;

  4. 4.

Type $D_{n}$: $\mathfrak{so}_{2n}(\mathbb{C})$;

  5. 5.

The Exceptionals: $E_{6}$, $E_{7}$, $E_{8}$, $F_{4}$, and $G_{2}$.

In the above $n\in\mathbb{Z}_{>1}$ and Types A to D are called the Classical
Lie algebras which come in infinite families, while Type E to G, are called
the Exceptionals. One could see an analogy to the classification of simple
finite groups!

Now, the correspondence between Lie algebras and Lie groups, for explicit
matrices, is the exponentiation map. Thus, Type A are traceless
$(n+1)\times(n+1)$ complex matrices, which, when exponentiated, give rise to
$(n+1)\times(n+1)$ unit-determinant matrices, i.e., $SL(n;\mathbb{C})$. The
real form is $\mathfrak{su}_{n+1}$ – in the sense that
$\mathfrak{sl}_{n+1}(\mathbb{C})=\mathfrak{su}_{n+1}\otimes_{\mathbb{R}}\mathbb{C}$
– and are traceless, anti-Hermitian matrices. These exponentiate to special
unitary matrices. In other words, Type A Lie algebra corresponds to the Lie
group $SU(n+1)$. Likewise, Type B and D Lie groups are the special orthogonal
groups of odd and even dimension respectively; Type C is the symplectic group.
The Exceptionals are more mysterious, with the root systems living in
dimensions 6, 7, 8, 4, and 2.

(a) $\alpha$$\beta$ (b)

Figure 3: (a)The root system and (b) Dynkin diagram of $A_{2}$.

Finally, Dynkin and Coxeter gave a marvellous graphical representation of
Theorem 2.3, by further reducing the problem of studying simple algebras to
root systems, using the so-called Cartan basis which we will not delve into
now. Suffice to say that the classification of Lie groups reduces first to
that of simple Lie groups, thence to that of simple Lie algebras, and finally
to that of positive simple roots in a root system:

######  Definition 2.4.

A root system $\Phi$ is a finite collection of non-zero vectors, called roots,
in Euclidean space $\mathbb{R}^{r}$ with the standard dot product
$(\cdot,\cdot)$ such that

  * •

The roots span $\mathbb{R}^{r}$; $r$ is called the rank;

  * •

The only scalar multiples of a root $\alpha\in\Phi$ that are still in $\Phi$
are $\pm\alpha$ and none other;

  * •

For any two roots $\alpha,\beta$, the vector
$\beta-2\frac{(\alpha,\beta)}{(\alpha,\alpha)}\alpha$, which is the reflection
through the hyperplane perpendicular to $\alpha$, is also a root 101010 For
Lie algebras, the factor $2\frac{(\alpha,\beta)}{(\alpha,\alpha)}$ in the
reflection is an integer.

In $\Phi$, we can always choose, potentially in many ways, the set $\Phi^{+}$
of positive roots which are such that

  * •

For each root $\alpha\in\Phi$, only one of $\pm\alpha$ is in $\Phi^{+}$;

  * •

For any two distinct roots $\alpha,\beta\in\Phi^{+}$ such that $\alpha+\beta$
is a root, then $\alpha+\beta\in\Phi^{+}$.

Finally, a simple root is a positive root which cannot be written as the sum
of two other positive simple roots. We denote the set of simple positive roots
as $\Delta\subset\Phi^{+}\subset\Phi$; the number of simple roots turns out to
be equal to the rank $r$.

The archetypical example of a root system in $\mathbb{R}^{2}$ is given in Fig.
3 (a), prescribed by the vertices of a regular hexagon. One can check that the
roots are $\\{\pm\alpha,\pm\beta,\pm(\alpha+\beta)\\}$, the positive roots are
$\\{\alpha,\beta,\alpha+\beta\\}$ and the positive simple roots are
$\\{\alpha,\beta\\}$, which subtend an angle of $\frac{2}{3}\pi$. This root
system is called $A_{2}$, and it does indeed correspond to the simple Lie
algebra $A_{2}$, and the associated Lie group $SU(3)$.

The root system representation can be even further simplified, using a
Coxeter–Dynkin diagram (or just Dynkin diagram), using the following rules:
(1) Let $\Delta$ be the simple roots and form a finite graph with each node
corresponding to a simple root – therefore there will be $r$ nodes where $r$
is the rank. (2) The edges between two nodes are determined by the angle
between the two corresponding simple roots using the rule:

$\pi/2$: no edge; $2\pi/3$: single edge; $3\pi/4$: double edge; $5\pi/6$:
triple edge.

It turns out that these are the only possible angles. (3) We introduce
direction, i.e., turn the edges into arrows - by letting the longer simple
root point to the shorter one; if the roots are of the same length, then
there’s no need for direction. In the above example of $A_{2}$, there are
therefore two nodes corresponding to $\alpha$ and $\beta$, there is a single
edge between them because they are at 120 degrees, and they are also of equal
length. The Dynkin diagram is in Fig. 3 (b).

Figure 4: The Dynkin diagrams of the simple Lie algebras/groups (source:
Wiki).

In summary, the graphical representation of Theorem 2.3 is Fig. 4. One readily
notices that of these diagrams, Types A-D-E have only single (and undirected)
edges, meaning that all the simple roots are of equal length, and are either
perpendicular, or at an angle of $2\pi/3$ to each other. These are called
simply-laced Dynkin diagrams, or simply A-D-E diagrams. They come in two
infinite families $A_{n}$ and $D_{n}$, as well as three exceptionals
$E_{6,7,8}$.

