Core abaci and Diophantine equations I: fundamental weight

TL;DR

This paper introduces core abaci for classical affine types, linking them to affine Grassmannians and Diophantine equations, providing new parameterizations and formulas.

Contribution

It extends core abaci concepts to arbitrary charge, connects them to affine Grassmannians, and solves related Diophantine equations with explicit parameterizations.

Findings

Height of β equals atomic length of associated Weyl group element.

Solutions to certain Diophantine equations are parameterized by core abaci.

Closed formulas for counting specific core abaci are provided.

Abstract

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 $(\lam, j)$ to a weight $\Lambda_j-\beta$ and an affine Weyl group element $w_{\lam, j}$, we prove that the height of $\beta$ is equal to the atomic length of $w_{\lam, j}$. 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.