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

TL;DR

This paper classifies when certain quadratic forms with prime coefficients are reflective, detailing the dimensions and primes for which they exhibit reflection properties, thus completing a comprehensive classification for these forms.

Contribution

It provides a complete classification of reflective quadratic forms of the specified type for all prime p, identifying the exact conditions and dimensions for reflection.

Findings

Reflective for p=5, n=2 to 8

Reflective for p=7,17, n=2 and 3

Reflective for p=11, n=2,3,4

Abstract

We present the classification of reflective quadratic forms $-px_0^2+x_1^2+\ldots+x_n^2$ 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.