# Irreducibility of root-height generating polynomial

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)?
• Last I checked, it is still open to prove that $R_{A_n}$ is irreducible for all $n$, although it is known for "most" n. See "Classes of polynomials having only one non-cyclotomic irreducible factor" by Borisov, Filaseta, Lam, Trifonov. – Gjergji Zaimi May 31 at 16:25