Past exam of the Lie algebras and representation theory of symmetric groups course of the University of Oxford / 2009 / A / 3

A base of a root system is a vector-space basis $\Delta$ of the ambient Euclidean space such that every root is an integer combination of its elements with all coefficients nonnegative or all nonpositive. The corresponding nonzero nonnegative combinations are the positive roots.

If distinct $\alpha ,\beta \in \Delta$ had positive inner product, the allowed root-difference fact would make $\alpha -\beta$ a root. Its simple-root coefficients have opposite signs, contradicting the definition. Hence distinct simple roots have nonpositive inner product.

For a positive root $\gamma =\sum n_{\alpha }\alpha$ which is not simple, the identity $(\gamma ,\gamma )=\sum n_{\alpha }(\gamma ,\alpha )>0$ gives a simple root $\alpha$ with $n_{\alpha }>0$ and $(\gamma ,\alpha )>0$. The allowed fact makes $\gamma -\alpha$ a root. Its coefficients remain nonnegative, and it is nonzero because $\gamma$ is not simple. It has smaller height $\sum n_{\alpha }$. Induction expresses it as a sum of simple roots with each partial sum a root; appending $\alpha$ gives the required expression for $\gamma$.

Finally, reflection in a simple root changes only that root's coefficient:For positive $\gamma \ne \alpha$, at least one other simple coefficient is positive: in a reduced root system the only positive root on the line of $\alpha$ is $\alpha$ itself. That other coefficient is unchanged. The reflected vector is a root, so its coefficients cannot all be nonpositive and must all be nonnegative. Thus ${\sigma }_{\alpha }$ preserves all positive roots except $\alpha$, which it sends to $-\alpha$.

Use the column-coroot convention $C_{ij}=2({\alpha }_i,{\alpha }_j)/({\alpha }_j,{\alpha }_j)$, consistent with the printed matrix and the stated shorter first root. It givesThe root string through $\beta$ in the $\alpha$ direction contains $\beta ,\beta +\alpha ,\beta +2\alpha ,\beta +3\alpha$. Reflecting $3\alpha +\beta$ in $\beta$ gives $3\alpha +2\beta$. Together with $\alpha$ and their negatives these giveThere are no further roots. Indeed the simple reflections act on coefficient pairs by $(a,b)\mapsto (-a+3b,b)$ and $(a,b)\mapsto (a,a-b)$; the displayed twelve-element set is closed under both. Every positive nonsimple root can be reflected to a positive root of smaller height: choose a simple root with positive inner product as in (a), and its positive integral Cartan pairing decreases height. Repetition reaches a simple root, so every root lies in an orbit already contained in the list. This is the G2 root system.

With $\alpha =e_1$, the inner product and length give the two reflected possibilities