Article ID Journal Published Year Pages File Type
9518082 Advances in Mathematics 2005 51 Pages PDF
Abstract
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 Poincaré-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.
Related Topics
Physical Sciences and Engineering Mathematics Mathematics (General)
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