This post continues my series on favorite theorems of the 21st century. For an overview of the categories and my previous selections, see this earlier post. In the category Between the Centuries in Algebra, my favorite theorem is the linearity of braid groups, established independently by Krammer (Krammer 2002) and Bigelow (Bigelow 2001).
The set of all invertible matrices with entries in a field forms a group under matrix multiplication. This group is denoted by and is called the general linear group of degree over . A group is said to be linear if it is isomorphic to a subgroup of for some natural number and some field . Linearity is a powerful property: linear groups enjoy a rich supply of algebraic and geometric tools, and showing that a group is linear grants access to these tools “for free.” Nevertheless, proving linearity can be highly nontrivial. For a long time, braid groups were a prominent example for which this question remained open.
We now briefly recall the definition of braid groups. Let be a set of points in the plane with coordinates and let be a parallel set of points with coordinates where . Consider (not necessarily straight) strands connecting points of to points of , such that each point is the endpoint of exactly one strand. The strands are required to run from left to right. Whenever two strands intersect, one is designated as passing over the other, and this over–under information is fixed. Such a configuration is called a braid. The precise geometric shape of the strands, as well as the specific values of and , are irrelevant: two configurations represent the same braid if one can be continuously deformed into the other without breaking or merging strands. What matters is the pattern of intertwining.
Now let be a braid from to , and let be another braid from to a third set of points with . By placing to the right of and concatenating the strands, we obtain a new braid from to . This operation defines the product of braids, written . The set of all braids with strands forms a group under this operation, called the braid group on strands.
In 1935, Burau proposed a representation of braid groups by matrices over a field, now known as the Burau representation. However, he did not determine whether this representation is faithful, that is, whether distinct braids are always sent to distinct matrices. In 1991, Moody showed that the Burau representation fails to be faithful in general. The problem of linearity of braid groups was finally resolved when Krammer (Krammer 2002) and, independently, Bigelow (Bigelow 2001) proved that a different representation—introduced earlier by Krammer (Krammer 2000)—is faithful. As a consequence, all braid groups are linear.