Question

I am trying to understand the interaction between Borel subgroups of $GL_n$ and its roots. Is it correct to say that for any choice of roots among each pair of reciprocal roots there is a Borel subgroup containing those root subgroups? (Meaning that exactly those roots appear on the decompsition of its Lie algebra) If not, what is the precise condition on the choice of roots that is needed for such a Borel to exist?

Answer 1

Not every collection of choices between roots $\alpha$ and $-\alpha$ is "allowed". Yes, there is a partition of the set of all roots into positive $\Phi$ and negative $-\Phi$, but, also, $\Phi$ must be closed under addition. In the case of $GL(n)$, the Weyl group (permutation matrices, if you like) acts simply-transitively on the set of such choices, and/so there are exactly $n!$ such choices, rather than the $2^{n(n-1)/2}$ choices of sign.

[Edit:] In response to @Brad H-D's comment/query: yes, I was a bit glib... Indeed, it is not the case that all possible (iterated) sums of positive roots are roots, since this would (erroneously) require that there be infinitely-many, etc. Rather, as Brad H-D leadingly-asked, it means that either the sum of two "positive" roots is a "positive" root, or is not a root at all. (Thx, Brad H-D.)

Answer 2

The question itself seems too elementary for this site, since it just involves the standard axiomatic treatment of root systems as in Bourbaki Groupes et algebres de Lie, VI.1.7. The question is really about an arbitrary reductive algebraic group (with nontrivial derived group) over an algebraically closed field, along with its Borel subgroups in natural bijection with systems of positive roots relative to a fixed maximal torus. Such a torus $T$ lies in exactly $|W|$ Borel subgroups, where $W=N_G(T)/T$ is the Weyl group. At this point the axiomatic theory takes over and provides straightforward criteria for a given "closed" set of roots to be the positive roots for some choice of simple roots: the set has to be disjoint from its negative and together with its negative exhaust all roots (Prop. 20, Cor. 1).