Isotropy group of Lotka-Volterra derivations

TL;DR

This paper investigates the structure of the isotropy group of specific Lotka-Volterra derivations, revealing finiteness for certain dimensions and conditions, and identifying cases with infinite groups or dihedral group isomorphisms.

Contribution

It characterizes the isotropy group of Lotka-Volterra derivations across different dimensions and parameter conditions, highlighting cases with finite, infinite, and dihedral group structures.

Findings

Isotropy group is finite for n=3 or n≥5.

Infinite isotropy group occurs when n=4 and C_i=-1.

Isotropy group is dihedral of order 2n when C_i=1.

Abstract

In this paper, we study the isotropy group of Lotka-Volterra derivations of $K[x_{1},\cdots,x_{n}]$, i.e., a derivation $d$ of the form $d(x_{i})=x_{i}(x_{i-1}-C_{i}x_{i+1})$. If $n=3$ or $n \geq 5$, we have shown that the isotropy group of $d$ is finite. However, for $n=4$, it is observed that the isotropy group of $d$ need not be finite. Indeed, for $C_{i}=-1$, we observed an infinite collection of automorphisms in the isotropy group of $d$. Moreover, for $n \geq 3, ~~\text{and}~~C_{i}=1$, we have shown that the isotropy group of $d$ is isomorphic to the dihedral group of order $2n$.