The left-to-right minima basis of the group algebra of the symmetric group (updated version)

TL;DR

This paper introduces a novel basis for the symmetric group algebra using left-to-right minima, revealing new algebraic actions and structures related to the descent algebra and Dynkin elements.

Contribution

It constructs a new basis based on left-to-right minima and demonstrates its properties, including a triangular action of the descent algebra, advancing understanding of symmetric group algebra structures.

Findings

The new basis is built from left-to-right minima sets of permutations.

The descent algebra acts by triangular operators on this basis.

The basis relates to Dynkin elements and free algebra interactions.

Abstract

We introduce a new basis of the group algebra of the symmetric group, built using the left-to-right minima sets of permutations. We show that on this basis, the descent algebra acts by triangular operators, thus making it an analogue of a cellular basis. The proof involves Dynkin elements (nested commutators) of the free algebra and their interactions with the $\mathbf B$-basis.