Comments on: the monster graph and McKay’s observation http://www.neverendingbooks.org/index.php/the-monster-graph-and-mckays-observation.html lieven le bruyn's blog Fri, 20 Jan 2012 16:50:41 +0100 hourly 1 http://wordpress.org/?v=3.3.1 By: John McKay http://www.neverendingbooks.org/index.php/the-monster-graph-and-mckays-observation.html/comment-page-1#comment-8041 John McKay Sun, 26 Apr 2009 01:08:51 +0000 http://www.neverendingbooks.org/?p=1541#comment-8041 <p>I prefer to work with the unfolded Dynkins since the highest root coefficients are not well-defined when folded because of the exchange of long and short roots. For E7 there are two F4's and for E6 there are two G2's. I work mod the centre of the sporadics extended by their centres.</p> I prefer to work with the unfolded Dynkins since the
highest root coefficients are not well-defined when folded because of the exchange of long and short roots. For E7 there are two F4′s and for E6 there are two G2′s. I work mod the centre of the sporadics extended by their centres.

]]> By: Daniel de França MTd2 http://www.neverendingbooks.org/index.php/the-monster-graph-and-mckays-observation.html/comment-page-1#comment-8039 Daniel de França MTd2 Sat, 25 Apr 2009 04:28:24 +0000 http://www.neverendingbooks.org/?p=1541#comment-8039 <p>I don't know about this observation. But what you are saying look similar to the derivation of octonions and the exceptionals F4, G2 and E8 from the symmetries of Spin(8).</p> <p>Week 90 of John Baez' "This Week's Finds in Mathematical Physics":</p> <p>http://math.ucr.edu/home/baez/week90.html</p> <p>Another, thing it is that although E(7) is not an exptional group, sometimes it insists on trying to become one. Like the Freudenthal magic square and in that MacKay observation.</p> I don’t know about this observation. But what you are saying look similar to the derivation of octonions and the exceptionals F4, G2 and E8 from the symmetries of Spin(8).

Week 90 of John Baez’ “This Week’s Finds in Mathematical Physics”:

http://math.ucr.edu/home/baez/week90.html

Another, thing it is that although E(7) is not an exptional group, sometimes it insists on trying to become one. Like the Freudenthal magic square and in that MacKay observation.

]]> By: lieven http://www.neverendingbooks.org/index.php/the-monster-graph-and-mckays-observation.html/comment-page-1#comment-8037 lieven Fri, 24 Apr 2009 20:40:17 +0000 http://www.neverendingbooks.org/?p=1541#comment-8037 <p>Daniel, </p> <p>about (3) : yes i do know of this vague and overly optimistic idea. in fact, i made [an attempt]( http://www.neverendingbooks.org/index.php/iguanodon-series-of-simple-groups.html) to put some of the easiest sporadics M(12) and M(24) into a series of simple groups and ive heard a talk by Bruce Westbury on attempts to do similar things with the exceptional Lie algebras (the Vogel plane and stuff like that if i remember well...)</p> <p>about (4) : i admit to be confused on these matters. i know of one text saying that McKay's E(7)-observation (about 2.B) is really about the double folding of extended E(7) giving F(4) and his E(6)-observation is on the triple folding giving G(2). i'd love to know more about this.</p> Daniel,

about (3) : yes i do know of this vague and overly optimistic idea. in fact, i made [an attempt](
http://www.neverendingbooks.org/index.php/iguanodon-series-of-simple-groups.html) to put some of the easiest sporadics M(12) and M(24) into a series of simple groups and ive heard a talk by Bruce Westbury on attempts to do similar things with the exceptional Lie algebras (the Vogel plane and stuff like that if i remember well…)

about (4) : i admit to be confused on these matters. i know of one text saying that McKay’s E(7)-observation (about 2.B) is really about the double folding of extended E(7) giving F(4) and his E(6)-observation is on the triple folding giving G(2). i’d love to know more about this.

