Uniformity in Higher class Free Lie algebras

TL;DR

This paper studies the enumeration of graded ideals in free class-c Lie rings on two generators over finite fields, revealing polynomial patterns for c ≤ 5 and deviations at c=6.

Contribution

It establishes uniform polynomial formulas for zeta functions of graded ideals in free Lie rings up to class 5 and identifies non-uniformity at class 6.

Findings

Polynomial formulas hold for c ≤ 5

Non-polynomial behavior at c=6

One-step graded ideal zeta functions are always polynomial

Abstract

Let $\mathfrak{f}_{c,2}$ denote a free class-$c$ Lie rings on $2$ generators. We investigate the zeta functions enumerating graded ideals in $\mathfrak{f}_{c,2}(\mathbb{F}_p)$ for $c\leq6$, prove that they are uniformly given by polynomials in $p$ for $c\leq5$ and not uniformly given by a polynomial in $p$ for $c=6$. We also show that the zeta functions enumerating one-step graded ideals $\mathfrak{f}_{c,2}(\mathbb{F}_p)$ is always given by a polynomial in $p$ for all $c$.