The complexity of the Lie module

TL;DR

This paper investigates the complexity of the Lie module in modular representation theory, establishing bounds based on the prime power divisors of n and relating complexities of related modules.

Contribution

It provides a new upper bound for the complexity of the Lie module and relates it to complexities of modules associated with prime power divisors.

Findings

Complexity of Lie(n) is bounded by the largest p-power dividing n.

If n is not a p-power, complexity equals the maximum of complexities of Lie(p^i) for i ≤ m.

The results connect module complexity with prime factorization of n.

Abstract

We show that the complexity of the Lie module $\mathrm{Lie}(n)$ in characteristic $p$ is bounded above by $m$ where $p^m$ is the largest $p$-power dividing $n$ and, if $n$ is not a $p$-power, is equal to the maximum of the complexities of $\Lie(p^i)$ for $1 \leq i \leq m$.