$\begin{array}[]{|c|c|c|}\hline\cr\mbox{Tetrahedron}&\mbox{Cube \qquad
Octahedron}&\mbox{Dodecahedron\qquad Icosahedron}\\\
\includegraphics[trim=0.0pt 0.0pt 0.0pt
0.0pt,height=72.26999pt]{tetra}&\includegraphics[trim=0.0pt 0.0pt 0.0pt
0.0pt,height=72.26999pt]{cube}\includegraphics[trim=0.0pt 0.0pt 0.0pt
0.0pt,height=72.26999pt]{octa}&\includegraphics[trim=0.0pt 0.0pt 0.0pt
0.0pt,height=72.26999pt]{dodeca}\includegraphics[trim=0.0pt 0.0pt 0.0pt
0.0pt,height=72.26999pt]{icosa}\\\ \hline\cr\end{array}$ Figure 5: The five
Platonic solids: the regular tetrahedron, cube, octahedron, dodecahedron and
icosahedron. We have grouped them according to duality.

####  2.2.2 Platonic Solids

Now, for something completely different. It is an old question what are the
discrete symmetries of Euclidean space. Phrased in a way ancient Greeks would
understand: what are the regular convex shapes? They are clearly (1) the
regular polygons and (2) the five Platonic solids, viz., the regular
tetrahedron, the cube, the regular octahedron, the regular dodecahedron and
the regular icosahedron. For reference, we show these five famous shapes in
Fig. 5.

Phrased in a more modern way: what are the discrete finite subgroups of
$SO(3)$, the group of special orthogonal matrices that constitute the
rotational group of symmetries of $\mathbb{R}^{3}$? These are the finite
groups that are the symmetries of the aforementioned regular convex shapes
(polytopes).

The regular $n$-gons are degenerate in the sense that they lie in the plane,
rather than being properly 3-dimensional. For them the symmetry groups are (1)
the cyclic group of size $n$ generated by $2\pi/n$ rotations; and (2) the
dihedral group of size $2n$, generated by $2\pi/n$ rotations and the flip
along any axis of symmetry. For the five platonic solids there are only three
symmetry groups. This is because of (graph) duality, in the sense of replacing
faces with vertices and edges with perpendicular edges. The cube–octahedron
are dual to each other, and likewise for the dodecahedron–icosahedron, while
the tetrahedron is self-dual. Hence, there are three symmetry groups:

  * •

(3a) the tetrahedral group which turns out to be the alternating group
$\mathcal{A}_{4}$, of size $4!/2=12$,

  * •

(3b) the cube–octahedral group (or simply the octahedral group) which is the
symmetric group $\mathcal{S}_{4}$, of size $4!=24$, and

  * •

(3c) the dodeca–icosahedral group (or simply the icosahedral group) which is
the alternating group $\mathcal{A}_{5}$, of size $5!/2=60$.

Everything I’ve said here would have been known to the 19th century
mathematician, and knowledge of the Lie groups and algebras, to those of the
early 20th. Yet, once again, it took waiting for many decades until John came
along to notice an extraordinary link between this set of symmetry groups of
$\mathbb{R}^{3}$: 2 infinite families and 3 outliers, and the seemingly
utterly unrelated set of simply-laced Dynkin diagrams: 2 infinite families and
3 exceptionals. In 1979, McKay announced at a conference on groups, his
observation which came to be known as the McKay Correspondence.

####  2.2.3 McKay Correspondence

John said to me several times that his favourite and proudest observation – in
his gentle humility he never used words such as ‘discovery’ when referring to
himself – was this 1979 correspondence. McKay first introduces a complex
representation by lifting $SO(3)$ to its double cover $SU(2)$ (incidentally,
these two Lie groups share the same Lie algebra). This lift is familiar to
physicists because it is a key difference between classical (vector) and
quantum-mechanical (spinor) angular momentum. On a group level, the discrete
symmetry groups are called binary and are double covers of what was mentioned
above. Specifically, we have (note that the cyclic groups, being Abelian, have
no non-trivial double cover)

######  Theorem 2.5.

The finite discrete subgroups of $SU(2)$ are:

$\begin{array}[]{|c|c|c|}\hline\cr\mbox{Name}&\mbox{Presentation}&|G|\\\
\hline\cr\hline\cr\mbox{Cyclic}&\left<R\ |\
R^{n}=\mathbb{I}\right>\simeq\mathbb{Z}/n\mathbb{Z}&n\\\ \hline\cr\mbox{Binary
Dihedral}&\left<R,S\ |\ R^{n}=S^{2}=(RS)^{2}\right>&4n\\\
\hline\cr\mbox{Binary Tetrahedral}&\left<R,S,T\ |\
RST=R^{2}=S^{3}=T^{3}\right>&24\\\ \hline\cr\mbox{Binary
Octahedral}&\left<R,S,T\ |\ RST=R^{2}=S^{3}=T^{4}\right>&48\\\
\hline\cr\mbox{Binary Icosahedral}&\left<R,S,T\ |\
RST=R^{2}=S^{3}=T^{5}\right>&120\\\ \hline\cr\end{array}$

Next, McKay takes the defining complex 2-dimensional representation ${\bf
R_{2}}$ (which necessarily exists because our groups live in $SU(2)$) of each
of these groups, and forms the tensor decomposition over all the irreps ${\bf
r_{i}}$

${\bf R_{2}}\otimes{\bf r_{i}}=\bigoplus\limits_{j}a_{ij}{\bf r_{j}}$ (9)

where $a_{ij}\in\mathbb{Z}_{\geq 0}$ denote the multiplicity. While such
tensor decompositions for groups, discrete or continuous, are standard,
somehow it occurred to no one that this should be done to the $SU(2)$
subgroups.

