Generalised core partitions and Diophantine equations

Abstract

We study generalised core partitions arising from affine Grassmannian elements in arbitrary Dynkin type. The corresponding notion of size is given by the atomic length in the sense of [CLG22]. In this paper, we first develop the theory for extended affine Weyl groups. In a series of applications, we give some remarkable parametrisations of the solutions of certain Diophantine equations resembling Pell's equation, by refining the results of [BN22] and [Alp14], and generalising them to further types.

Version 2, with some major additions and modifications: - Sections 1 and 2 revisited with important details added, - Section 3 is the fusion of previous Sections 3 and 4. Uniform proofs are provided and Theorem 3.9 is reformulated, - Section 8 is new and deals with 4-cores (10 pages), with Theorems 8.3, 8.5, 8.12 and 8.14 as new main results