Ars Mathematica

3 Responses to “Representations of GL(n)”

1. Jacob Freeze Says:
   August 3rd, 2008 at 7:48 pm

   This is only tangentially related to GL(n), but I’ve been trying to find an appropriate place to send up an S.O.S. for a while, so…

   Are there really only two elements of the absolute Galois group of the rational numbers that are given explicitly, with a complete description: the identity and complex conjugation?

   It’s just a bunch of automorphisms! Shouldn’t we be able to “see” a few more of them?

   Why not?

   I’m not looking for a link to some encyclopedia of Galois theory… just a little insight, like a child’s drawing of an elephant for someone who has never seen an elephant.

2. John Armstrong Says:
   August 3rd, 2008 at 10:45 pm

   I’m pretty sure dessins d’enfant are another subject entirely from Galois theory…

3. Jacob Freeze Says:
   August 3rd, 2008 at 11:45 pm

   I didn’t really expect my oblique reference to Grothendieck’s dessins d’enfant to attract Dr. Armstrong’s attention, but as long as we’re talking about the existence or non-existence of connections between Grothendieck’s little graphs and Galois groups…

   Belyi’s theorem implies that the absolute Galois group acts “faithfully” on dessins d’enfant, and in his brief introduction to dessins d’enfant, Leonardo Zapponi sketches some deeper interactions:

   Dessins correspond to covers of P1 defined over
   Q. The covers are permuted by the Galois group
   Gal(Q/Q), so this group also acts on the set of
   dessins, and one consequence of Belyı ˘’s theorem
   is that the action is faithful. The deepest open
   question in the theory of dessins is this: Can the
   Galois orbits of dessins be distinguished by combinatorial
   or topological invariants? That is, is there
   an effective way to tell whether two dessins belong
   to the same Galois orbit?

   At the beginning of his textbook on algebraic geometry, Serge Lang warns the reader that it’s possible to write endlessly on the subject, and read likewise, but every now and then a glimmer of actual insight would also be useful to orient the brain among millions of “interesting” results, so I’ll repeat my original question, now that the punning reference to Grothendieck has been taken care of:

   Why is it that we can only “see” two elements of the absolute Galois group over the rational numbers?

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