See also: J.H. Conway, "A simple construction for the Fischer-Griess monster group", Inventiones Mathematicae, 1985. Sections 8-14 are a progression, where first it is observed that: 98280 + 4096*24 + (1+2+3+...+24) = 98280 + 98304 + 300 = 196884 . and that this is a permutation representation of the monster group, minimal in some sense. By another general theorem, there exists a reduction of the reducible permutation rep. by one to the irrep of dimension 196884 - 1 = 196883. If you want to understand the underlying geometry of the Monster group, I would suggest that you don't worry about the numberphile video. Many of these videos I like, but the idea that you can go from triangular symmetry to the monster group in one jump is preposterous. Then again I don't think Conway's paper is a good place to start either. Here is a fun "middle-ground" problem for anyone who doesn't know representation theory: For each of the regular polyhedra: Tetrahedron, octahedron, Icosahedron, and for each of their irreducible representations as listed on the rows of the corresponding character tables, find a permutation representation on some combination of faces, edges, vertices, or subsets therein, such that the permutation representation P is the direct sum of the irreducible representation R and the trival representation 1, i.e. P = R(+)1. Again, I'm not an expert on the monster group, but it looks like this is the sort of construction that Conway was using to get to the smallest faithful representation. The structure in dim. 196883, while minimal, will be impossible to understand relative to the structure in dim. 196884. Is this why I.M. wanted to publish the Conway paper? Just now, I don't know. --Brad On Wed, Apr 15, 2020 at 5:35 PM Brad Klee <bradklee@gmail.com> wrote:
I haven’t studied monster in detail, but consider that if R is a minimum dimension representation, it must be irreducible. Thus we need more to prove that the dimension of an Irreducible representation divides the group order. I am fairly sure that this is a general feature of any character table, and that a proof of the divisibility theorem would be in most textbooks on Representation theory. But it’s been a while since I cared at all, and possibly I am mistaken. See also:
https://math.stackexchange.com/questions/243221/proofs-that-the-degree-of-an...
—Brad
On Apr 15, 2020, at 5:14 PM, Allan Wechsler <acwacw@gmail.com> wrote:
If I understand this correctly, the Monster is the symmetry group of some complicated object in a 196883-dimensional space.
Is it a coincidence that the dimension of the space divides the order of the Monster? Probably not. Is there an elementary explanation of that?
On Wed, Apr 15, 2020 at 4:12 PM Bill Gosper <billgosper@gmail.com> wrote:
I quickly found the 59 because 883 - 3・196 = = 883 - 600 +12 = 5⨉59,
but then I spazzed dividing it out, and erroneously concluded that it was
an error. Then I slogged thru primes until 47⨉4189. (It's becoming
infrequent that I am lucid enough to do this.) Then I cheated slightly by
opening my eyes and asking Mathematica if 4189 was prime. False?
Damn, that 59 was right after all!
Coincidentally, the final factor, 71, is the algebraic
degree of Conway's Constant, the asymptotic growth rate of the
Look-Say Sequence <https://www.youtube.com/watch?v=ea7lJkEhytA>.
Since JHC was a notorious mental factorer, he must have attacked
196883 as soon as it was revealed as the "dimension" of the
Monster Group <https://www.youtube.com/watch?v=jsSeoGpiWsw>. (After his
stroke, he admitted to Rich that he was
down to 4-digit numbers.) —rwg
In[1545]:= FiniteGroupData["Monster","Order"]
Out[1545]= 808017424794512875886459904961710757005754368000000000
In[1546]:= FI@%
Out[1546]= 2⁴⁶ 3²⁰ 5⁹ 7⁶ 11² 13³ 17 19 23 29 31 41 47 59 71
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