Subalgebras of Solomon’s descent algebra based on alternating runs

Matthieu Josuat-Vergès (1, 2) , C.Y. Amy Pang (3, 4, 5)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge
2 CNRS - Centre National de la Recherche Scientifique
3 HKBU - Hong Kong Baptist University
4 LaCIM - Laboratoire de combinatoire et d'informatique mathématique [Montréal]
5 UQAM - Université du Québec à Montréal = University of Québec in Montréal

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.

Mots clés: Descent algebra; Noncommutative symmetric functions; Permutation statistics

Domaines: Combinatoire [math.CO]; Mathématique discrète [cs.DM]

Matthieu Josuat-Vergès, C.Y. Amy Pang. Subalgebras of Solomon’s descent algebra based on alternating runs. Journal of Combinatorial Theory, Series A, 2018, 158, pp.36-65. ⟨10.1016/j.jcta.2018.03.012⟩. ⟨hal-02332676⟩