most recent 30 from http://mathoverflow.net 2013-05-23T06:44:11Z http://mathoverflow.net/feeds/user/10350 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/108758/an-idea-for-the-study-of-m24-group Paul Hjelmstad 2012-10-03T22:40:55Z 2012-10-03T22:40:55Z

<p>I am looking at the M24 Mathieu Group (basis for the Leech Lattice, Bosonic String Theory in 26 dimensions, and the Monster group).</p> <p>My idea is to look at different Steiner Systems that build up the M24 group, namely, S(5,8,12) which is isomorphic up to relabeling. I am measuring various things about the 759 blocks (which are octads):</p> <ol> <li>Their intervallic content</li> <li>Their translations </li> <li>Their mirror inverses</li> <li>Their isomericity (Two or more sets having the same intervallic content)</li> </ol> <p>Because of the extreme importance of 5,8, and 24 as pure numerical quantities (such as the E8 lattice, the Leech Lattice, octonians, etc). Particulary, in 26 dimensional bosonic string theory we have "Three pieces of 8" fitting into 24, I am thinking of various constructions of the Bi-Monster which is M X M wr 2, and of course string theory considering (I think 10 dimensions on the Left-moving string and 26 on the right - (8 + 2 and 24 + 2). And everything related to this, the Leech lattice orbifolded over a torus (with Z2). </p> <p>So, starting here, does anyone see any value in this type of investigation? So far, I know that there are 511 unique interval vector types, some translations, and some mirror images, and so forth. I want to see what patterns take hold across different labellings (different Steiner systems). For example, in M12, I have found that exactly 2/3 of all the sets are typically hexads which are Z-related (to borrow a term from music theory; i.e. isomeric sets, and always in pairs) With octads in Z24 there are doubles, triples, quads, and hextuplets....in M24 Steiner system there appear to be mostly pairs. I have also analyzed Z24 completely based on Polya necklace theory and related formulae.</p> <p>Another place to go would be looking at complementation, but here we have three-way complementation but with hexads they are always just simple complements, A and A' for example, and such that all isomericity is based on this complementaion (in any Z12 hexad for that matter)</p> <p>Please let me know if you think there is any value in this sort of set-theoretical investigation would be and any possible interesting avenues (or alleys) for this pursuit</p> <p>PGH</p>

http://mathoverflow.net/questions/88873/bosonic-string-theory Paul Hjelmstad 2012-02-19T00:53:14Z 2012-02-19T00:53:14Z

<p>I would like clarification of 26 dimensional Bosonic String Theory. A definition would be, that this is free bosons compactified on a torus and orbifolded by a 2-point reflection group (or asymmetrical Z2 grading), induced by the Leech Lattice.</p> <ol> <li>Is this "asymmetrical" because of the 25,1 signature, where Z2 contributes to 1 space and 1 time dimension</li> <li>Is this the Lorentzian form, which adds 2 dimensions of the worldsheet, 24 + 2, and how is this related to a vertex operator algebra, with Z2 grading, which I still don't understand, how VOAs work in the context of the Monster group (which is the symmetries of the vertex operator algebra of this structure, on an even II(25,1) lattice). I understand about the Leech Lattice, tori, orbifolding, etc. but everything together, is a bit overwhelming to understand. </li> </ol> <p>I may have some terms wrong, but I still haven't found a one paragraph explanation of how all this works, and how it relates to the Monster, Moonshine, VOAs, CFTs, etc. I do know that VOA's are to the Leech Lattice as the Lattice is to the Golay Code (and M24), and that this is analogous to E8 lattice being based on the Hamming code, and in turn, the E8 group being based on the E8 lattice, in similar fashion. Is the above an attempt to create an infinite dimensional structure (The Fake Monster Lie Algebra) with the Simple Monster group which somehow parallels what is going on with E8 and Lie Algebras?</p>

http://mathoverflow.net/questions/71732/bimonster-and-heterotic-string-theory Paul Hjelmstad 2011-07-31T15:58:54Z 2011-11-25T23:10:54Z

