2002/09/26 by Manfred Schocker · 2 citations
Mathematics · #math.RA #math.CO #msc:20C30 #msc:17B01 #msc:16N20 #msc:05E10
published as Adv. in Math. 192 (2005), No. 2, 259-309 · 60 pages
arxiv created 2002/09/26 · arxiv updated 2009/11/30
The linear span Pn 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 Dn; and the direct sum P of all Pn is a Hopf sub-algebra of the direct sum D of all Dn, 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 Pn; an algebraic characterization of Pn based on the action on the Poincar'e-Birkhoff-Witt basis of the free associative algebra; the display of peak variants of the classical Lie idempotents; an Eulerian-type sub-algebra of Pn; a description of the Jacobson radical of Pn and its nil-potency index, of the principal indecomposable and irreducible Pn-modules, and of the Cartan matrix of Pn. Furthermore, it is shown that the primitive Lie algebra of P is free, and that P is its enveloping algebra.