Kostant multiplicity in the sl3 adjoint module

Given: The Axiom of Choice, the realization of ${\mathfrak{s}\mathfrak{l}}_3$ with diagonal Cartan subalgebra and roots $\pm {\alpha }_1,\pm {\alpha }_2,\pm \theta$, where $\theta ={\alpha }_1+{\alpha }_2$, the Weyl group $W=S_3$ with simple reflections $s_1,s_2$, the Weyl vector $\rho$, and the adjoint module $L(\theta )$.

The Axiom of Choice is assumed; it enters through the Kostant formula and the highest-weight suppliers below (The Axiom of Choice).

In this realization the positive roots are ${\alpha }_1,{\alpha }_2,\theta$ and $\theta ={\alpha }_1+{\alpha }_2$ is the highest root; the simple reflections act by $s_1\theta ={\alpha }_2$, $s_2\theta ={\alpha }_1$, $s_1{\alpha }_1=-{\alpha }_1$, $s_2{\alpha }_2=-{\alpha }_2$, and $\rho ={\alpha }_1+{\alpha }_2=\theta$ (Root systems of the classical complex Lie algebras, Classical complex matrix Lie algebras, Height and highest root, The Weyl vector rho for a chosen positive system).

$W=S_3=\{1,s_1,s_2,s_1{s}_2,s_2{s}_1,w_0\}$ with lengths $0,1,1,2,2,3$, and the adjoint module is the irreducible module $L(\theta )$ of highest weight $\theta$; its weights are the roots together with $0$, with the root spaces ${\mathfrak{g}}_{\alpha }$ one-dimensional and $\mathfrak{h}$ of dimension $2$ (The adjoint highest weight is the highest root, Finite semisimple Cartan, root and string structure, Weyl length equals inversion number).

$P(\theta )=2$, because a family $(n_{{\alpha }_1},n_{{\alpha }_2},n_{\theta })$ with $n_{{\alpha }_1}{\alpha }_1+n_{{\alpha }_2}{\alpha }_2+n_{\theta }\theta =\theta$ is either $(0,0,1)$ or $(1,1,0)$, while $P(\nu )=0$ for $\nu \mathrm{\notin }{Q}_{+}$ (The Kostant partition function).

Kostant's formula reads $m_{\lambda }(\mu )={\sum }_{w\in W}(-1{)}^{\mathrm{\ell }(w)}P(w(\lambda +\rho )-(\mu +\rho ))$, and $\theta$ is dominant integral (Kostant's weight multiplicity formula, Integral, dominant, and strictly dominant weights).