Positive semidefiniteness of $h$-Shuhan matrices (analogs of Borcherds Cartan matrices) and associated determinant sequences

Abstract

We introduce an $h$-Shuhan matrix as a real square matrix $H=(h_{ij})$ with a common diagonal entry $h\ge 0$, nonpositive integral off-diagonal entries, and the following Cartan-type condition on each unordered pair $\{i,j\}$: either $h_{ij}=h_{ji}$, or $\max \{h_{ij},h_{ji}\}=-1$ and $\min \{h_{ij},h_{ji}\}\le -2$. These conditions imply that the scalar shift $H+(2-h)I_n$ is a generalized Cartan matrix; not every generalized Cartan matrix arises from an $h$-Shuhan matrix in this way. We classify, for $0\le h<2$, all indecomposable $h$-Shuhan matrices that are positive semidefinite, generalized positive semidefinite, or virtual positive semidefinite (the last notion means that all principal minors are nonnegative). For $h<2$, virtual positive semidefiniteness forces $H+(2-h)I_n$ to be of finite Cartan type, the positive-type case of Vinberg's trichotomy (1971). At $h=2$, we describe the boundary case in terms of finite and affine Cartan matrices. The associated determinant recurrences hold for general $h$. We study congruences for integral $h$ and asymptotic and Fibonacci-type identities for $h>2$.

arXiv comment29 pages