Reflection groups of the quadratic form -px_0^2+x_1^2+...+x_n^2

Abstract

We present the classification of reflective quadratic forms $-p{x}_0^2+x_1^2+\dots +x_n^2$ for $p$ prime. We show that for $p=5$, it is reflective for $2\le n\le 8$, 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.

14 pages, 10 figures