# The field with one element in a probability seminar

Something very exciting this afternoon. Professor Persi Diaconis was presenting at the Stanford probability seminar and the field with one element made an appearance. The talk was about joint work with Mackenzie Simper and Arun Ram. They had developed a way of “collapsing” a random walk on a group to a random walk on the set of double cosets. As an illustrative example, Persi discussed a random walk on $GL_n(\mathbb{F}_q)$ given by multiplication by a random transvection (a map of the form $v \mapsto v + ab^Tv$, where $a^Tb = 0$).

The Bruhat decomposition can be used to match double cosets of $GL_n(\mathbb{F}_q)$ with elements of the symmetric group $S_n$. So by collapsing the random walk on $GL_n(\mathbb{F}_q)$ we get a random walk on $S_n$ for all prime powers $q$. As Professor Diaconis said, you can’t stop him from taking $q \to 1$ and asking what the resulting random walk on $S_n$ is. The answer? Multiplication by a random transposition. As pointed sets are vector spaces over the field with one element and the symmetric groups are the matrix groups, this all fits with what’s expected of the field with one element.

This was just one small part of a very enjoyable seminar. There was plenty of group theory, probability, some general theory and engaging examples.

Update: I have written another post about some of the group theory from the seminar! You can read it here: Double cosets and contingency tables.