Subalgebras of Solomon's descent algebra based on alternating runs
Matthieu Josuat-Verg\`es, C.Y. Amy Pang

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.
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Algebraic structures and combinatorial models · Advanced Topics in Algebra
