[math-fun] I factored 196883 in bed with my eyes closed. Sort of.
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
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
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
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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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math-fun@mailman.xmission.com
https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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The Monster group doesn't have a permutation representation on fewer than 97239461142009186000 elements. But yes, you're correct that there's a particularly nice representation acting on real 196884-dimensional space (the automorphism group of the Griess algebra, or equivalently the representation constructed by Conway in the penultimate chapter of SPLAG, 2nd Edition). There's a line that's fixed by every element of the Monster group, so the ambient 196884-dimensional space splits as the direct sum of a 1-dimensional space (the aforementioned line) on which the Monster acts trivially, and a 196883-dimensional space (the orthogonal complement) on which the Monster acts faithfully. My favourite construction, also by Conway (and based on work by Ivanov and Norton), is as follows: -- take the (26-vertex) incidence graph G of the projective plane over the field of three elements; -- interpret this graph as a Coxeter-Dynkin diagram to get some infinite Coxeter group; -- for each 12-cycle that's an induced subgraph of G, the corresponding Coxeter subgroup is the symmetries of the lattice A^11. Add relations to quotient out by the translations in that Coxeter subgroup; -- the resulting quotient group is a finite group isomorphic to the wreath product of the Monster with the finite simple group of order 2. (The Monster has no small representations, but it does have small presentations.) Best wishes, Adam P. Goucher
Sent: Thursday, April 16, 2020 at 12:34 AM From: "Brad Klee" <bradklee@gmail.com> To: "math-fun" <math-fun@mailman.xmission.com> Subject: Re: [math-fun] I factored 196883 in bed with my eyes closed. Sort of.
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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https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
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Hi Adam, Thanks for correcting me. When skimming through a Conway paper, there is not too much hope to get it all right. It's funny to me that: "The Monster group doesn't have a permutation representation on fewer than 97239461142009186000 elements." Wow, how horrible. From A001379, it looks like there are 36 representations with smaller degree, too many to try and brute force the decomposition. "There's a line that's fixed by every element of the Monster group..." For permutation representations the fixed line can always be written (t,t,...,t), with parameter t. For representations 24, 4096, and 98280, is the fixed line comparatively simple to write? If so, how is it written? Also: Conway's comment about Christmas tree ornaments was at first incomprehensible to me, because I don't know that much about representation theory of Christmas tree ornaments. Maybe he is talking about an n-dimensional molecule? It would be nice to find some solid geometry with a decomposition to k-simplices, such that, for all k<n, the monster acts as permutation within the set of component simplices at dimension k. Has this been resolved, or is it Conway's "proof that never materialized"? --Brad On Thu, Apr 16, 2020 at 4:27 AM Adam P. Goucher <apgoucher@gmx.com> wrote:
The Monster group doesn't have a permutation representation on fewer than 97239461142009186000 elements.
But yes, you're correct that there's a particularly nice representation acting on real 196884-dimensional space (the automorphism group of the Griess algebra, or equivalently the representation constructed by Conway in the penultimate chapter of SPLAG, 2nd Edition).
There's a line that's fixed by every element of the Monster group, so the ambient 196884-dimensional space splits as the direct sum of a 1-dimensional space (the aforementioned line) on which the Monster acts trivially, and a 196883-dimensional space (the orthogonal complement) on which the Monster acts faithfully.
My favourite construction, also by Conway (and based on work by Ivanov and Norton), is as follows:
-- take the (26-vertex) incidence graph G of the projective plane over the field of three elements;
-- interpret this graph as a Coxeter-Dynkin diagram to get some infinite Coxeter group;
-- for each 12-cycle that's an induced subgraph of G, the corresponding Coxeter subgroup is the symmetries of the lattice A^11. Add relations to quotient out by the translations in that Coxeter subgroup;
-- the resulting quotient group is a finite group isomorphic to the wreath product of the Monster with the finite simple group of order 2.
(The Monster has no small representations, but it does have small presentations.)
Best wishes,
Adam P. Goucher
Sent: Thursday, April 16, 2020 at 12:34 AM From: "Brad Klee" <bradklee@gmail.com> To: "math-fun" <math-fun@mailman.xmission.com> Subject: Re: [math-fun] I factored 196883 in bed with my eyes closed. Sort of.
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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participants (4)
-
Adam P. Goucher -
Allan Wechsler -
Bill Gosper -
Brad Klee