The A2 Weyl denominator expansion

Given: The Axiom of Choice, the realization of ${\mathfrak{s}\mathfrak{l}}_3$ with diagonal Cartan subalgebra and roots $\pm {\alpha }_1,\pm {\alpha }_2,\pm ({\alpha }_1+{\alpha }_2)$, the positive system ${\mathrm{\Phi }}^{+}=\{{\alpha }_1,{\alpha }_2,{\alpha }_1+{\alpha }_2\}$, the Weyl vector $\rho$, the Weyl group $W=S_3$ with its elements and lengths, and the alternant $A(\rho )$.

The Axiom of Choice is assumed; it enters through the root-system and denominator suppliers below (The Axiom of Choice).

In this realization the positive roots are ${\alpha }_1,{\alpha }_2$ and ${\alpha }_1+{\alpha }_2$, the Weyl group acts by the simple reflections $s_1,s_2$ with $s_1{\alpha }_1=-{\alpha }_1$, $s_1{\alpha }_2={\alpha }_1+{\alpha }_2$, $s_2{\alpha }_2=-{\alpha }_2$, $s_2{\alpha }_1={\alpha }_1+{\alpha }_2$, so $s_1\rho ={\alpha }_2$, $s_2\rho ={\alpha }_1$, $s_1{s}_2\rho =-{\alpha }_1$, $s_2{s}_1\rho =-{\alpha }_2$ and $w_0\rho =-\rho$, while $\rho =\frac{1}{2}{\sum }_{\alpha \in {\mathrm{\Phi }}^{+}}\alpha ={\alpha }_1+{\alpha }_2$ (Root systems of the classical complex Lie algebras, Classical complex matrix Lie algebras, The Weyl vector rho for a chosen positive system, The root set is a reduced crystallographic root system).

The Weyl group of $A_2$ is $S_3=\{1,s_1,s_2,s_1{s}_2,s_2{s}_1,w_0\}$ with lengths $0,1,1,2,2,3$, length being the number of inversions and the least number of simple reflections in an expression (Weyl length equals inversion number).

The denominator identity states $A(\rho )=e^{\rho }{\prod }_{\alpha \in {\mathrm{\Phi }}^{+}}(1-e^{-\alpha })$, and $A(\nu )={\sum }_{w\in W}(-1{)}^{\mathrm{\ell }(w)}{e}^{w\nu }$ (The Weyl denominator identity, The Weyl alternation operator).