Let $\mathfrak{g}$ be a finite dimensional, simple, complex Lie algebra. Define the Coxeter adjacency matrix to be the matrix $A=2I-C$ where $C$ is the Cartan matrix of $\mathfrak{g}$. Let $a_n(x)$ be the characteristic polynomial of $A$, where $n$ is the size of $A$. Then the roots of $a_n(x)$ are \begin{equation} 2 \cos \frac{ m_i \pi}{h} \end{equation}

where $m_i$ are the exponents of $\mathfrak{g}$ and $h$ is the Coxeter number of $\mathfrak{g}$. Do you know where this result appeared first? I would also like to know if there is a proof of this fact which is not a case by case verification. I believe that the first such proof is via the Coxeter polymomial whose roots are well-known. I have computed $a_n(x)$ for all simple, complex Lie algebras in http://arxiv.org/abs/1110.6620 but that is a case by case computation.

  • $\begingroup$ I quote from: Kostant, Bertram. The principal three-dimensional subgroup and the Betti numbers of a complex simple Lie group. Amer. J. Math. 81 1959 973--1032. \bigskip A second empirical procedure for finding the exponents was discovered by H. M. Coxeter. He recognized that the exponents can be obtained from a particular transformation $\gamma$ in the Weyl group, which he had been studying, and which we take the liberty of calling a Coxeter-Killing transformation, in the following manner (see [5] ): Let h be the order of $\gamma$. Coxeter observed that $\endgroup$ – Pantelis Damianou Mar 23 '12 at 10:39
  • $\begingroup$ (1) $ h$ satisfies $ hl = 2r$, where $r$ is the number of positive roots, (2) $m_i \le h$ for all $i$ and (3) the eigenvalues of $\gamma$ are $\omega^{m_i}$ where $\omega=e^{\frac{2 \pi i}{h}}$ $\endgroup$ – Pantelis Damianou Mar 23 '12 at 10:42
  • $\begingroup$ A proof of (2) and (3) would provide, among other things, a proof of duality in the exponents $m_i$ observed by Chevalley (see [3], p. 24) since non- real eigenvalues of $\gamma$ necessarily occur in conjugate pairs. Requiring (1) $hl=2r$ as the only empirically observed fact such a proof was recently obtained by A. J. Coleman (see [4]). A proof that $hl=2r$ will be given in this paper. A second question posed in [4] of showing that $h=1 +o(\psi )$, where $\psi$ is the highest root, will also be settled here. $\endgroup$ – Pantelis Damianou Mar 23 '12 at 10:44

The formulation is somewhat out of focus, starting with the notation $a_n(x)$ for characteristic polynomial (what is $n$?). The roots indicated do occur in Coxeter's formulation, but not as the eigenvalues of your matrix $A$. It would help in any case to quote your own source and to describe the simplest nontrivial example involving a $2 \times 2$ matrix.

The original source is probably the influential paper by Coxeter in Duke Math. J. 18 (1951), which doesn't actually deal with simple complex Lie algebras and their Cartan matrices. Instead the framework is the study of finite real reflection groups (including Weyl groups of simple Lie algebras as a special case). So the results on Coxeter elements and exponents apply more broadly to reflection groups which need not be crystallographic. I'd have to look more carefully at the paper, but my impression is that what you are looking for doesn't require case-by-case study. There is a version of this development in the lengthy Exercise 3 (applied in Exercise 4) for Chapter V, Section 6, in Bourbaki Groupes et algebres de Lie (1968). Note that this chapter in Bourbaki just deals with reflection groups, before a treatment of crystallographic root systems and the related classification of finite Coxeter groups in Chapter VI.

P.S. While the case-by-case calculation of exponents for a finite Coxeter group (and their relationship with degrees of fundamental invariants) has evolved since Coxeter's original work, the matrix manipulations involved in the question here don't require knowing the explicit values of the $m_i$ in each case. Anyway, the question really has nothing directly to do with simple Lie algebras but only with finite Coxeter groups and Coxeter elements.

  • $\begingroup$ I finally bought the Bourbaki book. I mean the Springer edition. Indeed the last exercise of Chapter V is about this problem. In the same page there is also a footnote pointing to the paper of Coxeter in Duke Math. journal. Therefore, without doubt this result is due to Coxeter. $\endgroup$ – Pantelis Damianou Oct 7 '12 at 15:25

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