# highestRoot -- returns the highest root of a simple Lie algebra

## Synopsis

• Usage:
highestRoot(g), highestRoot("A",2)
• Inputs:
• Outputs:

## Description

Let R be an irreducible root system of rank m, and choose a base of simple roots $\Delta = \{\alpha_1,...,\alpha_m\}$. Then there is a unique root $\theta$ such that when $\theta$ is expanded in terms of the simple roots, i.e. $\theta= \sum c_i \alpha_i$, the sum $\sum c_i$ is maximized. The formulas implemented here are taken from the tables following Bourbaki's Lie Groups and Lie Algebras Chapter 6.

In the example below, we see that for $sl_3$, the highest root $\theta$ is $\omega_1+ \omega_2$, where $\omega_1$ and $\omega_2$ are the fundamental dominant weights.

 i1 : highestRoot("A",2) o1 = {1, 1} o1 : List

## Ways to use highestRoot :

• "highestRoot(LieAlgebra)"
• "highestRoot(String,ZZ)"

## For the programmer

The object highestRoot is .