New results on the peak algebra
New results on the peak algebra
复制标题
峰值代数的新结果
DOI：
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发表时间：
2004
期刊：
影响因子：
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通讯作者：
R. Orellana
中科院分区：
文献类型：
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作者：
M. Aguiar;Kathryn L. Nyman;R. Orellana
峰值代数$$mathfrak{P}_{n}$$是对称群代数的一个一元子代数，由具有公共峰集的置换和线性张成。通过利用[n−1]的稀疏子集的组合学（以及n的某些类别的组合称为近奇和薄），我们构造了$$mathfrak{P}_{n}$$的三个新的线性基。讨论了第一欧拉幂等的两个峰类，构造了峰代数的半幂等元基。我们使用这些基来描述$$mathfrak{P}_{n}$$的Jacobson根，并根据对称群在矢量空间张量代数上的正则作用来描述$$mathfrak{P}_{n}$$的元素。我们定义一个理想链$$mathfrak{P}_{n}^{j}$$，其中$$mathfrak{P}_{n}$$, j = 0，…， $${lfloor frac{n}{2}
floor}$$，使得$$mathfrak{P}_{n}^{0}$$是具有一组共同内峰的置换和的线性张成，$$smash{mathfrak{P}_{n}{lfloor frac{n}{2}
floor}}$$是峰代数。我们将上述结果推广到$$mathfrak{P}_{n}^{j}$$，推广了Schocker （j = 0）的结果。
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,..., $${lfloor frac{n}{2}
floor}$$, such that $$mathfrak{P}_{n}^{0}$$ is the linear span of sums of permutations with a common set of interior peaks and $$smash{mathfrak{P}_{n}{lfloor frac{n}{2}
floor}}$$ is the peak algebra. We extend the above results to $$mathfrak{P}_{n}^{j}$$, generalizing results of Schocker (the case j = 0).