New results on the peak algebra.

Aguiar, Marcelo ; Nyman, Kathryn ; Orellana, Rosa

Journal of Algebraic Combinatorics, Tome 23 (2006) no. 2, pp. 149-188.

Résumé

Summary: The peak algebra ${\mathfrak{P}}_n$ mathfrakP_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 ${\mathfrak{P}}_n$ mathfrakP_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 ${\mathfrak{P}}_n$ mathfrakP_n and to characterize the elements of ${\mathfrak{P}}_n$ mathfrakP_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 ${\mathfrak{P}}_n^j$ mathfrakP_n^j of ${\mathfrak{P}}_n$ mathfrakP_n , $j=0,\dots$, ë$\frac{n}{2}$ û $\lfloor$fracn2rfloor , such that ${\mathfrak{P}}_n^0$ mathfrakP_n^0 is the linear span of sums of permutations with a common set of interior peaks and ${\mathfrak{P}}_n$ ë$\frac{n}{2}$ û smashmathfrakP_n$\lfloor$fracn2rfloor is the peak algebra. We extend the above results to ${\mathfrak{P}}_n^j$ mathfrakP_n^j , generalizing results of Schocker (the case $j=0$).

Détail

Mots-clés : keywords Solomon's descent algebra, peak algebra, signed permutation, type B, Eulerian idempotent, free Lie algebra, Jacobson radical