The *height* $ht(\alpha)$ of a positive root $\alpha$ in a (finite, crystallographic) root system $\Phi$ is $\sum\_{i=1}^n c\_i$ where $\alpha = \sum\_{i=1}^n c\_i \alpha\_i$ is its decomposition as a sum of simple roots. I am interested in the polynomial
$$R\_{\Phi}(x):=\sum\_{\alpha \in \Phi^+} x^{ht(\alpha)-1}.$$


For example (using the usual Cartan-Killing nomenclature), we have:


* $R\_{A\_n}(x)=n+(n-1)x+\cdots +2x^{n-2}+x^{n-1}$
* $R\_{B\_n}(x)=R\_{A\_n}(x^2)+xR\_{A\_{n-1}}(x^2)$
* $R\_{D\_n}(x)=R\_{B\_n}(x)-\sum\_{i=n-1}^{2n-2} x^i$


**Questions:**


1. When $\Phi$ is an irreducible root system, is $R\_{\Phi}(x)$ an irreducible polynomial over $\mathbb{Q}$? This has been checked for the exceptional types ($G\_2, F\_4, E\_6, E\_7,$ and $E\_8$) and for the infinite families listed above for $n \leq 500$.
2. If the answer to (1) is "yes", is there some uniform Lie-theoretic reason that this is true (that is, a proof not relying on the classification)?



Irreducibility of root-height generating polynomial
(Turning Gjergji Zaimi's comment into a community wiki answer.)


A problem equivalent to the case of $R\_{A\_n}$ is discussed in [Classes of polynomials having only one non-cyclotomic irreducible factor](http://doi.org/10.4064/aa-90-2-121-153), by Borisov, Filaseta, Lam, and Trifonov (*Acta Arithmetica* **90** (1999), 121–153). They prove irreducibility in many special cases but the problem remains open (or at least, none of the papers that Google Scholar lists as citing this paper solves the problem).


