A New Descent Algebra of Weyl Groups of Type An

TL;DR

This paper introduces a novel basis for a new, commutative, and semi-simple descent algebra associated with Weyl groups of type A_n, constructed via an equivalence relation on certain elements.

Contribution

It defines a new equivalence relation on elements to form a basis for a new descent algebra of type A_n, leading to a commutative and semi-simple algebra.

Findings

Constructed a new basis for the descent algebra

Established the algebra's commutative and semi-simple properties

Provided a framework for algebraic analysis of Weyl groups of type A_n

Abstract

In this paper we define an equivalence relation on the set of all $x_{J}$ in order to form a basis for a new descent algebra of Weyl groups of type $A_{n}$. By means of this, we construct a new commutative and semi-simple descent algebra of Weyl groups of type $A_{n}$ generated by equivalence classes arising from this equivalence relation.