The peak algebra of the symmetric group revisited

By M. Schocker. 2002

49 citations recorded by the source corpus.

Abstract The linear span P n of the sums of all permutations in the symmetric group Sn with a given set of peaks is a sub-algebra of the symmetric group algebra, due to Nyman. This peak algebra is a left ideal of the descent algebra D n ; and the direct sum P of all P n is a Hopf sub-algebra of the direct sum D of all D n , dual to the Stembridge algebra of peak functions. In our self-contained approach, peak counterparts of several results on the descent algebra are established, including a simple combinatorial characterization of the algebra P n ; an algebraic characterization of P n based on the action on the Poincare–Birkhoff–Witt basis of the free associative algebra; the display of peak variants of the classical Lie idempotents; an Eulerian-type sub-algebra of P n ; a description of the Jacobson radical of P n and its nil-potency index, of the principal indecomposable and irreducible P n -modules, and of the Cartan matrix of P n . Furthermore, it is shown that the primitive Lie algebra of P is free, and that P is its enveloping algebra.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/6643ee39549d662a57223220e658f4019087d7ba

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Citations in the corpus

Permutations with given peak set (2012, background): “There has been a great deal of research into peaks of permutations, see [1, 2, 4, 6, 9, 10, 11, 12, 13, 15, 16, 17].”
The peak algebra and the descent algebras of types B and D (2003, background): “ained this result from the fact that its type B analog Θ± is (clearly) a morphism of Hopf algebras (Proposition 8.2). (2) The Hopf algebra structure of ◦ P (or its dual) has been carefully studied in [29] and [17]. One ofthe important results obtained inthese works is thatthe elements ◦ P{2,4,...,2k”
Enriched P-partitions and peak algebras (2005, background): “The study of algebras related to peaks began with John Stembridge’s paper [30] on enriched P -partitions, followed by others, including [1, 2, 7, 8, 9, 21, 27].”