The peak algebra of the symmetric group revisited
The peak algebra of the symmetric group revisited
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DOI：
10.1016/j.aim.2004.04.007
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发表时间：
2002-09
影响因子：
1.7
通讯作者：
M. Schocker
中科院分区：
文献类型：
--
作者：
M. Schocker
对称群Sn中所有置换之和的线性跨度Pn是对称群代数的一个子代数，由于Nyman。这个峰代数是下降代数Dn的左理想;所有Pnis的直和是所有Dn的直和D的Hopf子代数，对偶于峰函数的Stembridge代数。在我们的自包含方法中，建立了几个关于下降代数的结果的峰对应，包括代数Pn的一个简单的组合刻画，基于自由结合代数的Poincaré-Birkhoff-Witt基作用的Pn的一个代数刻画，经典李幂等元的峰变体的显示，Pn的一个欧拉型子代数，描述的Jacobson根的Pn的幂零指数，主要的不可分解和不可约Pn-模块，和Cartan矩阵的Pn。进一步证明了P的本原李代数是自由的，P是它的包络代数。
The linear span Pnof the sums of all permutations in the symmetric group Snwith a given set of peaks is a sub-algebra of the symmetric group algebra, due to Nyman. This peak algebra is a left ideal of the descent algebra Dn; and the direct sum P of all Pnis a Hopf sub-algebra of the direct sum D of all Dn, dual to the Stembridge algebra of peak functions. In our self-contained approach, peak counterparts of several results on the descent algebra are established, including a simple combinatorial characterization of the algebra Pn; an algebraic characterization of Pnbased on the action on the Poincaré–Birkhoff–Witt basis of the free associative algebra; the display of peak variants of the classical Lie idempotents; an Eulerian-type sub-algebra of Pn; a description of the Jacobson radical of Pnand its nil-potency index, of the principal indecomposable and irreducible Pn-modules, and of the Cartan matrix of Pn. Furthermore, it is shown that the primitive Lie algebra of P is free, and that P is its enveloping algebra.