Bases for Root Systems
We don’t always want to deal with a whole root system . Indeed, that’s sort of like using a whole group when all the information is contained in some much smaller generating set. For a vector space we call such a small generating set a basis. For a root system, we call it a base. Specifically, a subset is called a base if first of all is a basis for , and if each vector can be written as a linear combination
where the coefficients are either all nonnegative integers or all nonpositive integers.
Some observations are immediate. Because is a basis, it contains exactly vectors of . It also tells us that the decomposition of each is unique. In fact, as for any basis, every vector in can be written uniquely as a linear combination of the vectors in . What we’re emphasizing here is that for vectors in , the coefficients are all integers, and they’re either all nonnegative or all nonpositive.
Another thing a choice of base gives us is a partial order on the root system . We say that is a “positive root” with respect to (and write ) if all of its coefficients are nonnegative integers. Similarly, we say that is a “negative root” with respect to (and write ) if all of its coefficients are nonpositive integers. We extend this to a partial order by defining if .
Every root is either positive or negative. We write for the collection of positive roots with respect to a base and for the collection of negative roots. It should be clear that , and also that — the negative roots are exactly the negatives of the positive roots.
We can also define a kind of size of a vector . Given the above (unique) decomposition, we define the “height” of relative to as
This will be useful when it comes to proving statements about all vectors in by induction on their heights.
If are two vectors in a base , then we know that and . Indeed, our lemma tells us that if then would be in . But this is impossible, because every vector in can only be written as a linear combination of vectors in in one way, and that way cannot have some positive signs and some negative signs like does.
What this tells us (among other things) is that must be one end of the root string through . The other end must be , and the root string must be unbroken between these two ends. Every vector with must be in .