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 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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