8

Let $R\subset V$ be an abstract root system in a finite-dimensional vector space $V$. Let $\Pi \subset R$ be a base of $R$. Define the Cartan matrix of $R$ as $C(R)=(\langle \alpha,\beta \check{}\rangle)_{\alpha,\beta \in \Pi}$.

My lecture notes claim:

Since we may assume $V$ to be a Euclidean vector space, $\det(C(R)) > 0$.

Why is that? Why do we have to assume that $V$ is a Euclidean vector space?

Since $\langle\alpha,\beta \check{}\rangle=2 \frac{(\alpha,\beta)}{(\beta,\beta)}$ (for an inner product $(-,-)$ on $V$ that is invariant under the Weyl group of $R$) the definition of the Cartan matrix is reminiscent to the Gram matrix of an inner product? Are the two related?

Peter
  • 851

1 Answers1

3

As you say yourself, for an inner product $( \cdot, \cdot)$ on $V$ compatible with the root system i.e. its Weyl group, and $\Pi =\{\alpha_1, ..., \alpha_n\}$ a basis of the root system (which is also a vector space basis of $V$), the Cartan matrix $C$ is given by:

$$\left(\dfrac{2(\alpha_i, \alpha_j)}{(\alpha_j, \alpha_j)} \right)_{i, j}$$

Now just notice that $$C = S \cdot D$$ with the symmetric matrix

$$S=\left((\alpha_i, \alpha_j)\right)_{i,j}$$

and $D$ the diagonal matrix whose $(j,j)$-th entry is $\dfrac{2}{(\alpha_j, \alpha_j)}$.

Because $( \cdot , \cdot)$ is a ("Euclidean") inner product, $S$ is positive definite, which (is much stronger than, but certainly) implies $\det(S) > 0$; and also, all those diagonal entries of $D$ are positive numbers.

Some people flip rows and columns in their definition of the Cartan matrix, in that case one would multiply the other way around.

Note: Of course, if one already knows the classification, one can also just compute the determinants of the Cartan matrices of all simple root systems by hand. The results are to be found e.g. in Wikipedia. Of course they are integers (the Cartan matrix has integer entries), we have shown they are posoitive, and with more theory one can show that they give exactly the index of the root lattice in the weight lattice, which via even more theory (or definition) is the order of the corresponding fundamental group. See Fundamental group of a Root System and determinant of the Cartan matrix and/or comments by Tobias Kildetoft to Determinant of a cartan matrix. The calculations are basically the ones at the end of https://math.stackexchange.com/a/3877651/96384.

Torsten Schoeneberg
  • 26,051