Finally, McKay recognises the matrices $a_{ij}$, which happen to be symmetric
and with 2 on the diagonal and 0 or 1 otherwise! Explicitly, they can be
expressed [27] in terms of the character table $\chi_{\gamma}^{(i)}$ of the
finite group $G$ as

$a_{ij}=\frac{1}{|G|}\sum\limits_{\gamma=1}^{r}C_{\gamma}\chi_{\gamma}^{{\bf
R_{2}}}\chi_{\gamma}^{(i)}\overline{\chi}_{\gamma}^{(j)}\ ,$ (10)

where the sum is over the $r$ conjugacy classes and $C_{\gamma}$ is the size
of the $\gamma$-th conjugacy class. He considers them to be the adjacency
matrices of some finite graphs – now called the McKay Quivers – by associating
a node to each row/column and an edge between vertices $i$ and $j$ whenever
$a_{ij}=1$. The result is presented in Fig. 6. These are almost the simply-
laced ADE Dynkin diagrams, just with an extra node! It turns out that these
are called the affine ADE Dynkin diagrams and encode an infinite
generalisation of the corresponding Lie algebras called affine Lie algebras.
The extra affine node, in the McKay Correspondence, is associated to the
identity element in the finite group.

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&\widehat{E_{8}}\sim\mbox{Binary Icosahedral}\par\end{array}$ Figure 6: The
McKay Correspondence between the affine simply-laced Dynkin diagrams and the
discrete finite subgroups of $SU(2)$, or between the affine Lie algebras and
the Platonic solids.

Once again, John managed to link two fields of mathematics which had not
talked to each other for at least half a century, Lie theory and symmetry
groups of the Platonic solids. Perhaps mathematicians had missed this because
both sides needed some manipulation: affinisation on the Lie side and lift to
$SU(2)$ on the rotational groups side; in addition, one needed the
genius111111To demystify all this a little, what inspired John was, first, the
pattern of 2 infinite families and 3 outliers, and second, that the sum of
squares of certain Coxeter labels in the Dynkin diagrams (which I will not
discuss here for sake of space) gives rise to the size of the finite groups.
of Eq. (9). One might wonder whether the Moonshine sporadics and the McKay
Correspondence exceptionals might be related. The initial seeds were indeed
planted by John and elaborated in [111], awaiting further clarification.

My very first paper and a good part of my PhD thesis was to find
generalisations of the McKay Correspondence and especially to see their
realisations in superstring theory in physics. Thus from the very outset of my
career, I worshipped McKay. To complete the circle, the very last thing on
which I worked with him was a textbook [115], together with Cameron and
Dechant, aimed at a wide mathematical audience, entitled “A-D-E”. Tragically,
John will never see the completion of this project.

The McKay Correspondence did much more than bridge two fields. It prompted
mathematicians to look for the ADE meta-pattern everywhere. It seems
ubiquitous across mathematical disciplines ranging from algebraic geometry to
conformal field theory, and has gained almost a religious status. V. Arnol’d
found some 20 disparate instances of what he appropriately called the
“Trinity”, $E_{6,7,8}$ [28]. If you ever run into a classification problem in
mathematics where there are 2 infinite families and 3 exceptional cases, then
beware! Chances are, they fall into an A-D-E type of meta-pattern.

###  2.3 Characters & Primes

We finally proceed to John’s earliest miracle [49, 50], made between 1971-2,
which still remains one of the most important open problems in representation
theory (for a recent review see [29]). This is commonly known as the McKay
Conjecture, later generalised and formalised [30] to the Alperin-McKay
Conjecture. In the words of Markus Linckelmann,

> “The McKay conjecture, from 1972, and its block theoretic generalisation
> known as the Alperin-McKay conjecture, from 1976, are instances of a host of
> on-the-surface miraculous numerical coincidences that haunt modular
> representation theory of finite groups.”

Once again, it begins with a miraculous observation by John, made shortly
after he completed his PhD. Consider the complex characters
$\chi_{\gamma}^{(i)}$ of a finite group $G$ – the computer calculation of
which was the subject of his thesis. Recall that characters are traces of the
matrix representations and $\chi_{\gamma}^{(i)}$ is an $r\times r$ square
table where $r$ is equal to both the number of conjugacy classes of $G$ and
the number of irreducible representations, termed ‘irreps’. The rows, indexed
by $\gamma$, go through the irreps, and the first row is always just all 1s,
corresponding to the trivial representation where all group elements are
represented by 1. That is, $\chi_{1}^{(i)}=1$. The columns, indexed by $i$, go
through the conjugacy classes, and the first column is always the list of
dimensions of the irreps, being the trace of the identity matrix
representation of $\mathbb{I}_{G}$. That is, $\chi_{\gamma}^{(1)}$ are the
dimensions of the irreps, like the $r_{i}$ in Eq. (7) for the Monster. We made
use, in the foregoing, of the character table for both the McKay-Thompson
series in Eq. (8) and in inverting the McKay decomposition in Eq. (9).

