David Madore's WebLog: Gratuitous Literary Fragment #147 (referee report)

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Gratuitous Literary Fragment #147 (referee report)

To whom it may concern:

This is a referee report on the thesis titled The Character Table of the Weyl Group of E8: Applications to the Arcane Arts, a dissertation submitted by M. Parry Hotter in partial fulfillment of the degree of Magiæ Doctor at 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 E8 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 E8 — 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(E8) 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 E8 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 E8 (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 E8 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 E8 (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 E8, 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 A7 (i.e., all but the top node) and D7 (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 E8 to those of W(E8)).

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