To whom it may concern:

This is a referee report on the thesis titled The Character Table of the Weyl Group of E

_{8}: Applications to the Arcane Arts, a dissertation submitted by M. Parry Hotter in partial fulfillment of the degree ofMagiæ Doctorat the University of Hogsbridge.

Context:To put this study in its proper historical perspective, which M. Hotter himself does at length in the first chapter of the thesis under review, would require more space than can be afforded here. As the author aptly recalls, the E_{8}perspective on the arcane arts can be traced back to the unification, proposed by Leibniz in 1710, of six of the seven classical schools of magic (Earth, Water, Air, Fire, Macrocosm, Life and Spirit, arranged linearly by Paracelsus) with six of the seven oriental phases (Earth, Water, Wind, Fire, Heaven, Change and Unchange, with Change and Unchange branching from Heaven), by equating Heaven with Macrocosm and Change with Life (and renaming Unchange as Time). The asymetrical nature of the resulting diagram — which we now know as the Dynkin diagram of E_{8}— prompted a number of attempts to identify at least one more house — attempts that we presently understand to be misguided.But it is in the year 1918, which saw the publication of Hermann Weyl's now classic Earth, Water, Air, Fire, Space, Time, Life and Spirit, that the 240 directions of the mysticohedron were put upon a firm theoretical footing. This represents a considerable paradigm shift, whose practical consequences were slow to come to fruition (starting with the startling realization of where the level grades, 1, 248, 3 875, 27 000, 30 380, 147 250, 779 247… appear). And as examined in more detail in Aldus Bumblebore's The Eight Elements (or: What's so Special about the Number 696 729 600?), it was only considerably recently that any attention was given to the profond interconnection between the largest exceptional Weyl group and the transmutations of magic.

As explained in the abstract, M. Hotter's work consists of two main parts. The first explores applications of the « pure » character theory of W(E

_{8}) beyond the mere, and previously known, identification of the 112 representations (lines of the character table) and conjugacy classes (columns) with the arcane circles and astral configurations. The second, and much deeper, part of this thesis, develops an invariant theory for E_{8}that is analogous to the classical Schur-Weyl duality between representations of the linear group and those of the symmetric group, and then applies this duality to obtain esoterica. Eight specific and illustrative examples are given in an appendix. We now review each of the chapters in greater detail.

**Some commentary:** (Generally I don't discuss the
references in my texts but I've made
exceptions before.)

I've often said
that E_{8}
(and the cohort of related objects) is
so deeply fascinating and
profoundly beautiful
(see here
and there) that if the
Universe has any sense of æsthetics, it really should involve
E_{8} in some way (some people have
indeed tried
to find it, but with dubious success: it is possible that the Universe
we live in does not have the same notion of æsthetics as
mathematicians). At any rate, in a world in which magic is real and
wizards who (after their elementary and high school years spent doing
more applied work like fighting supervillain wizards) go to university
and write doctoral theses in pure and theoretical magic, I cannot
conceive the mathematical foundations of magic to be
anything *but* E_{8} (OK, maybe I'll take
the largest Mathieu group as an
acceptable substitute). Of course, in my vision of magic, the head
magician does not look so
much like
this
as like
that (seriously, if anyone is a real world magician, it's John
Conway). Anyway.

So if magic is to be built on E_{8}, then the system at the
root of all arcana is represented by the following diagram:

—and there should definitely be some labels attached here. (They
would actually be of some use to real world mathematicians because
nobody can agree on how to number the vertices of this diagram.
Unfortunately, Conway, the great inventor of witty names, did not do
his job here.) So I propose to name the seven on the bottom line,
from left to right: Spirit, Life, Macrocosm, Fire, Air, Water and
Earth, and the top one, Time. There isn't much rationale to my
suggestion that the Europeans and Chinese(?) should have discovered
the A_{7} (i.e., all but the top node) and D_{7}
(i.e., all but the leftmode node) subdiagrams of the above, but it is
true that Leibniz was fascinated by
the Yi Jing and
popularized it in Europe. (It seemed right to make Leibniz play a
role here when I had had fun with
Newton in a previous fragment.)

More importantly, Hermann Weyl, something of a magician himself, to whom we owe much of the theory of representations of compact Lie groups (and in particular the formula which allows to compute the sequence I mention), wrote a book called Space, Time, Matter, one of the first expositions of Einstein's theory of general relativity (and indeed one of the books — found in my father's library — through which I myself learned the subject): in an alternate universe, it would certainly have been a book on magic.

Incidentally, I wish someone would tell me how one can construct
some kind of analogue of Schur-Weyl duality for the exceptional groups
(or in any way relate the representations of E_{8} to those of
W(E_{8})).