Now, fix a prime $p$ and consider a Sylow $p$-subgroup $S_{p}$ of $G$. Recall
that a $p$-subgroup is a subgroup of size which is a prime-power $p^{n}$, and
a Sylow $p$-subgroup is maximal in the sense that it is not a proper subgroup
of any other $p$-subgroup. Next, consider the normaliser of $S_{p}$ in $G$,
i.e., $N_{G}(S_{p}):=\\{g\in G:gS_{p}g^{-1}=S_{p}\\}$. The McKay Conjecture,
which remains open for arbitrary finite groups121212 Interestingly, John’s
original “McKay Conjecture” was for finite simple groups and the prime $p=2$,
which is now proved [33]). , states

######  Conjecture 2.6.

(McKay) Denote by $Irr_{p^{\prime}}(H)$ the set of characters such that $p$
does not divide the dimension of the irreps of a group $H$, i.e.,
$p\nmid\chi_{\gamma}^{(1)}$. Then

$|Irr_{p^{\prime}}(G)|=|Irr_{p^{\prime}}(N_{G}(S_{p}))|\ .$

That is, the two sets have the same cardinality.

Let us see this conjecture in action. It should work for any finite group, so
we might as well take one with which we are familiar. From Theorem 2.5,
consider the binary icosahedral group (which turns out to be isomorphic to
$SL(2;5)$, the 2-dimensional special linear group defined over the finite
field $\mathbb{F}_{5}$). The character table of this group is presented in
Table 1 (a). In the fine tradition of McKay, one could readily use computer
algebra [31] to obtain this. We see that there are $r=9$ irreps and conjugacy
classes, and we have added an extra row on top to denote the sizes of the
conjugacy classes for completeness. We can check that this row sums to 120,
the size of the group and, by standard Schur orthgonality for character
tables, that the square sum of the first column
$1^{2}+2^{2}+2^{2}+3^{2}+3^{2}+4^{2}+4^{2}+5^{2}+6^{2}=120$ also131313Indeed,
on an unrelated note, one can further check that substituting this character
table into Eq.(10) gives the adjacency matrix of the affine $E_{8}$ Dynkin
diagram. As one can check that the character table in part (b) gives the
affine $E_{6}$ Dynkin diagram..

(a) $\begin{array}[]{|c|c|c|c|c|c|c|c|c|c|}\hline\cr
1&12&12&20&30&20&12&1&12\\\ \hline\cr\hline\cr 1&1&1&1&1&1&1&1&1\\\ \hline\cr
2&-\phi^{\prime}&-\phi&1&0&-1&\phi^{\prime}&-2&\phi\\\ \hline\cr
2&-\phi&-\phi^{\prime}&1&0&-1&\phi&-2&\phi^{\prime}\\\ \hline\cr
3&\phi&\phi^{\prime}&0&-1&0&\phi&3&\phi^{\prime}\\\ \hline\cr
3&\phi^{\prime}&\phi&0&-1&0&\phi^{\prime}&3&\phi\\\ \hline\cr
4&-1&-1&1&0&1&-1&4&-1\\\ \hline\cr 4&-1&-1&-1&0&1&1&-4&1\\\ \hline\cr
5&0&0&-1&1&-1&0&5&0\\\ \hline\cr 6&1&1&0&0&0&-1&-6&-1\\\ \hline\cr\end{array}$
(b) $\begin{array}[]{|c|c|c|c|c|c|c|}\hline\cr 1&4&4&6&1&4&4\\\
\hline\cr\hline\cr 1&1&1&1&1&1&1\\\ \hline\cr
1&\omega_{3}^{2}&\omega_{3}^{2}&1&1&\omega_{3}&\omega_{3}\\\ \hline\cr
1&\omega_{3}&\omega_{3}&1&1&\omega_{3}^{2}&\omega_{3}^{2}\\\ \hline\cr
2&-1&1&0&-2&1&-1\\\ \hline\cr
2&-\omega_{3}&\omega_{3}&0&-2&\omega_{3}^{2}&-\omega_{3}^{2}\\\ \hline\cr
2&-\omega_{3}^{2}&\omega_{3}^{2}&0&-2&\omega_{3}&-\omega_{3}\\\ \hline\cr
3&0&0&-1&3&0&0\\\ \hline\cr\end{array}$

Table 1: The character tables of (a) the binary icosahedral group and (b) the
normalizer of its Sylow 2-group. Here, $\phi=\frac{1+\sqrt{5}}{2}$,
$\phi^{\prime}=\frac{1-\sqrt{5}}{2}$ and $\omega_{3}=\exp(2\pi i/3)$.

Now let’s pick the prime $p=2$. The Sylow 2-group $S_{2}$ turns out to be the
quaternion group $Q_{8}$ of size 8. Its normaliser $N_{G}(Q_{8})$ turns out to
be $SL(2;3)$, which happens to be the binary tetrahedral group of size 24,
whose character table is given in Table 1 (b). Here, there are $r=7$ irreps
and conjugacy classes, and again we add a top row for the sizes of the
conjugacy classes.

The characters for $G$ for which the entries in the first column are not
divisible by $p=2$ are the rows which begin with $\\{1,3,3,5\\}$ from Table 1
(a). Likewise, from Table 1 (b), the characters for $N_{G}(Q_{8})$ for which
the entries of the first column are not divisible by $2$ are the rows
beginning with $\\{1,1,1,3\\}$. Indeed, the cardinality of both sets is 4, as
the conjecture demands.

