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 the transition matrix, and its support properties. We determine the transition matrix on every weight space of $E \otimes E$ and compare it with the transition matrix for the dual sequence $(ω_2, ω_2)$ under the Schützenberger bijection. This is, to the best of our knowledge, the first verification of Conjecture 5.15 beyond type $A_1$.