Annals of Combinatorics ( IF 0.5 ) Pub Date : 2020-07-31 , DOI: 10.1007/s00026-020-00504-5 Jean-Christophe Novelli , Jean-Yves Thibon , Frédéric Toumazet
We define a new basis of the algebra of quasi-symmetric functions by lifting the cycle-index polynomials of symmetric groups to noncommutative polynomials with coefficients in the algebra of free quasi-symmetric functions, and then projecting the coefficients to QSym. By duality, we obtain a basis of noncommutative symmetric functions, for which a product formula and a recurrence in the form of a combinatorial complex are obtained. This basis allows to identify noncommutative symmetric functions with the quotient of \(\mathrm{FQSym}\) induced by the pattern-replacement relation \(321\equiv 231\) and \(312\equiv 132\).
中文翻译:
非交换环指数和拟对称函数与非交换对称函数的新基础
通过将对称群的循环指数多项式提升为具有自由准对称函数代数中系数的非交换多项式,然后将系数投影到QSym,我们为准对称函数代数定义了新的基础。通过对偶性,我们获得了非交换对称函数的基础,为此,可以得到乘积公式和组合复数形式的递归。该基础允许识别由模式-替换关系\(321 \ equiv 231 \)和\(312 \ equiv 132 \)诱导的商\\\\ mathrm {FQSym} \的非可交换对称函数。