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 n2$ 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$\frac n2$ 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

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