On the Complexity of Multiplication in the Iwahori--Hecke Algebra of the Symmetric Group

TL;DR

This paper introduces a novel nested coefficient list data structure for elements of the Iwahori--Hecke algebra of the symmetric group, significantly improving multiplication efficiency while maintaining addition costs.

Contribution

The paper proposes a new nested coefficient list data structure for Iwahori--Hecke algebra elements, enhancing multiplication performance compared to standard coefficient list representations.

Findings

Multiplication in $H(A_m)$ is significantly faster with the new data structure.

Addition costs remain unchanged between the old and new data structures.

The new data structure offers practical efficiency improvements for algebraic computations.

Abstract

We present new efficient data structures for elements of Coxeter groups of type $A_m$ and their associated Iwahori--Hecke algebras $H(A_m)$. Usually, elements of $H(A_m)$ are represented as simple coefficient list of length $M = (m+1)!$ with respect to the standard basis, indexed by the elements of the Coxeter group. In the new data structure, elements of $H(A_m)$ are represented as nested coefficient lists. While the cost of addition is the same in both data structures, the new data structure leads to a huge improvement in the cost of multiplication in~$H(A_m)$.