F_un and braid groups

Posted by on Sep 22, 2008 in KapranovSmirnov, undergraduate2 comments

This text was posted on June 15th, 2008 at neverendingbooks

Recall that an n-braid consists of n strictly descending elastic strings connecting n inputs at the top (named 1,2,…,n) to n outputs at the bottom (labeled 1,2,…,n) upto isotopy (meaning that we may pull and rearrange the strings in any way possible within 3-dimensional space). We can always change the braid slightly such that we can divide the interval between in- and output in a number of subintervals such that in each of those there is at most one crossing.

n-braids can be multiplied by putting them on top of each other and connecting the outputs of the first braid trivially to the inputs of the second. For example the 5-braid on the left can be written as B=B_1.B_2 with B_1 the braid on the top 3 subintervals and B_2 the braid on the lower 5 subintervals.

In this way (and using our claim that there can be at most 1 crossing in each subinterval) we can write any n-braid as a word in the generators \sigma_i (with 1 \leq i < n) being the overcrossing between inputs i and i+1. Observe that the undercrossing is then the inverse \sigma_i^{-1}. For example, the braid on the left corresponds to the word


Clearly there are relations among words in the generators. The easiest one we have already used implicitly namely that \sigma_i.\sigma_i^{-1} is the trivial braid. Emil Artin proved in the 1930-ies that all such relations are consequences of two sets of ‘obvious’ relations. The first being commutation relations between crossings when the strings are far enough from each other. That is we have

\sigma_i . \sigma_j = \sigma_j . \sigma_i whenever |i-j| \geq 2


The second basic set of relations involves crossings using a common string

\sigma_i.\sigma_{i+1}.\sigma_i = \sigma_{i+1}.\sigma_i.\sigma_{i+1}


Starting with the 5-braid at the top, we can use these relations to reduce it to a simpler form. At each step we have outlined to region where the relations are applied

= = =

These beautiful braid-pictures were produced using the braid-metapost program written by Stijn Symens.

Tracing a string from an input to an output assigns to an n-braid a permutation on n letters. In the above example, the permutation is ~(1,2,4,5,3). As this permutation doesn’t change under applying basic reduction, this gives a group-morphism

\mathbb{B}_n \rightarrow S_n

from the braid group on n strings \mathbb{B}_n to the symmetric group. We have seen before that the symmetric group S_n has a F-un interpretation as the linear group GL_n(\mathbb{F}_1) over the field with one element. Hence, we can ask whether there is also a F-un interpretation of the n-string braid group and of the above group-morphism.

Kapranov and Smirnov suggest in their paper that the n-string braid group \mathbb{B}_n \simeq GL_n(\mathbb{F}_1[t]) is the general linear group over the polynomial ring \mathbb{F}_1[t] over the field with one element and that the evaluation morphism (setting t=0)

GL_n(\mathbb{F}_1[t]) \rightarrow GL_n(\mathbb{F}1) gives the groupmorphism \mathbb{B}_n \rightarrow S_n

The rationale behind this analogy is a theorem of Drinfeld‘s saying that over a finite field \mathbb{F}_q, the profinite completion of GL_n(\mathbb{F}_q[t]) is embedded in the fundamental group of the space of q-polynomials of degree n in much the same way as the n-string braid group \mathbb{B}_n is the fundamental group of the space of complex polynomials of degree n without multiple roots.

And, now that we know the basics of absolute linear algebra, we can give an absolute braid-group representation

\mathbb{B}_n = GL_n(\mathbb{F}_1[t]) \rightarrow GL_n(\mathbb{F}_{1^n})

obtained by sending each generator \sigma_i to the matrix over \mathbb{F}_{1^n} (remember that \mathbb{F}_{1^n} = (\mu_n)^{\bullet} where \mu_n = \langle \epsilon_n \rangle are the n-th roots of unity)

\sigma_i \mapsto \begin{bmatrix}
1_{i-1} & & & \\
& 0 & \epsilon_n & \\
& \epsilon_n^{-1} & 0 & \\
& & & 1_{n-1-i} \end{bmatrix}

and it is easy to see that these matrices do indeed satisfy Artin’s defining relations for \mathbb{B}_n.

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  1. Very interesting post; this analogy is a novelty to me. Inspired by this, I blogged about GL_n(F_1[t,1/t]) over at my place.

  2. [...] Quomodocumque has an interesting post on F_un, buildings, the braid group and , starting from the Kapranov-Smirnov observation that should be considered as the braid group on n-strings (see also this post). [...]

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