[GAP Forum] "Affine Monster"
Robert Eckert
eckert at math.wayne.edu
Thu Jul 1 17:32:21 BST 2004
In looking over that big long middle-of-the-night post to Stefan, I see
that I didn't really quite around to answering his question, whether I
would use a constructive or non-constructive way to prove existence of the
"Affine Monster", and if constructive what kind of construction.
I would certainly want to construct it. One style would be an "alphabet"
for it, an infinite tuple of Z_2's, Z_3's, Z_5's, Z_7's, Z_11's, Z_19's,
whatever primes it likes, presumably recurring in a repetitive pattern,
some long batch of them over and over, or maybe each batch is longer than
the previous but systematically describable and indexable. And there
would be some pseudo-groups, but except for the final one all of these
would occur in the initial, non-repeating, description of the Finite
Monster (like a decimal expansion of a fraction where the first digits are
not part of the final loop). Each would have an action as it passes any
element from a more leftward Z_p, many of these actions trivial but in
resolving the product each Z_p will have an infinite number of more
rightward places from which it could receive a non-trivial action: thus,
the usual kind of restriction that all but a finite number of the Z_p's
are at the trivial value must be imposed or there is no way the product
could be defined. The element from the rightmost atomic, the capping
pseudo-group, has to pass everything, receiving a back-action and spitting
back a carry each time; the defining rule, I am sure, is very difficult to
express.
Or: the "Minotaur" could be defined in a more permutational style, as
mappings on the infinite set of vertices in the multi-sheeted cover of the
4D-icosahedron (120 vertices in each sheet), doing reflections and
rotations on the hypericosahedron but also moving some vertices up or down
some number of sheets. I think the simple core of this symmetry group is
the "Affine Monster" I am hunting; but maybe the "Minotaur" and "Affine
Monster" are not actually the same thing, just cousins of some kind.
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