# 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

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

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 […]

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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 […]

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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 […]

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

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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 […]

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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 […]

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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. […]

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

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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 […]

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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 […]

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13. […] that forms a base for […]

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

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

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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 […]

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

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