23/02/11 23:42:38.02 cDdl8Z4s.net
>>350
つづき
Let us fix a light-cone element c ∈ Δ such that there are no real roots
orthogonal to it. Such a vector exists and the set L = [a ∈ ΔR: (a, c) =1} is isomorphic to the unique even unimodular lattice of rank 24, which
does not contain elements of length √2 [2]. We denote by V1,c, the space
Σα∈L V1,c,α. Then the character of V1,c, is
j(q) = θL(q)/η(q)^24 =q^-1 + 24 + 196884q +・・ (4.21)
It was noticed by McKay that the number 196884 exceeds by only one
the dimension of the minunal representation of F1. Conway and Norton
[3] conjectured that there is a natural graded representation of F] with the
character (4.21) minus 24. First Garland [12] and Kac [17] independently
tried to construct i7i in a space isomorphic to ^ . The first problem was
to obtain a representation of one important subgroup C=2^+l
' ・(・0)/±1, where -0 is the automorphism group of the Leech lattice. It is
easy to construct another group C' = 224 ・ (-0) (= (2M+1/±1) ・ (-0)).
Using one observation of Griess, Kac [18] succeeded in passing from C'
to C. The last question is: Where is the whole group F\1 Recently,
important progress has been made in answer to this question [10].
Turning again to the dual resonance models gives a hint as to the answer.
Physicists know that m the contmuous version of F; ^ the obvious action
of the group 0(24) can be extended to the bigger group 0(25). This
extension becomes apparent only if we return to the bigger space V1+.
Whether this unusual phenomenon corresponds to the extension of C to
F1 will become clear in the future.
(引用終り)
以上