Question by John Baez

John Baez made an interesting question in n-Category-Cafe. The question reads as follows:

Is every representation of every finite group definable on the field Q^ab obtained by taking the field Q of rational numbers and by adding all possible roots of unity?

Since every finite group can appear as Galois group the question translates to the question whether one can represent all possible Galois groups using matrices with elements in Q^ab.

This form of question has an interesting relation to Langlands program. By Langlands conjecture the representations of the Galois group of algebraic closure of rationals can be realized in the space of functions defined in GL(n,F)\GL(n,Gal(Q^ab/Q)), where Gal(Q^ab/Q) is the maximal Abelian subgroup of the Galois group of the algebraic closure of rationals. Thus one has group algebra associated with the matrix group for which matrix elements have values in Gal(Q^ab/Q). Something by several orders of more complex than matrices having values in Q^ab.

This relates interestingly also to my own physics inspired ideas about Langlands program (see this). For some time ago I went on to propose that the Galois group of algebraic numbers could be regarded as the permutation group S_∞ of infinite number of objects generated by permutations for finite numbers of objects.

1. The corresponding group algebra is hyper-finite factor of type II₁ (briefly HFF) and this led to quite fascinating physics inspired ideas about Langlands program and a connection with what I call number theoretic braids. In particular, Galois groups would have interpretation as physical symmetries in TGD Universe and would act on the infinite-D spinors of the world of classical worlds realized as infinite-D fermionic Fock algebra.

2. The representations of finite Galois groups G are the physically interesting ones. They could be interpreted as representations an infinite diagonal subgroup consisting of elements g×g×...., where g belongs to G. This requires the completion of the S_∞ to contain this kind of permutations. At the level of infinite-D spinors the action would be periodic with respect to the lattice defined by the tensor factors and belonging to the completion of HFF. The motivation for this representation is that infinite braid cannot be realized at space-time level and the periodicity would allow to reduce the situation to that for a finite braid since there is not need to say same thing again and again (I hope I could learn this too;-)!).

3. The interpretation in terms of gauge symmetries and spontaneous symmetry breaking suggests itself: number theoretic counterparts for local gauge transformations would correspond to S_∞ and those for global gauge transformations to elements of form g×g×....

What this has then to do with John's question and Langlands program? S_∞ contains any finite group G as a subgroup. If all the representations of finite-dimensional Galois groups could be realized as representations in Gl(n,Q^ab), same would hold true also for the proposed symmetry breaking representations of the completion of S_∞ reducing to representations of finite Galois groups. There would be an obvious analogy with Langlands program using functions defined in the space Gl(n,Q)\Gl(n, Gal(Q^ab/Q)). Be as it may, mathematicians are able to work with incredibly abstract objects! A highly respectful sigh is in order!