There are collaborative efforts attempting to prove the conjectures using the
classification of finite simple groups. For $p=2$, Conjecture 2.6 was proved
by Malle and Späth [32] in 2016, and the Alperin-McKay generalisation, again
for $p=2$, was only proved last year by Ruhstorfer [33], a week before John
died. Continuing to quote Linckelmann,

> “In parallel to the attempts using the classification of finite simple
> groups, these numerical conjectures have generated a lot of work searching
> for structural explanations in terms of module categories, their derived
> categories, drawing on fields further afar, such as homotopy theory. Apart
> for some special cases, structural explanations remain largely elusive –
> translating mystery into insight seems still out of reach.”

##  3 Legacy

> “Aside from his energy and broad curiosity, the thing that struck me most
> was his phenomenal instinct for finding the singular mathematical pearl in a
> field of marbles. I think that what characterised his remarkable
> contributions to disparate mathematical areas, was his incredible intuition
> for identifying those (seemingly mundane) facts which were keys to unlocking
> deep underlying structures. These drew extensively on his ability to adapt
> computational methods to abstract mathematical issues …I will always be
> grateful for having had John as a friend and colleague. He made Concordia
> and the world a more interesting place.”

Officially, McKay, who was never into formalities, had only one PhD student,
K.-C. Young from McGill in 1974. Yet he was enormously generous with his time
and shared his ideas gregariously, especially with the younger generation.
Therefore, many of us feel as if we are in his intellectual family. These
have, over the years, included: Leonard Soicher, a Master’s student of John’s,
who on his recommendation went on to pursue his PhD with Conway, and then
returned to be McKay’s postdoc; Abdellah Sebbar who was his postdoc; Abdelkrim
Elbasraoui who was a student at Concordia; Roland Friedrich who worked with
him on free probability theory and who called him regularly. Additionally,
there is myself, my students James Read and Valdo Tatitscheff, and many more.
We all felt like John’s students. Over his career spanning half a century,
McKay had many influential publications, which I have collated here [35] \-
[115]. Of course, of all these, his three miracles distinguish themselves, and
to their account I have devoted this obituary.

I have heard some non-mathematicians, and even an occasional mathematician,
dismiss what I have described here. They presume that if you make a conjecture
a day, surely you will hit on something important sooner or later. To these
pompous and tiresome people, I mumble, “Well, why don’t you try it!” It has
been a tremendous privilege to have done what they could not, and that is to
have worked with John McKay on somewhat of an equal footing. To have had a
glimpse of McKay’s methods and thoughts has made me realise that to hit on one
of his miracles is already astounding, but to do so thrice requires super-
human powers. I should also point out that everything John did was decades
before the internet, when one could not simply google at will. Today, if you
typed ‘196884’ into OEIS [34], you would find $j$-invariant coefficients and
moonshine-related material.

John had nothing but his vast knowledge and perspicacious insight. His mind
was divine.

###### Acknowledgements.

I am grateful to John’s wife Trinh McKay-Vo, his sister Liz Hurt, and his ex-
wife Wendy McKay for all their tirelessly answering my questions on his life.
Above all, Trinh, together with John, had very much become a part of my
family; to her this obituary is dedicated. Sincere thanks also go to John’s
friends Leonard Soicher, John Harnad, Hershy Kisilevsky, and John Ball for
providing various snippets of his life over different periods, as well as
Madeleine Hall, John Harnad, Leonard Soicher, Pierre-Philippe Dechant, Thomas
Hodgkinson, and Markus Linckelmann for many helpful comments.

