# The Unapologetic Mathematician

## Bases for Root Systems

We don’t always want to deal with a whole root system $\Phi\subseteq V$. 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 $\Delta\subseteq\Phi$ is called a base if first of all $\Delta$ is a basis for $V$, and if each vector $\beta\in\Phi$ can be written as a linear combination

$\displaystyle\beta=\sum\limits_{\alpha\in\Delta}k_\alpha\alpha$

where the coefficients $k_\alpha$ are either all nonnegative integers or all nonpositive integers.

Some observations are immediate. Because $\Delta$ is a basis, it contains exactly $n=\dim(V)$ vectors of $\Phi$. It also tells us that the decomposition of each $\beta$ is unique. In fact, as for any basis, every vector in $V$ can be written uniquely as a linear combination of the vectors in $\Delta$. What we’re emphasizing here is that for vectors in $\Phi$, 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 $\preceq$ on the root system $\Phi$. We say that $\beta$ is a “positive root” with respect to $\Delta$ (and write $\beta\succeq0$) if all of its coefficients are nonnegative integers. Similarly, we say that $\beta$ is a “negative root” with respect to $\Delta$ (and write $\beta\preceq0$) if all of its coefficients are nonpositive integers. We extend this to a partial order by defining $\beta\preceq\alpha$ if $\beta-\alpha\preceq0$.

Every root is either positive or negative. We write $\Phi^+$ for the collection of positive roots with respect to a base $\Delta$ and $\Phi^-$ for the collection of negative roots. It should be clear that $\Delta\subseteq\Phi^+$, and also that $\Phi^-=-\Phi^+$ — the negative roots are exactly the negatives of the positive roots.

We can also define a kind of size of a vector $\beta\in\Phi$. Given the above (unique) decomposition, we define the “height” of $\beta$ relative to $\Delta$ as

$\displaystyle\mathrm{ht}(\beta)=\sum\limits_{\alpha\in\Delta}k_\alpha$

This will be useful when it comes to proving statements about all vectors in $\Phi^+$ by induction on their heights.

If $\alpha\neq\beta$ are two vectors in a base $\Delta\subseteq\Phi$, then we know that $\langle\alpha,\beta\rangle\leq0$ and $\alpha-\beta\notin\Phi$. Indeed, our lemma tells us that if $\langle\alpha,\beta\rangle>0$ then $\alpha-\beta$ would be in $\Phi$. But this is impossible, because every vector in $\Phi$ can only be written as a linear combination of vectors in $\Delta$ in one way, and that way cannot have some positive signs and some negative signs like $\alpha-\beta$ does.

What this tells us (among other things) is that $\beta$ must be one end of the $\alpha$ root string through $\beta$. The other end must be $\sigma_\alpha(\beta)$, and the root string must be unbroken between these two ends. Every vector $\beta+k\alpha$ with $0\leq k\leq-\beta\rtimes\alpha$ must be in $\Phi^+$.