An Improvement of Rota's Straightening Algorithm

TL;DR

This paper introduces an improved straightening algorithm based on dual brackets that reduces computational complexity and overcomes limitations of Rota's original algorithm, especially for higher dimensions and degrees.

Contribution

The paper proposes a new straightening algorithm utilizing dual brackets, enhancing efficiency and applicability over Rota's algorithm in bracket algebra computations.

Findings

The new algorithm requires fewer straight bracket monomials per step.

It performs better as dimension and degree increase.

It remains effective where Rota's algorithm fails.

Abstract

In bracket algebra, the calculation of invariant division and invariant Gr\"{o}bner basis proposed in \cite{li 2014} rely on straightening algorithm. Until now, there are at least three different types of straightening algorithms, among which Rota's straightening algorithm has the best efficiency. However, there exists a flaw in Rota's straightening algorithm, i.e., it needs find all the straight bracket monomials with the same content as the input beforehand, which is quite difficult. So in this paper, we will propose a new straightening algorithm based on dual bracket, which is a new concept of Young tableau. In this new straightening algorithm, we only need to find a few number of straight bracket monomials in each step instead of finding them all in one step. And so it is an improvement of Rota's straightening algorithm. According to our tests, this new straightening algorithm reflects more advantages when the dimension and the degree increase. Moreover, this straightening algorithm still works when Rota's straightening algorithm fails.