Since I wrote this post, I’ve become less sure about this assertion that the braid group can be thought of as GL_n(F_1[t]). Here are three reasons to be doubtful:

- As Jim points out in commments, GL_n(F_q) embeds in GL_n(F_q[t]), but S_n doesn’t embed in the braid group. This has to be counted against the braid group, I think. Jim also says that in his version of F_1 geometry, which comes out of lambda-rings, GL_n(F_1[t]) is just S_n.
- Terry Tao observed that one might expect GL_n(F_1[t]) to embed into GL_n(F_q[t]), just as GL_n(F_1) embeds into GL_n(F_q). But this doesn’t appear to be the case, at least not for any obvious reason. Keep in mind, it was quite hard to prove that the braid group had any faithful linear representations at all! The n-dimensional linear representations developed by Lawrence and Krammer, and proven faithful by Bigelow (and then again by Krammer) have coefficients in Z[t,1/t,u,1/u]. So the idea that one might find the braid group inside GL(F_1[t,1/t,u,1/u]) remains, from this point of view, alive! But I wonder whether Jim thinks this latter group is also S_n…?
- Finally, the argument given by Kapranov and Smirnov looks like it’s making a case that the braid group admits a map to GL_n(F_1[[t]]), not so much that it should be thought of as GL_n(F_1[t]).

GL_n(F_1[[t]]), by the way, seems a little easier to get our hands on. Note that the order of the finite group GL_n(F_q[t]/t^k) is just a power of q times |GL_n(F_q)|. So, setting q = 1, one might expect

|GL_n(F_1[t]/t^k)| = |GL_n(F_1)| = n!

and in particular

GL_n(F_1[[t]]) = GL_n(F_1[t]/t^k) = S_n.

(Short version of this argument: “Pro-1 groups are trivial.”) In this case, the braid group certainly does map to GL_n(F_1[[t]])!

By the argument in the previous post, one would then want to say GL_n(F_1((t))) is the affine Weyl group Z^{n-1} semidirect S_n. Which means that the Hecke algebra

GL_n(F_1[[t]]) \ GL_n(F_1((t))) / GL_n(F_1[[t]])

is a pretty standard object — the double cosets above are in bijection with S_n-orbits on Z^{n-1}, which can be identified with the symmetrized monomials in n variables whose degree is a multiple of n, up to multiplication by x_1, … x_n. So the Hecke algebra should just be some version of the algebra of symmetric functions on n variables.

Believers that the braid group is GL_n(F_1[t]) are strongly encouraged to revivify my faith in comments.

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