## References

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  * [2] A. Masters, ‘Simon: The genius in my basement’, Harper Collins 2011, ISBN 9780007445264
  * [3] S. Roberts, ‘Genius At Play: The Curious Mind of John Horton Conway’, Bloomsbury 2015, ISBN 9781620405949.
  * [4] Y.-H. He. ‘The Genius on Skype: the exceptional and sporadic mind of John McKay’, in progress.
  * [5] J. McKay, ‘Computing with Finite Groups’, PhD Thesis, University of Edinburgh, 1970, https://era.ed.ac.uk/bitstream/handle/1842/6615/McKay1971.pdf Note that McKay got his PhD in 1967 but the full write-up of the thesis was completed in 1970, after his first research fellowship.
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  * [7] J. Harnad, ‘John McKay, Obit’, Le Bulletin du Centre de Recherches Mathématiques, https://www.crmath.ca/wp-content/uploads/2023/02/bulletin_crm_2023_28_1.pdf
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  * [9] Y.-H. He ‘John McKay, Old Alleynian, Obit.’ Dulwich College, https://www.dulwich.org.uk/old-alleynians-home/obituaries/~board/1950s/post/john-mckay-oa-obituary
  * [10] J. Harnad, F. Lalonde, P. Winternitz, organizers, April 27-29, 2007 http://www.crm.umontreal.ca/McKay07/index_e.shtml. Proceedings published in AMS CRM Proceedings & Lecture Notes Volume: 47; 2009, ISBN 978-0-8218-4481-6.
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  * [14] R. Griess, ‘The structure of the Monster simple group’. In Scott, Gross, Fletcher (eds.). Proceedings of the Conference on Finite Groups (Univ. Utah, 1975). Boston, ISBN 978-012633650-4
  * [15] R. Griess, U. Meierfrankenfeld, Y. Segev. “A Uniqueness Proof for the Monster.” Annals of Mathematics 130, no. 3 (1989): 567-602. https://doi.org/10.2307/1971455.
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  * [18] T. Gannon Moonshine beyond the Monster: The Bridge Connecting Algebra, Modular Forms and Physics, ISBN 978-0-521-83531-2. Cambridge monographs, 2006.
  * [19] M. Du Sautoy, Finding Moonshine, A Mathematician’s Journey Through Symmetry, ISBN 978-0-00-721461-7, 2008
  * [20] V. Tatitscheff, “Monstrous Moonshine: A Short Introduction,” Reson. 27 (2022) no.12, 2107-2126 [arXiv:1902.03118 [math.NT]].
  * [21] V. G. Kac, “Infinite-Dimensional Algebras, Dedekind’s q-Function, Classical Mobius Function and the Very Strange Formula,” Advances in Math. 30 (1978), 85-136.
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-, “Monstrous Moonshine and monstrous Lie superalgebra” Invent. Math, 109 (1992) pp405 - 444
  * [25] M. Bhargava, et al., eds. Finite simple groups: thirty years of the Atlas and beyond, Contemporary Maths. Vol 694. AMS, 2017 (“Celebrating the atlases and honoring John Conway”, November 2 - 5, 2015, Princeton University)
  * [26] H. Kisilevsky, “Remarks on Knowing John McKay”, CRM Obituaries, 2022.
  * [27] A. Hanany and Y.-H. He, “Non-Abelian finite gauge theories,” JHEP 02 (1999), 013 [arXiv:hep-th/9811183].
  * [28] V. Arnold, ‘Symplectization, Complexification and Mathematical Trinities.’, Toronto Lectures, 1997. https://www.maths.ed.ac.uk/~v1ranick/papers/arnold4.pdf
  * [29] A. Evseev, ‘The McKay conjecture and Brauer’s induction theorem’, Proc. LMS, 106 (6) 2013, 1248 - 1290. [arXiv:1009.1413 [math.RT]]
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  * [32] G. Malle, B. Späth, Characters of odd degree, arXiv:1506.07690 [math.RT]
  * [33] L. Ruhstorfer, The Alperin-McKay conjecture for the prime 2, arXiv:2204.06373 [math.RT]
  * [34] OEIS Foundation Inc. (2023), The On-Line Encyclopedia of Integer Sequences, http://oeis.org

