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I repeatedly heard that K(F_1) is the sphere spectrum. Does anyone know about the proof and what that means?

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In case you're not aware of this feature, you can change which answer you accepted, shall you decide to. –  Ilya Nikokoshev Oct 24 '09 at 16:50
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Let's be serious. Anything may arise from the heuristics on the field with one element. It's numerology. I don't think anyone has real hope to prove the Riemann hypothesis from these heuristics. –  Fernando Muro Jun 23 '11 at 17:25
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3 Answers

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I understand that this is because GLn(F1) is supposed to be Sigman, the symmetric group on n letters. Thus K(F1) = K(finite sets) which is the sphere spectrum by the Barratt-Priddy-Quillen-Segal theorem.

But I have no idea why GLn(F1) should be Sigman...

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Yes, taking GL _n(F _1) to be Sigma _n one can make sense both of the Q- and the +-construction and both yield the same answer as shown by Deitmar in http://arxiv.org/abs/math/0605429.

GL _n(F _1)=Sigma _n is suggested by several observations. One is that counting formulas for subspaces of n- dimensional vector spaces over F _q turn into counting formulas for subsets of n-element sets, if one sets q=1. So one could say that an n-dimensional vector space over F_1 is an n-element set and GL _n(F _1)=Aut(F _1^n)=Sigma _n. See Cohn's very nicely written http://arxiv.org/abs/math/0407093 for this.

One gets another hint by looking at the Tits building for GL _n(F _q) (that is a simplicial complex where the group acts). There is a natural limit for q going to one 1 - what then remains is the so-called chamber of the building and the symmetry group of that is Sigma _n.

Further hints that one should just drop addition (in comparison to the usual notion of module) come from arithmetic geometry, but that is maybe less convincing and a longer story...

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I don't have a direct reference to this, but if you compare representation theory and the theory of group actions on finite sets, it seems to me that you can see more evidence that the correct way of thinking about F_1 involves forgetting about addition. –  Harrison Brown Oct 21 '09 at 17:58
    
Is GLn(F1) = Sigma_n right? The cardinality doesn't work out when I take the q->1 limit of |GLn(Fq)| (off by a factor of (q-1)^n). –  Reid Barton Oct 21 '09 at 18:10
    
@harrison: Yes, burnside ring and representation ring, marks and characters, etc. behave very similarly. Even some proofs carry over, e.g. Steinberg adapted a proof of Frobenius about the irreducibility of some permutation representation to the linear representation analogon (I saw this in a lecture by C. Soule, Nashville 2009) –  Peter Arndt Oct 24 '09 at 13:41
    
@Reid:What one usually does to count F1-points, when the number of points of a variety is given by a polynomial (which is not unusual), is to "multiply away" the zeros at q=1 and then insert 1. This then is consistent with the symmetry group of the chamber view point - for any Chevalley group not only GLn. –  Peter Arndt Oct 24 '09 at 13:42
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Here is another heuristic, related to what Randal-Williams said above. The sphere spectrum is the unit object in nice categories of spectra. That is, ring spectra are algebras over the sphere spectrum. Now, to every scheme X you can associate a K-theory ring spectrum K(X), and this is contravariant. So, in the usual theory there is a morphism K(Z)->K(X) for all schemes X. So, finding F_1 also means finding something (its K-theory spectrum) that maps to the (homotopy) limit of all K-theory spectra. That this should be the unit object of the category of spectra doesn't seem very surprising.

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Thanks. I just looked where the idea comes from: Manin mentions it in "Lectures on zeta functions and motives (according to Deninger and Kurokawa)" shortly and refers to Priddy "Transfer, symmetric groups, and stable homotopy theory", fitting to the first answer. –  Thomas Riepe Oct 25 '09 at 10:25
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