# 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 $\Phi^+$.

February 1, 2010 - Posted by | Geometry, Root Systems

## 17 Comments »

1. Does every root system have a base (I’m guessing yes)? Is the answer to that question obvious?

Comment by Chad | February 1, 2010 | Reply

2. Good eye, Chad. It’s not obvious, and that’s exactly the question I’m set to take up tomorrow.

Comment by John Armstrong | February 1, 2010 | Reply

3. […] Existence of Bases for Root Systems We’ve defined what a base for a root system is, but we haven’t provided any evidence yet that they even exist. Today […]

Pingback by The Existence of Bases for Root Systems « The Unapologetic Mathematician | February 2, 2010 | Reply

4. […] That is, and are either both positive or both negative. So this means that , and thus the induced bases are equal: . We see, then, that we have a natural bijection between the Weyl chambers of a root […]

Pingback by Weyl Chambers « The Unapologetic Mathematician | February 3, 2010 | Reply

5. […] Lemmas on Simple Roots If is some fixed base of a root system , we call the roots “simple”. Simple roots have a number of nice […]

Pingback by Some Lemmas on Simple Roots « The Unapologetic Mathematician | February 4, 2010 | Reply

6. […] and the group itself is generated by the reflections corresponding to the simple roots in any given base […]

Pingback by The Action of the Weyl Group on Weyl Chambers « The Unapologetic Mathematician | February 5, 2010 | Reply

7. […] all are simple roots for some choice of a base . In general we can do this in many ways, and some will have larger […]

Pingback by Lengths of Weyl Group Elements « The Unapologetic Mathematician | February 8, 2010 | Reply

8. […] When we first discussed Weyl chambers, we defined the fundamental Weyl chamber associated to a base as the collection of all the vectors satisfying for all simple roots . Today, I want to discuss […]

Pingback by The Fundamental Weyl Chamber « The Unapologetic Mathematician | February 9, 2010 | Reply

9. […] of roots so that each root in is perpendicular to each one in . I assert that, for any base , is reducible if and only if can itself be broken into two collections in just the same way. […]

Pingback by Properties of Irreducible Root Systems I « The Unapologetic Mathematician | February 10, 2010 | Reply

10. […] a base of the root system. Since is finite-dimensional and is a basis of , must be finite, and so […]

Pingback by Cartan Matrices « The Unapologetic Mathematician | February 16, 2010 | Reply

11. […] on the other hand, must all be negative or zero. Indeed, our simple roots must be part of a base , and any two vectors must satisfy . Even better, we have a lot of information about pairs of […]

Pingback by From Cartan Matrix to Root System « The Unapologetic Mathematician | February 17, 2010 | Reply

12. […] of this proof, we will call such a set of vectors “admissible”. The elements of a base for a root system , divided by their lengths, are an admissible […]

Pingback by Proving the Classification Theorem I « The Unapologetic Mathematician | February 22, 2010 | Reply

13. […] that forms a base for […]

Pingback by Construction of A-Series Root Systems « The Unapologetic Mathematician | March 2, 2010 | Reply

14. […] from this we can always build for . Then if we can write . This proves that is a base for […]

Pingback by Construction of D-Series Root Systems « The Unapologetic Mathematician | March 3, 2010 | Reply

15. […] any vector can be written as the sum of two of these vectors. And so is a base for […]

Pingback by Construction of B- and C-Series Root Systems « The Unapologetic Mathematician | March 4, 2010 | Reply

16. […] remember that and , as vertices, really stand for vectors in some base of a root system, and the number of edges connecting them encodes their Cartan integers. If we […]

Pingback by Transformations of Dynkin Diagrams « The Unapologetic Mathematician | March 5, 2010 | Reply

17. […] a basis of a vector space, a base of a vector space contains enough information to reconstruct the whole root system. Further, any […]

Pingback by Root Systems Recap « The Unapologetic Mathematician | March 12, 2010 | Reply