Combinatorial Wall-Crossing and the Mullineux Involution

TL;DR

This paper introduces combinatorial wall-crossing and generalized column regularization transformations on partitions, demonstrating their effects on one-row partitions and characterizing partitions that remain regular throughout the process.

Contribution

It defines new combinatorial transformations and proves their properties, including the invariance of one-row partitions and explicit descriptions of resulting quotients.

Findings

The composition of transformations preserves the one-row partition $(n)$.

One-row partition is uniquely regular at all steps of wall-crossing.

Explicit descriptions of partition quotients in the process.

Abstract

In this paper, we define the combinatorial wall-crossing transformation and the generalized column regularization on partitions and prove that a certain composition of these two transformations has the same effect on the one-row partition $(n)$. As corollaries we explicitly describe the quotients of the partitions which arise in this process. We also prove that the one-row partition is the unique partition that stays regular at any step of the wall-crossing transformation.