Date: Mon, 23 Dec 2002 01:12:43 -0500

From: Dylan Thurston <>

I'm afraid it will be a little time before I

can make substantive comments, as it is quite a long book...  I was

particularly interested in your "Magic Triangle".  If I understand

correctly, the central claim/conjecture of this triangle is that there

are diagrammatic calculi which specialize to give each row in the

triangle.  This is quite remarkable, especially if these diagrammatic

calculi themselves fall into a series, as suggested by two things: the

symmetry of the magic triangle around the main diagonal; and the

similarity between the Diophantine equations you end up solving.  The

first part is mainly missing a proof that the elementary graphical

relations you write down are enough to reduce arbitrary diagrams; and

the second part seems to still be numerology (but highly intriguing

numerology).  Is this a good summary?

(PC 1 Aug 2007: that's fair.)

There are a couple of references that might be relevant to your work

(and that weren't in your bibliography):

Hans Wenzl, "On Tensor Categories of Lie Type E_N, N \ne 9", to appear

in Advances in Mathematics, available from

Jacob Lurie, "On simply laced lie algebras and their minuscule

representations", undergraduate thesis, Harvard University, 2000;

also Comment. Math. Helv. 76 (2001), no. 3, 515--575.

These both consider exceptional Lie algebras from the point of view of

their _miniscule representations_: the representation with the

smallest dimension.  Hans shows that the invariants (for the "quantum"

groups in addition to the classical ones) are nearly generated by a

single tensor in dimension N (on the nose for N=6,7, and in a direct

summand for other N, N != 9).  I think Jacob does something similar,

but more explicitly, for E_6 and E_7.

(PC 1 Aug 2007: all of this now promoted to a higher level of limbo, into the \Preliminary{ parts of the webbook manuscript} )

Dylan Thurston