<p>A. The Bimonster and the Complex Lorentzian Leech Lattice involves a construction that extends Y555 from 16 to 26, and relates to Incidence(P^2/F3) among other things. (13 projective points + 13 projective lines), Complex Lorentzian Leech Lattice as Hyperbolic cell 2 + 24 dimensions, etc. </p> <p>B. How does this relate to the 26 dimensions of Bosonic String Theory (Where 26 dimensions of the Lorentzian Leech Lattice (IIsub(25,1)) which is a torus in 24 dimensions orbifolded over a 2-point reflection group....and collapses from 26 to 10 dimensions using E8 X E8, one version of heterotic string theory...</p> <p>The confusion lies in that this (B.) is 10 + 16 = 26 (or 26 dimensions collapsed down to 10), and the Bimonster situation (A.) is 16 nodes of the Dynkin diagram Y555 (or M666) of M X M wreath built up (with some difficulties) to 26 nodes of Inc(P2/F3) if I understand it right. </p> <ol> <li>Is this 16 homomorphic (or related) to E8 X E8 in string theory?</li> <li>Is the 2-point reflection group in bosonic string theory the same as the hyperbolic cell discussed in the Bimonster construction? Z2 in E8 X E8 is apparently the identification of the common boundaries of E8 X E8... I am fascinated by both A and B and would like more clarification of their relationships.</li> </ol> <p>Thanks</p> <p>PGH</p>

http://mathoverflow.net/questions/62084/m12-simple-sporadic-group/71733#71733 Paul Hjelmstad 2011-07-31T16:33:24Z 2011-07-31T16:33:24Z

<p>I studied how M12 is built up from M9, in stages, and also got proficient with the tetracodeword construction. An interesting fact is that the Ternary Golay Code and the tetracodeword construction in SPLAG for S(5,6,12) (doubly = 264) are not so easily related as the Binary Golay Code and the hexacodeword construction for S(5,8,24), which is more direct. This of course relates to the MiniMOG and the MOG, respectively. </p>

http://mathoverflow.net/questions/62084/m12-simple-sporadic-group Paul Hjelmstad 2011-04-18T04:18:06Z 2011-07-31T16:33:24Z

<p>This is in regards to Chapter 11 of SPLAG. The tetracode construction of M12 is based on col-col, col+tet, tet-tet, col+col-tet, which are 6 + 36 + 36 + 54 = 132. (Unsigned hexads in the C12 code, of the Ternary Golay Code). Now I noticed the coincidence, that the cols are the inverse of the tets in the C12 code, and also S3 in C4 X S3 is the inverse of D4 in D4 X C3, in terms of generators, that is, S3 applied to a set will have the inverse effect of D4 applied there. (And cols=3 elements while tets=4 elements). Perhaps just a coincidence---</p> <p>Anyway, 12P5 = 95040 and this is also the order of M12, the stabilizer of a S(5,6,12) Steiner system, with the tetracode construction being one possible construction (with various labellings possible). Now I know that it is also a stabilizer of the C12 code, and that M12 is quintuply sharply transitive on these sets. It sends blocks to blocks in S(5,6,12), My question being, how a single g (member of) M12 accomplishes this, does each permutation of 12P5, (which sends every pentad to every possible pentad) correspond to a g, so that there are 95040 elements, acting regularly transitive on the Steiner system, and/or C12..</p> <p>The best example would be the hexads (264) of the Ternary Golay Code, constructed with the Tetracode, how does this stabilize C12?</p> <p>Thanks </p>

http://mathoverflow.net/questions/59329/m12-simple-sporadic-group Paul Hjelmstad 2011-03-23T17:13:07Z 2011-03-23T20:49:35Z

<p>I've spent quite a bit of time studying the Mathieu Groups, and I own the ATLAS.</p> <p>My question is about M12. It is based on the ternary Golay Code, and is the automorphism group of a Steiner S(5,6,12) system. Now, all of these Steiner systems are isomorphic up to labelling. The order of M12 is 95040, which is 132 x 720. Since there are 132 blocks in this Steiner system, one can see that the 720 or S6 piece merely scrambles the six elements of the hexad. Then, the 132 part is just sending the elements of one hexad to another, of which there are 132 ways to do this. </p> <p>Can someone give me an intuitive construction for this, not just generators...would it make sense to say that the (sharply) quintuple transitive action might be to send block 1 to 2, and 2 to 3, and perhaps another action to send block 1 to 3, 3 to 5 etc. or something of this nature? Is there a hard and fast way to look at this action (M12) in terms of the blocks? Or was I wrong about the 720 X 132 decomposition of the order of the group...Thanks, Paul.</p>

http://mathoverflow.net/questions/43466/monstrous-moonshine Paul Hjelmstad 2010-10-25T03:28:38Z 2010-12-12T05:30:38Z

