# Straightening rule for an $m'$-truncated polynomial ring

## TL;DR

This paper introduces a new rule for simplifying elements in an $m'$-truncated polynomial ring, linking categorified Green rings of symmetric groups and Schur algebras through explicit formulas.

## Contribution

It provides an explicit formula to rewrite signed Young permutation modules and mixed powers in terms of bases parametrized by partitions with specific divisibility properties.

## Key findings

- Derived a formula for rewriting modules in terms of basis elements.
- Counted compositions related to partitions with non-divisible partial sums.
- Connected categorified algebraic structures with combinatorial partition properties.

## Abstract

We consider a certain quotient of a polynomial ring categorified by both the isomorphic Green rings of the symmetric groups and Schur algebras generated by the signed Young permutation modules and mixed powers respectively. They have bases parametrised by pairs of partitions whose second partitions are multiples of the odd prime $p$ the characteristic of the underlying field. We provide an explicit formula rewriting a signed Young permutation module (respectively, mixed power) in terms of signed Young permutation modules (respectively, mixed powers) labelled by those pairs of partitions. As a result, for each partition $\lambda$, we discovered the number of compositions $\delta$ such that $\delta$ can be rearranged to $\lambda$ and whose partial sums of $\delta$ are not divisible by $p$.