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Home > Vol 23, No 2 (2006)

Marcelo Aguiar¹ , Kathryn Nyman¹ and Rosa Orellana³

¹Texas A\&M University Department of Mathematics College Station TX 77843 USA College Station TX 77843 USA
³Dartmouth College Department of Mathematics Hanover NH 03755 USA Hanover NH 03755 USA

The peak algebra \mathfrak P _n \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 \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 \mathfrak{P}_{n} and to characterize the elements of \mathfrak P _n \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 \mathfrak{P}_{n}^{j} of \mathfrak P _n \mathfrak{P}_{n} , j = 0,..., ë\frac n2 û {\lfloor \frac{n}{2}\rfloor} , such that \mathfrak P _n ⁰ \mathfrak{P}_{n}^{0} is the linear span of sums of permutations with a common set of interior peaks and \mathfrak P _n ë\frac n2 û \smash{\mathfrak{P}_{n}{\lfloor \frac{n}{2}\rfloor}} is the peak algebra. We extend the above results to \mathfrak P _n ^j \mathfrak{P}_{n}^{j} , generalizing results of Schocker (the case j = 0).

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