### John McKay’s Works

* [35] Atkin, A., Bratley, P., Macdonald, I. & McKay, J. Some computations for m-dimensional partitions. Proc. Cambridge Philos. Soc.. 63 pp. 1097-1100 (1967)
* [36] Bratley, P. & McKay, J. More amicable numbers. Math. Comp.. 22 pp. 677-678 (1968)
* [37] McKay, J. A method for computing the character table of a finite group. Computers In Mathematical Research. pp. 140-148 (1968)
* [38] Higman, G. & McKay, J. On Janko’s simple group of order 50, 232, 960. Theory Of Finite Groups (Symposium, Harvard Univ., Cambridge, Mass., 1968). pp. 65-77 (1969)
* [39] Higman, G. & McKay, J. On Janko’s simple group of order 50,232,960. Bull. London Math. Soc.. 1 pp. 89-94; correction, ibid. 1 (1969), 219 (1969)
* [40] McKay, J. Table errata: The groups of order $2^{n},(n\leq 6)$ (Macmillan, New York, 1964) by M. Hall, Jr. and J. K. Senior. Math. Comp.. 23, 691-692 (1969)
* [41] Higman, G. & McKay, J. Correction: On Janko’S Simple Group of Order 50,232,960. Bull. London Math. Soc.. 1, 219 (1969)
* [42] Higman, G. & McKay, J. On Janko’s Simple Group of Order 50,232,960. Bull. London Math. Soc.. 1, 89-94 (1969)
* [43] McKay, J. The construction of the character table of a finite group from generators and relations. Computational Problems In Abstract Algebra (Proc. Conf., Oxford, 1967). pp. 89-100 (1970)
* [44] Bratley, P., Lunnon, F. & McKay, J. Amicable numbers and their distribution. Math. Comp.. 24 pp. 431-432 (1970)
* [45] McKay, J. & Wales, D. The multipliers of the simple groups of order 604,800 and 50,232,960. J. Algebra. 17 pp. 262-272 (1971)
* [46] McKay, J. Multi-dimensional partition. Combinatorial Mathematics And Its Applications (Proc. Conf., Oxford, 1969). pp. 183-184 (1971)
* [47] McKay, J. & Wales, D. The multiplier of the Higman-Sims simple group. Bull. London Math. Soc.. 3 pp. 283-285 (1971)
* [48] McKay, J. Subgroups and permutation characters. Computers In Algebra And Number Theory (Proc. SIAM-AMS Sympos. Appl. Math., New York, 1970). pp. 177-181 (1971)
* [49] J. McKay, A new invariant for simple groups. Notices AMS 18 (1971), 397
* [50] McKay, J. Irreducible representations of odd degree. J. Algebra. 20 pp. 416-418 (1972)
* [51] McKay, J. A setting for the Leech lattice. Finite Groups ’72 (Proc. Gainesville Conf., Univ. Florida, Gainesville, Fla., 1972). pp. 117-118 (1973)
* [52] McKay, J. Computing with finite simple groups. Proceedings Of The Second International Conference On The Theory Of Groups (Australian Nat. Univ., Canberra, 1973). pp. 448-452 (1974)
* [53] McKay, J. The largest degrees of irreducible characters of the symmetric group. Math. Comp.. 30, 624-631 (1976)
* [54] Jansson, W. & McKay, J. More about the Mathieu group $M_{22}$. Canadian J. Math.. 28, 929-937 (1976)
* [55] Fischer, J. & McKay, J. The nonabelian simple groups G, $|G|<10^{6}$ \- maximal subgroups. Math. Comp.. 32, 1293-1302 (1978)
* [56] McKay, J. & Young, K. The nonabelian simple groups G, $|G|<10^{6}$-minimal generating pairs. Math. Comp.. 33, 812-814 (1979)
* [57] Erbach, D., Fisher, J. & McKay, J. Polynomials with PSL(2, 7) as Galois group. J. Number Theory. 11, 69-75 (1979)
* [58] McKay, J. Some remarks on computing Galois groups. SIAM J. Comput.. 8, 344-347 (1979)
* [59] Cannon, J., McKay, J. & Young, K. The nonabelian simple groups G, $G<10^{5}$ \- presentations. Comm. Algebra. 7, 1397-1406 (1979)
* [60] McKay, J. The non-abelian simple groups G, $|G|<10^{6}$ \- character tables. Comm. Algebra. 7, 1407-1445 (1979)
* [61] McKay, J. Graphs, singularities, and finite groups. The Santa Cruz Conference On Finite Groups (Univ. California, Santa Cruz, Calif., 1979). 37 pp. 183-186 (1980)
* [62] McKay, J. Cartan matrices, finite groups of quaternions, and Kleinian singularities. Proc. Amer. Math. Soc.. 81, 153-154 (1981)
* [63] McKay, J., Patera, J. & Sharp, R. Second and fourth indices of plethysms. J. Math. Phys.. 22, 2770-2774 (1981)
* [64] Ford, D. & McKay, J. Representations and Coxeter graphs. The Geometric Vein. pp. 549-554 (1981)
* [65] Butler, G. & McKay, J. The transitive groups of degree up to eleven. Comm. Algebra. 11, 863-911 (1983)
* [66] Lam, C., Thiel, L., Swiercz, S. & McKay, J. The nonexistence of ovals in a projective plane of order 10. Discrete Math.. 45, 319-321 (1983)
* [67] McKay, J. Graphs, singularities and finite groups. Uspekhi Mat. Nauk. 38, 159-162 (1983), Translated from the English by S. A. Syskin
* [68] Kolesova, G. & McKay, J. Practical strategies for computing Galois groups. Computational Group Theory (Durham, 1982). pp. 297-299 (1984)
* [69] McKay, J. & Regener, E. Actions of permutation groups on r-sets. Comm. Algebra. 13, 619-630 (1985)
* [70] Soicher, L. & McKay, J. Computing Galois groups over the rationals. J. Number Theory. 20, 273-281 (1985)
* [71] Dummit, D., Kisilevsky, H. & McKay, J. Multiplicative products of L-functions. Finite Groups: coming Of Age (Montreal, Que., 1982). 45 pp. 89-98 (1985)
* [72] Ford, D. & McKay, J. From polynomials to Galois groups. EUROCAL ’85, Vol. 2 (Linz, 1985). 204 pp. 535-536 (1985)
* [73] McKay, J. & Stauduhar, R. Coda to a theorem of Schur. J. Reine Angew. Math.. 377 pp. 219-220 (1987)
* [74] McKay, J. On computing discriminants. Amer. Math. Monthly. 94, 523-527 (1987)
* [75] Conder, M. & McKay, J. A necessary condition for transitivity of a finite permutation group. Bull. London Math. Soc.. 20, 235-238 (1988)
* [76] McKay, J. Advances in computational Galois theory. Computers In Algebra (Chicago, IL, 1985). 111 pp. 99-101 (1988)
