Let $\mathbf{g}$ be a Lie algebra. We give three equivalent descriptions of an involution * on the weights of $\mathbf{g}$:
1. The involution * is given by $-w_0$, where $w_0$ is the longest word in the Weyl group $W(\mathbf{g})$.
2. If $\mu$ is a dominant integral weight, and $V_{\mu}$ is the irreducible Lie algebra module with highest weight $\mu$, then $\mu^*$ is the highest weight of the dual module $(V_{\mu})^*$.
3. If the Dynkin diagram of $\mathbf{g}$ has an involution, then * corresponds to the action of this involution on weights.
The formulas implemented have been adapted from Di Francesco, Mathieu, and Senechal, Conformal Field Theory, Springer Graduate Texts in Theoretical Physics, p. 511. Some changes are needed because we use the Bourbaki ordering of the roots in type E instead of the [DMS] ordering.
In the example below, we see that for $sl_3$, $\omega_1^* = \omega_2.$
i1 : g=simpleLieAlgebra("A",2) o1 = g o1 : LieAlgebra |
i2 : starInvolution({1,0},g) o2 = {0, 1} o2 : List |
The object starInvolution is a method function.