Dual Weight and Monodromy of Dual Affine RS Correspondence

The dual affine Robinson–Schensted correspondence and the affine matrix-ball construction give two related parametrizations of extended affine permutations. From the stable-window data of the dual correspondence, we introduce a dual weight $\beta$ and prove that it is consistent with the original pair $(\lambda ,N_0)$. We prove that $\beta$ and the AMBC weight $\rho$ have identical monodromy along affine Knuth paths. We further give an explicit relation between $\beta$ and $\rho$, showing that their difference depends only on the associated tabloids.