Explanation for E_8's torsion - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T04:35:19Z http://mathoverflow.net/feeds/question/3700 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/3700/explanation-for-e-8s-torsion Explanation for E_8's torsion Ilya Nikokoshev 2009-11-01T20:06:22Z 2012-12-18T01:27:17Z <p>To study the topology of Lie groups, you can decompose them into the simple compact ones, plus some additional steps, such as taking the cover if necessary. After that, the structure of $SO(n)$'s is rather straightforward, but the exceptional groups are more interesting.</p> <p>Any simple compact Lie group, by means of Hopf algebra theory, has the <em>rational</em> homology of a product $$S^a \times S^b \times \dots \times S^z$$ where the numbers are called <strong>exponents</strong>. Other than that, their cohomology could also have <strong>torsion</strong>. Now the torsion for all groups is known:</p> <ul> <li>Among classical groups, only 2-torsion is possible and only for $Spin(n)$</li> <li>Exceptional groups can only have 2 and 3-torsion (most do), with the exception of:</li> <li>$E_8$ which has 2-, 3-, and 5- torsion.</li> </ul> <p>Well, this is <em>bound</em> to be related to $E_8$'s Coxeter number, which is 30, but are there any hints as to why? My reference would be <a href="http://arxiv.org/abs/math-ph/0212067" rel="nofollow">math-ph/0212067</a> but it can't relate this to Coxeter number either. </p> <p>For the reference, <em>exponents</em> are known to be related to Coxeter number, see Kostant, <em><a href="http://www.google.com/search?hl=en&amp;safe=off&amp;client=safari&amp;rls=en&amp;q=The+principal+three-dimensional+subgroup+and+the+Betti+Numbers+of+a+Complex+Simple+Lie+Group&amp;aq=f&amp;oq=&amp;aqi=" rel="nofollow">The Principal Three-Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group</a></em> (google search).</p> <p>Is this an open problem? Maybe yes, but maybe it's been explained, so I'm posting it as it is for now.</p> http://mathoverflow.net/questions/3700/explanation-for-e-8s-torsion/3803#3803 Answer by Will Orrick for Explanation for E_8's torsion Will Orrick 2009-11-02T15:28:51Z 2009-11-02T15:28:51Z <p>I don't know the answer to your question, so the following may simply be a repackaging of the mystery, or may be wholly related, and at any rate, is probably already well-known to you. The numbers 2, 3, 5 remind one of the symmetries of the icosahedron, which is related to E<sub>8</sub> by the McKay correspondence (see <a href="http://math.ucr.edu/home/baez//ADE.html" rel="nofollow">http://math.ucr.edu/home/baez//ADE.html</a>). The ADE Dynkin diagrams are related to finite subgroups of SU(2) which include cyclic groups, dihedral groups, and the three exceptional symmetries: tetrahedral, octahedral, and icosahedral.</p> http://mathoverflow.net/questions/3700/explanation-for-e-8s-torsion/3837#3837 Answer by B S for Explanation for E_8's torsion B S 2009-11-02T19:01:06Z 2009-11-02T19:01:06Z <p>JP Serre, in his june 1999 Bourbaki talk "Sous-groupes finis des groupes de Lie", gives the following two references for torsion in Lie groups</p> <p>R. STEINBERG - Torsion in reductive groups, Adv. in Math. 15 (1975), 63-92</p> <p>and</p> <p>A. BOREL - Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes, T^ohoku Math. J. 13 (1961), 216-240.</p> <p>Btw, the Bourbaki talk is on Serre's College de France page </p> <p><a href="http://www.college-de-france.fr/media/ins%5Fpro/UPL61366%5FSerre%5FBourbaki%5F864.pdf" rel="nofollow">http://www.college-de-france.fr/media/ins_pro/UPL61366_Serre_Bourbaki_864.pdf</a></p> <p>Hope this helps.