<p>Wikipedia claims that the group of units of Z24 (1,5,7,11,13,17,19,23), which all have order 2, and are isomorphic to (Z/2Z)^3 have an important connection to Monstrous Moonshine theory, however, I cannot find any other reference besides Wikipedia that claims this --- It was recommended on sci.math that I pose this question here.</p> <p>Perhaps it's a mistake? And he meant that the primes of the Monster, which continue to 71, are what are considered in Moonshine.</p> <p>Paul Hjelmstad, B.M, B.A.</p> <p>[<b>Edit</b> (PLC): Here is the relevant passage from wikipedia:]</p> <blockquote> <p>24 is the highest number $n$ with the property that every element of the group of units $(\mathbb{Z}/n\mathbb{Z})^{\times}$ of the commutative ring $\mathbb{Z}/n\mathbb{Z}$, apart from the identity element, has order $2$; thus the multiplicative group <code>$(\mathbb{Z}/24\mathbb{Z})^{\times} = \{1,5,7,11,13,17,19,23\}$</code> is isomorphic to the additive group $(\mathbb{Z}/2\mathbb{Z})^3$. This fact plays a role in monstrous moonshine.</p> </blockquote>

http://mathoverflow.net/questions/46696/complex-lorentzian-leech-lattice-and-the-bimonster Paul Hjelmstad 2010-11-20T00:52:17Z 2010-11-20T00:52:17Z

<p>I'm reading an excellent paper on the complex Lorentzian Leech Lattice and the bimonster (Tathagata Basak). Instead of using the binary Golay Code, the author uses the ternary Golay code and the complex (lorentzian) leech lattice.</p> <p>Many references in the paper are to SPLAG, however, I am having trouble finding (2) on page 6 of the paper in SPLAG. When I master LaTex I will post it here. Basically, it defines the ternary Golay code over the Eisenstein integers (Z(exp(2pi*i/3), the Complex Lorentzian Leech Lattice as vectors over E^12, and C12, the ternary Golay code in F3^12. I've gotten pretty good with the MOG and the MINIMOG, so I grasp the use of the code, just not in this context. </p> <p>And then, of course, how it all leads to the Bimonster, and the Inc(P^2(F3)) with 26 nodes. Also reading Conway's (26 Implies the Bimonster) which also discusses the 13 line 13 point projective plane of order 3, in the same context. And deflating the 12-gons, etc. </p> <p>But for today, just want to make sense of this use of the TGC in conjunction with the CLLL:</p> <p>The complex Leech Lattice ^ consists of the set of vectors in E^12 of the form</p> <p>{(m + theta*c(sub i) + 3z(sub i))i=1...12 : m = 0, or +/- 1, (c(sub i) member of C12, </p> <p>Sigma z(sub i) =cong= m mod theta} where </p> <p>E = the ring of Z[exp(2 pi*i/3)] of Eisenstein integers C12 = Ternary Golay Code over F(sub 3)^12 and Theta = sqrt(-3) or w - w bar where w = exp(2*pi*i)</p> <p>If someone could help my tie in my (better) understanding of the TGC with this more novel use of a Lattice over the Eisenstein integers, and new concepts of Weyl vectors, Coxeter-Dynkins Diagrams, and the rest, (I've read the whole paper) it would be greatly appreciated. But I would be happy just to understand the formula above, which is in SPLAG, I just cannot find it.</p> <p>PGH</p>

http://mathoverflow.net/questions/45376/z-48-and-moonshine-beyond-the-monster Paul Hjelmstad 2010-11-08T22:52:23Z 2010-11-09T02:26:18Z

<p>I am interested in pursuing an understanding of K-theory. Primarily, the $K_3(\mathbb{Z})$ algebraic K-group over ring of integers of an algebraic number field and its relationship to the $\mathbb{Z}/48$ ring of integers modulo 48. </p> <p>This is (of course), again, from Terry Gannon's "Moonshine Beyond the Monster" where he talks about many amazing coincidences with the number 24, the Riemann Zeta Function $\Sigma_{n=1}^\infty (1/n)^{-1} = -1/12$, Apery's constant, where $\Sigma_{n=1}^\infty (1/n)^2 = \pi/6$ (which he states are both synonomous in their relationship to $K_3(\mathbb{Z})\leftrightarrow \mathbb{Z}/48$....)</p> <p>A little harder to discern is the (possible) relationship of the Bimonster, $(M \times M) \rtimes \mathbb{Z}/2 \to M \wr 2$, and the Incidence Graph of the M-13 pseudogroup with 13 points and lines, (the 13 point, 13 line projective plane, where here, the coincidence would appear to be the number 26, which is the dimension of Bosonic String Theory (2 + 24 dimensions, the quantum harmonic oscillator on a 2-dimbrane, which relates to -1/12 above per John Baez "My Favorite Number is 24")). It's tempting to see the resemblance of 24 relating to the Monster, and 48 to the BiMonster, but that seems to obvious. Finally, is there any relevance in bringing in the M12-Mathieu group here, being so close to the M13-pseudogroup? I apologize ahead of time if this last paragraph is "shooting the moon" but hopefully my first two paragraphs are well-stated questions.</p>

http://mathoverflow.net/questions/118399/moonshine-and-the-riemann-zeta-hypothesis-or-function Paul Hjelmstad 2013-01-09T21:37:19Z 2013-01-09T21:37:19Z

