The Baumann--Gaussent--Littelmann Symmetry Conjecture for $SL_3$: The Case $(ω_1,ω_1)$

We prove a conjecture of Baumann--Gaussent--Littelmann for $G = SL_3(\mathbb{C})$ and $λ= (ω_1, ω_1)$. The proof uses the $L$-perfectness of the Mirković--Vilonen basis, the Cartan quotient onto the symmetric square ${\rm Sym}^2 E$ of the standard representation $E$, the diagonal normalization of th