</p> http://mathoverflow.net/questions/3700/explanation-for-e-8s-torsion/3921#3921 Answer by Pasha Zusmanovich for Explanation for E_8's torsion Pasha Zusmanovich 2009-11-03T11:14:23Z 2009-11-03T11:14:23Z <p>To throw in a bit more numerology from another Serre's Bourbaki seminar: Cohomologie galoisienne : progrès et problèmes. Séminaire Bourbaki, 36 (1993-1994), Exposé No. 783, 29 p. [<a href="http://www.numdam.org/item?id=SB_1993-1994__36__229_0" rel="nofollow">available at numdam</a>]. In \S 2.2 he refers to torsions related to Lie groups in two senses, as far as I understand, the second one is related to group of automorphisms of the completed Dynkin diagram.</p> http://mathoverflow.net/questions/3700/explanation-for-e-8s-torsion/3946#3946 Answer by B S for Explanation for E_8's torsion B S 2009-11-03T15:06:15Z 2009-11-03T15:06:15Z <p>Sorry, I forgot to mention in my answer that only \S 1.3 in Serre 99 is about torsion in Lie or algebraic groups in zero characteristic (which doesn't prevent him to try embed some finite groups of Lie Type into them !).</p> <p>PS: didn't found how to append this to your answer to my answer.</p> http://mathoverflow.net/questions/3700/explanation-for-e-8s-torsion/5306#5306 Answer by Jason DeVito for Explanation for E_8's torsion Jason DeVito 2009-11-13T02:58:23Z 2009-11-13T02:58:23Z <p>This doesn't directly address your question, but it does give you a way of thinking about torsion in the cohomology of Lie groups in general.</p> <p>(This is all coming from Borel and Serre's Sur certains sous-groupes des groupes de Lie, which can be found in Commentarii mathematici Helvetici Volume 27, 1953)</p> <p>As you mentioned above, every compact lie group is rationally a product of odd spheres. But how many odd spheres? Turns out, if G is compact and rank k, then it is rationally a product of k spheres (of various dimensions).</p> <p>There is an analogous result for torsion. That is, one can define the 2-group of G to be any subgroup which is isomorphic to $(\mathbb{Z}/2\mathbb{Z})^n$ or some n. One defines the 2-rank of a group as the maximal $n$ of any 2-group in G. (On can show that for connected $G$, the 2-rank is bounded by twice the rank, and is thus finite).</p> <p>Just to point out something that really threw me when I first learned of these - while the rank is an invariant of the algebra (i.e., all Lie groups with the same algebra have the same rank), the 2-rank of a Group is NOT an invariant of the algebra. For example, the 2-rank of SU(2) is 1 (in fact, -Id is the UNIQUE element of SU(2) of order 2), while the 2-rank of SO(3) is 2 (generated by diag(-1,-1,1) and diag(-1,1,-1) ). The 2-rank of O(3) is 3 (generated by diag(-1,1,1), diag(1,-1,1), and diag(1,1,-1) ).</p> <p>Now, given $T\subseteq G$, the maximal torus, it's clear that simply by taking the maximal 2-group in T, that the 2-rank of G is AT LEAST the rank of G. When is it strictly bigger? Precisely when the group G contains 2-torsion.</p> <p>The analogous result for p-groups and p-torsion (p any prime) also holds.</p> <p>In short, to understand the existence of the 5-torsion in $E_{8}$, one need only understand why there is a subgroup isomorphic to $(\mathbb{Z}/5\mathbb{Z})^n\subseteq E_8$ for some $n\geq 9$.</p> http://mathoverflow.net/questions/3700/explanation-for-e-8s-torsion/116672#116672 Answer by Jeff Harvey for Explanation for E_8's torsion Jeff Harvey 2012-12-18T01:27:17Z 2012-12-18T01:27:17Z <p>Take a look at "Finite H-spaces and Lie Groups" by Frank Adams, particularly the letter from E8 and the appendix which follows it. </p>