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New results on the peak algebra

Authors:	Marcelo Aguiar Kathryn Nyman Rosa Orellana

Institution:	(1) Department of Mathematics, Texas A&M University, College Station, TX 77843, USA;(2) Department of Mathematics, Dartmouth College, Hanover, NH 03755, USA

Abstract:	The peak algebra $\mathfrak{P}_{n}$ is a unital subalgebra of the symmetric group algebra, linearly spanned by sums of permutations with a common set of peaks. By exploiting the combinatorics of sparse subsets of n−1] (and of certain classes of compositions of n called almost-odd and thin), we construct three new linear bases of $\mathfrak{P}_{n}$ . We discuss two peak analogs of the first Eulerian idempotent and construct a basis of semi-idempotent elements for the peak algebra. We use these bases to describe the Jacobson radical of $\mathfrak{P}_{n}$ and to characterize the elements of $\mathfrak{P}_{n}$ in terms of the canonical action of the symmetric groups on the tensor algebra of a vector space. We define a chain of ideals $\mathfrak{P}_{n}^{j}$ of $\mathfrak{P}_{n}$ , j = 0,..., ${\lfloor \frac{n}{2}\rfloor}$ , such that $\mathfrak{P}_{n}^{0}$ is the linear span of sums of permutations with a common set of interior peaks and $\smash{\mathfrak{P}_{n}{\lfloor \frac{n}{2}\rfloor}}$ is the peak algebra. We extend the above results to $\mathfrak{P}_{n}^{j}$ , generalizing results of Schocker (the case j = 0). Aguiar supported in part by NSF grant DMS-0302423 Orellana supported in part by the Wilson Foundation

Keywords:	Solomon's descent algebra Peak algebra Signed permutation Type B Eulerian idempotent Free Lie algebra Jacobson radical
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