* [77] Ford, D. & McKay, J. Computation of Galois groups from polynomials over the rationals. Computer Algebra (New York, 1984). 113 pp. 145-150 (1989)
* [78] Ford, D. & McKay, J. Ramifications of Ramanujan’s work on Î·-products. Proc. Indian Acad. Sci. Math. Sci.. 99, 221-229 (1989)
* [79] McKay, J. & Strauss, H. The q-series of monstrous moonshine and the decomposition of the head characters. Comm. Algebra. 18, 253-278 (1990)
* [80] Darmon, H. & McKay, J. A continued fraction and permutations with fixed points. Amer. Math. Monthly. 98, 25-27 (1991)
* [81] Alexander, D., Cummins, C., McKay, J. & Simons, C. Completely replicable functions. Groups, Combinatorics & Geometry (Durham, 1990). 165 pp. 87-98 (1992)
* [82] Casperson, D. & McKay, J. Symmetric functions, m-sets, and Galois groups. Math. Comp.. 63, 749-757 (1994)
* [83] Ford, D., McKay, J. & Norton, S. More on replicable functions. Comm. Algebra. 22, 5175-5193 (1994)
* [84] Cohn, H. & McKay, J. Spontaneous generation of modular invariants. Math. Comp.. 65, 1295-1309 (1996)
* [85] Casperson, D., Ford, D. & McKay, J. Ideal decompositions and subfields. J. Symbolic Comput.. 21, 133-137 (1996)
* [86] McKay, J. A note on the elliptic curves of Harada-Lang. Groups, Difference Sets, And The Monster (Columbus, OH, 1993). 4 pp. 409-410 (1996)
* [87] Conder, M. & McKay, J. Markings of the Golay code. New Zealand J. Math.. 25, 133-139 (1996)
* [88] Mattman, T. & McKay, J., Comp. of Galois groups over function fields. Math. Comp.. 66, 823-831 (1997)
* [89] McKay, J. & Stauduhar, R. Finding relations among the roots of an irreducible polynomial. Proc. 1997 International Symposium On Symbolic And Algebraic Computation (Kihei, HI). pp. 75-77 (1997)
* [90] Conway, J., Hulpke, A. & McKay, J. On transitive permutation groups. LMS J. Comp. Math.. 1 pp. 1-8 (1998)
* [91] McKay, J. & Sebbar, A. Fuchsian groups, Schwarzians, and theta functions. C. R. Acad. Sci. Paris I Math.. 327, 343-348 (1998)
* [92] McKay, J. Semi-affine Coxeter-Dynkin graphs and $G\subseteq SU(2,C)$. Canad. J. Math.. 51 pp. 1226-1229 (1999), Dedicated to H. S. M. Coxeter on the occasion of his 90th birthday
* [93] Harnad, J. & McKay, J. Modular invariants and generalized Halphen systems. SIDE III: symmetries And Integrability Of Difference Equations (Sabaudia, 1998). 25 pp. 181-195 (2000)
* [94] J. McKay and A. Sebbar editors, Moonshine and related topics, Proc. of the Moonshine Workshop, Montreal, QC, May 1999. CRM Proceedings & Lecture Notes 30 AMS 2001 ISBN: 0-8218-2879-7
* [95] McKay, J. & Sebbar, A. Fuchsian groups, automorphic functions and Schwarzians. Math. Ann.. 318, 255-275 (2000)
* [96] Harnad, J. & McKay, J. Modular solutions to equations of generalized Halphen type. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci.. 456, 261-294 (2000)
* [97] McKay, J. Linear dependencies among completely replicable functions. Proceedings On Moonshine And Related Topics (Montreal, QC, 1999). 30 pp. 117-118 (2001)
* [98] McKay, J. & Sebbar, A. Arithmetic semistable elliptic surfaces. Proceedings On Moonshine And Related Topics (Montreal, QC, 1999). 30 pp. 119-130 (2001)
* [99] McKay, J. The essentials of Monstrous Moonshine. Groups And Combinatorics:in Memory Of Michio Suzuki. 32 pp. 347-353 (2001)
* [100] Cox, D., McKay, J. & Stevenhagen, P. Principal moduli and class fields. Bull. LMS. 36, 3-12 (2004)
* [101] Conway, J., McKay, J. & Sebbar, A. On the discrete groups of Moonshine. Proc. AMS. 132, 2233-2240 (2004)
* [102] McKay, J. & Sebbar, A. Replicable functions: an introduction. Frontiers In Number Theory, Physics, And Geometry. II. pp. 373-386 (2007)
* [103] McKay, J. & Sevilla, D. Decomposing replicable functions. LMS J. Comput. Math.. 11 pp. 146-171 (2008)
* [104] Conway, J., McKay, J. & Trojan, A. Galois groups over function fields of positive characteristic. Proc. Amer. Math. Soc.. 138, 1205-1212 (2010)
* [105] Friedrich, R. & McKay, J. Free probability and complex cobordism. C. R. Math. Acad. Sci. Soc. R. Can.. 33, 116-122 (2011)
* [106] He, Y.-H. & McKay, “N=2 gauge theories: congruence subgroups, coset graphs, and modular surfaces. ” J. Math. Phys.. 54, 012301, 24 (2013) [arXiv:1201.3633 [hep-th]].
* [107] He, Y.-H., McKay, J. & Read, J., “Modular subgroups, dessins d’enfants, and elliptic K3 surfaces” LMS J. Comput. Math.. 16 pp. 271-318 (2013) [arXiv:1211.1931 [math.AG]].
* [108] Y.-H. He and J. McKay, “Eta Products, BPS States and K3 Surfaces,” JHEP 01 (2014), 113 [arXiv:1308.5233 [hep-th]].
* [109] He, Y.-H. & McKay, J. Moonshine and the meaning of life. Finite Simple Groups: Thirty Years Of The Atlas And Beyond. 694 pp. 1-2 (2017), https://doi.org/10.1073/pnas.81.10.3256 [arXiv:1408.2083 [math.NT]].
* [110] El Basraoui, A. & McKay, J. The Schwarzian equation for completely replicable functions. LMS J. Comput. Math.. 20, 30-52 (2017)
* [111] Y.-H. He and J. McKay, “Sporadic and Exceptional,” [arXiv:1505.06742 [math.AG]].
* [112] A. Chen, Y.-H. He and J. Mckay, “Erland Samuel Bring’s ‘Transformation of Algebraic Equations’,” [arXiv:1711.09253 [math.HO]].
* [113] Tatitscheff, V., He, Y.-H. & McKay, J. Cusps, congruence groups and monstrous dessins. Indag. Math.. 31, 1015-1065 (2020) [arXiv:1812.11752 [math.NT]]
* [114] J. McKay & Y.-H. He, Kashiwa lectures on new approaches to the monster. ICCM Not. 10, 71-88 (2022) [arXiv:2106.01162 [math.HO]].
* [115] P.-P. Dechant, P. Cameron, Y.-H. He, J. McKay, A-D-E, Cambridge University Press, to appear. Yang-Hui He
London Institute for Mathematical Sciences, Royal Institution of Great
Britain, 21 Albemarle Street, London W1S 4BS, UK & Merton College, Oxford
University, OX14JD, UK