Well, it's closed, but response to the first response, diametrically opposed things, can relate, but i get the point, and in response to the second response, I was referring to Freeman Dyson's discovery relating the energy levels of the nucleus with the statistical moments, not the original formulation of RZH (anyway this is regarding the function not the hypothesis) and in response to the third response, I will reformulate based on this, this is the core of my inquiry

http://mathoverflow.net/questions/18568/leech-lattice-decomposition/18626#18626 Paul Hjelmstad 2011-12-31T22:24:24Z 2011-12-31T22:24:24Z

I see you are also posting on the GAP forum. I would be interested in working with this too, since I also work with the GAP software. Could you possibly bring me up to speed with respect to the 4095 crosses in the Conway group construction, is this like the 54 crosses of Curtis's Kitten for M12 (or the related MOG construction for M24?) Thanks PGH

http://mathoverflow.net/questions/71732/bimonster-and-heterotic-string-theory Paul Hjelmstad 2011-11-26T19:53:25Z 2011-11-26T19:53:25Z

Yes, I am trying to get my ducks in a row here and supply all the correct definitions, framework, and context. @Percy Paul, please send me your email in your next response, I wasn't able to find a way to contact you. Thanks so much. Paul.

http://mathoverflow.net/questions/71732/bimonster-and-heterotic-string-theory/73146#73146 Paul Hjelmstad 2011-09-03T23:50:04Z 2011-09-03T23:50:04Z

The Monstrous Branes paper is very good also. However, I am stuck on something, I cannot find any definition of the S-modular transform. A question for Dr. Harvey (coauthor of the paper) or whomever has the answer. I need to also go over quite a bit of string theory and brane theory but answering this question would be very helpful. I imagine it relates to S-duality, but I am not sure pgh. PS The short paper &quot;An Elementary Approach to the Monster&quot; by Christopher Simons, and &quot;26 Implies the Bimonster' By J. H. Conway proved to be helpful introductory papers for tackling these longer ones.

http://mathoverflow.net/questions/71732/bimonster-and-heterotic-string-theory/73146#73146 Paul Hjelmstad 2011-08-19T17:21:32Z 2011-08-19T17:21:32Z

I've read the papers Dr. Borcherds cited. The Miyamoto paper is interesting in that by removing 4 points and 1 line from the 13 points and lines of the Bimonster (Inc(P^2/F3) you obtain a representation of the Monster, in the 21 involutions of the Affine Plane (3), which has 9 points and 12 lines. I don't understand all of it yet, but apparently these 4 points are on the part of the Bimonster that is identified by the Z2 portion of M X M wr 2 (The wreath product of M with itself). The Monster acts on the LL, which is E8 three times Thanks for the citation - I will try to get to it today. pgh.

http://mathoverflow.net/questions/71732/bimonster-and-heterotic-string-theory/71751#71751 Paul Hjelmstad 2011-07-31T23:27:48Z 2011-07-31T23:27:48Z

Thanks. I've read the two Basak papers on the Bimonster, I will download the Miyamoto paper, which I may have read but I don't recall it. Note: I am not a string theorist nor an expert on this subject either. I can try to make my posts more accessible, but part of that is due to the limitations of my own understanding on the subject matter. PGH

http://mathoverflow.net/questions/59329/m12-simple-sporadic-group Paul Hjelmstad 2011-03-23T21:19:29Z 2011-03-23T21:19:29Z

These are fun! Thanks. The 24 puzzle reminds me alot of the M13-game of John Conway, which also uses colored circles (26 of them). PGH

http://mathoverflow.net/questions/59329/m12-simple-sporadic-group/59335#59335 Paul Hjelmstad 2011-03-23T20:25:49Z 2011-03-23T20:25:49Z

Actually I have read that (and other chapters of SPLAG, which I own). hexacodeword construction is fun with 1,0,w,w-bar. I should probably be able to answer my own question with SPLAG, I just hoped perhaps there was a more direct method. PGH