New results on the peak algebra期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

New results on the peak algebra

Authors:	Marcelo Aguiar Kathryn Nyman Rosa Orellana

Affiliation:	(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
本文献已被 SpringerLink 等数据库收录！

设为首页 | 免责声明 | 关于勤云 | 加入收藏