Reflection groups of the quadratic form $-px_0^2+x_1^2+\dotsb+x_n^2$ with $p$ prime

Abstract

We present the classification of reflective quadratic forms $-px_0^2+x_1^2+\dotsb+x_n^2$ for $p$ prime. We show that for $p = 5$, it is reflective for $2\leq n\leq 8$, for $p = 7\text{ and }17$ it is reflective for $n = 2\text{ and }3$, for $p=11$ it is reflective for $p=2$, $3,\text{ and } 4$, and it is not reflective for higher values of $n$. We also show that it is non-reflective for $n > 2$ when $p = 13$, $19,\text{ and }23$. This completes the classification of these forms with $p$ prime.