# The Unapologetic Mathematician

## Flag varieties

As another part of preparing for the digestion of the $E_8$ result, I need to talk about flag vareties. You’ll need at least some linear algebra to follow from this point.

A flag in a vector space is a chain of nested subspaces of specified dimensions. In three-dimensional space, for instance, one kind of flag is a choice of some plane through the origin and a line through the origin sitting inside that plane. Another kind is just a choice of a plane through the origin. The space of all flags of a given kind in a vector space can be described by solving a certain collection of polynomial equations, which makes it a “variety”. It’s sort of like a manifold, but there can be places where a variety intersects itself, or comes to a point, or has a sharp kink. In those places it doesn’t look like $n$-dimensional space.

Flag varieties and Lie groups have a really interesting interaction. I’ll try to do the simplest example justice, and the rest are sort of similar. We take a vector space $V$ and consider the group $SL(V)$ of linear transformations $T:V\rightarrow V$ with $\det(T)=1$. Clearly this group acts on $V$. If we pick a basis $\{b_1,b_2,...,b_n\}$ of $V$ we can represent each transformation as an $n\times n$ matrix. Then there’s a subgroup of “upper triangular” matrices of the form
$\left(\begin{array}{cccc}1&a_{1,2}&\cdots&a_{1,n}\\ 0&1&\cdots&a_{2,n}\\\vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&1\end{array}\right)$
check that the product of two such matrices is again of this form, and that their determinants are always $1$. Of course if we choose a different basis, the transformations in this subgroup are no longer in this upper triangular form. We’ll have a different subgroup of upper triangular matrices. The subgroups corresponding to different bases are related, though — they’re conjugate!

Corresponding to each basis we also have a flag. It consists of the line spanned by the first basis element contained in the plane spanned by the first two elements, contained in… and so on. So why do we care about this flag? Because the subgroup of upper triangular matrices with respect to this basis fixes this flag! The special line is sent back into itself, the special plane back into itself, and so on. In fact, the group $SL(V)$ acts on the flag variety “transitively” (there’s only one orbit) and the stabilizer of the flag corresponding to a basis is the subgroup of upper triangular matrices with respect to that basis! The upshot is that we can describe the flag variety from the manifold of $SL(V)$ by picking a basis, getting the subgroup of upper triangular matrices $U$, and identifying elements of $SL(V)$ that “differ” by an element of $U$. The subgroup $U$ is not normal in $SL(V)$, so we can’t form a quotient group, but there’s still a space of cosets: $SL(V)/U$.

So studying the flag variety in $V$ ends up telling us about the relationship between the group $SL(V)$ and its subgroup $U$. In general if we have a Lie group $G$ and a subgroup $B$ satisfying a certain condition we can study the relation between these two by studying a certain related variety of flags.