Subdivision of KLRW Algebras in Affine Type A

TL;DR

This paper studies the subdivision map between KLRW algebras of affine type A, demonstrating how idempotents and partitions transform, and establishing relations between graded decomposition numbers.

Contribution

It introduces a new understanding of how idempotents and partitions behave under the subdivision map in affine type A KLRW algebras.

Findings

Idempotents map to idempotents indexed by partitions.

Explicit form of the new partition after subdivision.

Equality of certain graded decomposition numbers.

Abstract

In this paper, we consider the subdivision map between two KLRW algebras of type $A^{(1)}_e$ and $A^{(1)}_{e+1}$. We show that the image of an idempotent indexed by a partition under this map is still an idempotent indexed by a partition, and give the form of this new partition. Moreover, we give an equality of some graded decomposition numbers.