Core abaci and Diophantine equations I: fundamental weight

In the light of a series of papers on moving vectors, we define and study core abaci of classical affine types for arbitrary charge. This greatly extends the concept of cores with charge zero, and make us being able to parameterize the affine Grassmannian $W^j$ by core abaci of charge $j$ for arbitrary classical affine types. By associating a core abacus $(\text{lam},j)$ to a weight ${\mathrm{\Lambda }}_j-\beta$ and an affine Weyl group element $w_{\text{lam},j}$, we prove that the height of $\beta$ is equal to the atomic length of $w_{\text{lam},j}$. This solves a generalized version of the open problem raised by Brunat, Chapelier-Laget and Gerber. Moreover, Diophantine equations of classical affine types are established by using the height formula that given by Uglov vector. The solutions of certain classes of these Diophantine equations are proved to be completely parameterised by core abaci. As another application, closed formulae for computing the number of certain kinds of core abaci are given.