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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.

Any simple compact Lie group, by means of Hopf algebra theory, has the rational homology of a product $$S^a \times S^b \times \dots \times S^z$$ where the numbers are called exponents. Other than that, their cohomology could also have torsion. Now the torsion for all groups is known:

  • Among classical groups, only 2-torsion is possible and only for $Spin(n)$
  • Exceptional groups can only have 2 and 3-torsion (most do), with the exception of:
  • $E_8$ which has 2-, 3-, and 5- torsion.

Well, this is bound to be related to $E_8$'s Coxeter number, which is 30, but are there any hints as to why? My reference would be math-ph/0212067 but it can't relate this to Coxeter number either.

For the reference, exponents are known to be related to Coxeter number, see Kostant, The Principal Three-Dimensional Subgroup and the Betti Numbers of a Complex Simple Lie Group (google search).

Is this an open problem? Maybe yes, but maybe it's been explained, so I'm posting it as it is for now.

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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.

(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)

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).

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).

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) ).

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.

The analogous result for p-groups and p-torsion (p any prime) also holds.

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 $n\geq 9$.

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  • $\begingroup$ There is a famous $5^3 \subset E_8$ maximal abelian, in particular not contained in any torus, and it is the unique 5-subgroup which is maximal abelian. What is this group $5^9$ that you know? $\endgroup$ – Theo Johnson-Freyd Oct 21 '18 at 23:59
  • $\begingroup$ @Theo: I know next to nothing about $E_8$. When I wrote the answer above, I was thinking in the reverse: I have heard (but don't know of a reference!) that $H^\ast(E_8)$ contains $5$-torsion. It then follows from the above result of Borel and Serrie that there is a $5^9\subseteq E_8$. $\endgroup$ – Jason DeVito Oct 22 '18 at 2:09
  • $\begingroup$ IIRC, maximal tori are maximal abelian. So I should probably start with a $5^8 \subset T$, where $T$ is a maximal torus, and then hope to find another $5$ that commutes with $5^8$ although not with the rest of the torus. $\endgroup$ – Theo Johnson-Freyd Oct 22 '18 at 3:56
  • $\begingroup$ So I should do an easier case. You are telling me that there is a $p$-dimensional projective representation of $p^p$ (this is the $PSU(p)$ case). Do I know this representation? I know a projective representation of $p^3$: the only irreducible projective representation (up to outer automorphism) of $p^2$ has dimension $p$, and you can get a third copy of $p$ to act nonprojectively by $p$th roots of unity. $\endgroup$ – Theo Johnson-Freyd Oct 22 '18 at 13:01
  • $\begingroup$ But I'm having trouble representing $p^4$. Any element in $H^2(p^4; U(1)) = \wedge^2(p^4)$ is consistent with a splitting $p^4 = p^2 \times p^2$, with the first acting projectively. So over the first $p^2$, our $p$-dimensional representation is already irreducible. But then the second $p^2$ must act centrally, and all maps $p^2 \to U(1)$ have kernel. $\endgroup$ – Theo Johnson-Freyd Oct 22 '18 at 13:02
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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

R. STEINBERG - Torsion in reductive groups, Adv. in Math. 15 (1975), 63-92

and

A. BOREL - Sous-groupes commutatifs et torsion des groupes de Lie compacts connexes, T^ohoku Math. J. 13 (1961), 216-240.

Btw, the Bourbaki talk is on Serre's College de France page

http://www.college-de-france.fr/media/ins_pro/UPL61366_Serre_Bourbaki_864.pdf

Hope this helps.

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  • $\begingroup$ The last one, at least, seems to be about G(F_q) where G is a simple algebraic group and F_q a field of q elements, which is a very different story, as long as I know! I haven't seen the first two though. $\endgroup$ – Ilya Nikokoshev Nov 2 '09 at 19:34
  • $\begingroup$ 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 !). PS: didn't found how to append this to your answer to my answer. $\endgroup$ – B S Nov 3 '09 at 15:06
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Take a look at "Finite H-spaces and Lie Groups" by Frank Adams, particularly the letter from E8 and the appendix which follows it.

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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 E8 by the McKay correspondence (see http://math.ucr.edu/home/baez//ADE.html). 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.

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  • $\begingroup$ I've heard about McKay, but indeed how to apply it here would be a mystery! $\endgroup$ – Ilya Nikokoshev Nov 2 '09 at 18:48
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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. [available at numdam]. 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.

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