# weylAlcove -- the dominant integral weights of level less than or equal to l

## Synopsis

• Usage:
weylAlcove(l,g)
• Inputs:

## Description

Let $\mathbf{g}$ be a Lie algebra, and let $l$ be a nonnegative integer. Choose a Cartan subalgebra $\mathbf{h}$ and a base $\Delta= \{ \alpha_1,\ldots,\alpha_n\}$ of simple roots of $\mathbf{g}$. These choices determine a highest root $\theta$. (See highestRoot). Let $\mathbf{h}_{\mathbf{R}}^*$ be the real span of $\Delta$, and let $(,)$ denote the Killing form, normalized so that $(\theta,\theta)=2$. The fundamental Weyl chamber is $C^{+} = \{ \lambda \in \mathbf{h}_{\mathbf{R}}^* :$(\lambda,\alpha_i)$>= 0, i=1,\ldots,n \}$. The fundamental Weyl alcove is the subset of the fundamental Weyl chamber such that $(\lambda,\theta) \leq l$. This function computes the set of integral weights in the fundamental Weyl alcove.

In the example below, we see that the Weyl alcove of $sl_3$ at level 3 contains 10 integral weights.

 i1 : g=simpleLieAlgebra("A",2) o1 = g o1 : LieAlgebra i2 : weylAlcove(3,g) o2 = {{0, 0}, {0, 1}, {0, 2}, {0, 3}, {1, 0}, {1, 1}, {1, 2}, {2, 0}, {2, 1}, ------------------------------------------------------------------------ {3, 0}} o2 : List

## Ways to use weylAlcove :

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

## For the programmer

The object weylAlcove is .