]]> By: Daniel de França MTd2 http://www.neverendingbooks.org/index.php/the-monster-graph-and-mckays-observation.html/comment-page-1#comment-8036 Daniel de França MTd2 Fri, 24 Apr 2009 19:17:53 +0000 http://www.neverendingbooks.org/?p=1541#comment-8036 <p>"McKay’s observation goes beyond E(8) and covers also (extended) E(6) and E(7) which correspond resp. to 3.Fi(24) and 2.B." </p> <p>So, there is nothing left for G2 and F4? Why not?</p> “McKay’s observation goes beyond E(8) and covers also (extended) E(6) and E(7) which correspond resp. to 3.Fi(24) and 2.B.”

So, there is nothing left for G2 and F4? Why not?

]]> By: Daniel de França MTd2 http://www.neverendingbooks.org/index.php/the-monster-graph-and-mckays-observation.html/comment-page-1#comment-8035 Daniel de França MTd2 Fri, 24 Apr 2009 19:15:52 +0000 http://www.neverendingbooks.org/?p=1541#comment-8035 <p>My second question was not mine, but an adaptaion from elsewhere, that is, from Israel Gelfand. This is how David Corfield summarized at n-category cafe:</p> <p>"See what the claim is? Finite simple sporadic groups do not form a natural kind. First, as they are currently defined they exclude close relations which are non-sporadic, i.e., which appear in the infinite families of the classification. Second, even with these additional groups they are part of a much larger kind, the ones appearing in the classification just ‘happening’ to be groups."</p> <p>http://golem.ph.utexas.edu/category/2006/09/mathematical_kinds.html</p> <p>It's nice to read all the post.</p> My second question was not mine, but an adaptaion from elsewhere, that is, from Israel Gelfand. This is how David Corfield summarized at n-category cafe:

“See what the claim is? Finite simple sporadic groups do not form a natural kind. First, as they are currently defined they exclude close relations which are non-sporadic, i.e., which appear in the infinite families of the classification. Second, even with these additional groups they are part of a much larger kind, the ones appearing in the classification just ‘happening’ to be groups.”

http://golem.ph.utexas.edu/category/2006/09/mathematical_kinds.html

It’s nice to read all the post.

]]> By: lieven http://www.neverendingbooks.org/index.php/the-monster-graph-and-mckays-observation.html/comment-page-1#comment-8033 lieven Fri, 24 Apr 2009 16:14:27 +0000 http://www.neverendingbooks.org/?p=1541#comment-8033 <p>McKay's observation goes beyond E(8) and covers also (extended) E(6) and E(7) which correspond resp. to 3.Fi(24)' and 2.B. The cute things being that the order of the extension one needs to take of these sporadics is the same as the order of the corresponding extended Dynkin diagram.</p> <p>As to your final paragraph : no idea!</p> McKay’s observation goes beyond E(8) and covers also (extended) E(6) and E(7) which correspond resp. to 3.Fi(24)’ and 2.B. The cute things being that the order of the extension one needs to take of these sporadics is the same as the order of the corresponding extended Dynkin diagram.

As to your final paragraph : no idea!

]]> By: Daniel de França MTd2 http://www.neverendingbooks.org/index.php/the-monster-graph-and-mckays-observation.html/comment-page-1#comment-8030 Daniel de França MTd2 Thu, 23 Apr 2009 19:17:58 +0000 http://www.neverendingbooks.org/?p=1541#comment-8030 <p>Is it only a coincidence the fact that E8 and the monster are the largest in their own classifications, that is, in the exceptional and sporadic gropus and the fact that the nodes of the Dynkin diagram of E8 fits the conjugacy classes of the Monster?</p> <p>So, we have this kind of unique structures among simple Lie groups, which do not follow an infinite sequence, as well we can find similar, the sporadics, in the broader definition, the simple groups. Following this pattern, is there any unique structure, contained in an broader definition than the simple groups? Did anyone look for it in the nodes of the monster dynkin diagram in the same way one finds the conjugacy classes in the E8 diagram?</p> Is it only a coincidence the fact that E8 and the monster are the largest in their own classifications, that is, in the exceptional and sporadic gropus and the fact that the nodes of the Dynkin diagram of E8 fits the conjugacy classes of the Monster?

So, we have this kind of unique structures among simple Lie groups, which do not follow an infinite sequence, as well we can find similar, the sporadics, in the broader definition, the simple groups. Following this pattern, is there any unique structure, contained in an broader definition than the simple groups? Did anyone look for it in the nodes of the monster dynkin diagram in the same way one finds the conjugacy classes in the E8 diagram?

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