Flagging and colossus convergence. The rectangle (2d) was a start for the learning algorithm (in 5d). And the flag was essentially a start for the rectangle.

Originally posted on The Unapologetic Mathematician:

As another part of preparing for the digestion of the $latex E_8$ 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 $latex n$-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 $latex V$ and consider the group $latex SL(V)$ of linear transformations $latex T:V\rightarrow V$ with $latex \det(T)=1$. Clearly this group acts on $latex V$. If we pick a basis $latex \{b_1,b_2,…,b_n\}$ of $latex V$ we can represent each transformation as an $latex n\times n$ matrix. Then there’s a subgroup of “upper triangular” matrices of the form
$latex \left(\begin{array}{cccc}1&a_{1,2}&\cdots&a_{1,n}\\ 0&1&\cdots&a_{2,n}\\\vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&1\end{array}\right)$
check that the product of two such matrices is again of this form, and that their determinants are always $latex 1$. 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!

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