Description of root systems with given Weyl group (Q1290812)

The paper deals with the discrete groups generated by reflections. A criterion for the existence of a root system with a given Weyl group is derived in \S 2. Let \(G= (g_{ij})\) be an indecomposable \((n\times n)\) matrix with \(g_{ii}=2\) for all \(i\), \(i= 1,\dots, n\) and \(g_{ij}= 0\iff g_{ji}=0\). Let \(({\mathfrak H},\Pi,\Pi^\vee)\) be the realization of the matrix \(G\). That is, \({\mathfrak H}\) is a vector space, \(\Pi= \{\alpha_1,\dots, \alpha_n\}\subset{\mathfrak H}^*\), \(\Pi^\vee= \{\alpha_1^\vee,\dots, \alpha_n^\vee\}\subset{\mathfrak H}\) and \(\alpha_i (\alpha_j^\vee)= g_{ij}\), \(i,j= 1,\dots, n\). The Weyl group for \(g\) is the group of transformations of the space \({\mathfrak H}^*\) generated by the reflections \(R_i\), such that \(R_i(x)= x- x(\alpha_i^\vee) \alpha_i\). The matrix \(A\) is said to be diagonally conjugate to \(G\) if \(A= D\cdot G\cdot D^{-1}\), \(D= \text{diag} \{\lambda_1,\dots, \lambda_n\}\), \(\lambda_i\neq 0\), where the relation holds up to a permutation of indices. The Weyl groups for \(A\) and \(G\) coincide. A criterion for the existence of a generalized Cartan matrix \(A\) diagonally conjugate to \(G\) is obtained. Hence it is possible to connect \(G\) with the appropriate Kac-Moody algebra. The description of all root systems for a Weyl group \(W\) of the above mentioned matrix \(G\) is derived in \(\S 3\).