Subalgebras of Solomon's descent algebra based on alternating runs

TL;DR

This paper introduces new commutative subalgebras of the symmetric group algebra based on the permutation statistic of alternating runs, connecting them to Eulerian peak algebras and noncommutative symmetric functions.

Contribution

It defines novel subalgebras of the descent algebra using alternating runs and computes their orthogonal idempotents in terms of noncommutative symmetric functions.

Findings

Subalgebras are commutative and related to alternating runs

Eulerian peak algebras are identified as subalgebras

Orthogonal idempotents are explicitly calculated

Abstract

The number of alternating runs is a natural permutation statistic. We show it can be used to define some commutative subalgebras of the symmetric group algebra, and more precisely of the descent algebra. The Eulerian peak algebras naturally appear as subalgebras of our run algebras. We also calculate the orthogonal idempotents for run algebras in terms of noncommutative symmetric functions.