Let $\frak{g}$ be a semisimple complex Lie algebra and let $\Delta$ be its associated root system with $\{\alpha_1, \dotsc, \alpha_l\}$ a choice of positive roots. As we all know - $\Delta$ admits a partial order defined by $\alpha > \beta$ if $\alpha - \beta$ is a positive root. Do there exist other interesting, or well motivated, partial (or total) orders on $\Delta$. For example take the following ordering: For any root $\beta = \sum_i m_i \alpha_i$ and take the length function $$ \lambda(\beta) := \sum_i m_i. $$ Now declare $\beta > \beta'$ if $\lambda(\beta) > \lambda(\beta')$. Is this partial ordering of any importance?
1 Answer
Another partial order which comes up a lot is to take the transitive closure of the relation that, if $\langle \alpha_i, \beta \rangle < 0$, then $\beta \prec s_i(\beta)$.
For example, consider $B_2$ with short root $\alpha$ and long root $\beta$. The partial order you describe in your question is $$\alpha,\ \beta < \alpha+\beta < 2 \alpha+ \beta.$$ The partial order I have described above is the weaker (fewer relations) order: $$\alpha \prec \alpha+\beta,\ \ \beta \prec 2 \alpha+\beta.$$
This latter order has the advantage of being defined in the non-crystallographic types, and being independent of the Cartan matrix.
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$\begingroup$ Sorry for a foolish misunderstanding—what does "independent of the Cartan matrix" mean? Doesn't the root system determine the Cartan matrix? $\endgroup$– LSpiceOct 12, 2022 at 14:48
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1$\begingroup$ @LSpice Sorry, that was shorthand for the following: A Coxeter group is a group with generators $s_i$ and relations $s_i^2 = 1$, $(s_i s_j)^{m_{ij}} = 1$. A reflection is an element of $W$ conjugate to an $s_i$. The set of reflections is called $T$; the set of simple reflections is called $S$. $\endgroup$ Oct 12, 2022 at 14:49
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$\begingroup$ A Cartan matrix for $W$ is a matrix obeying certain conditions, like $A_{ij} A_{ji} = 4 \cos^2 \tfrac{\pi}{m_{ij}}$. It gives rise to a representation $V$ of $W$ and (if $A$ is crystallographic) a root system in $V$. The positive roots are then in bijection with the reflections of $W$. $\endgroup$ Oct 12, 2022 at 14:51
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1$\begingroup$ Exactly. "Independent of the Cartan matrix" is usually abusive shorthand for "can be defined as a relation on $T$, solely in terms of $W$ and $S$. $\endgroup$ Oct 12, 2022 at 14:52
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1$\begingroup$ In this case, I think the relation on $T$ is the transtitive closure of $t \prec sts$ if $\ell(t) < \ell(sts)$, though I'm not sure. $\endgroup$ Oct 12, 2022 at 14:52