Highest weight

Let $\mathfrak g$ be a complex semisimple Lie algebra , fix a Cartan subalgebra $\mathfrak h\subset \mathfrak g$, and choose a set of positive roots, hence a decomposition into root spaces (see root space decomposition ).

Definition (Highest-weight vector and highest weight).
Let $V$ be a finite-dimensional $\mathfrak g$-module. A nonzero vector $v\in V$ is a highest-weight vector if

1. $v$ is a weight vector: there exists $\lambda\in \mathfrak h^*$ such that $h\cdot v=\lambda(h)v$ for all $h\in \mathfrak h$ (compare weights ), and
2. $v$ is killed by the positive root spaces: $\mathfrak n^+\cdot v=0$, where $\mathfrak n^+$ is the sum of positive root spaces.

The corresponding weight $\lambda$ is called a highest weight of $V$.

In the finite-dimensional semisimple setting, the weights of $V$ are partially ordered by the positive root cone; a highest weight is maximal in this order, and (for irreducible $V$) it is unique.

Context.
The highest weight encodes the representation: irreducible finite-dimensional modules are classified by dominant integral highest weights via the highest-weight theorem . The action of the Weyl group permutes weights, but the highest weight is singled out by the choice of positive roots.