论文标题
非公共变量中的拟函数函数和准对称函数中的Powersum碱基
Powersum Bases in Quasisymmetric Functions and Quasisymmetric Functions in Non-commuting Variables
论文作者
论文摘要
我们引入了一个新的$ p $基础，以改进对称幂基础的准对称功能的HOPF代数。与类型1和2的准对称功率总和不同，我们的基础是合并定义的：矩阵的填充物给出了其在准对称单函数中的扩展。此基础具有洗牌产物，一个脱凝结的共同体，并且通过使用丝带的元素将基本规则更改为准对称基本基础。我们通过在非交通变量中通过不相交的填充物引入填充物，将准对称的Powersum $ p $基础提高到甲壳度功能的HOPF代数。这个新的基础具有变化的混洗产品和标准的去又催化剂，并且某些基础要素与Malvenuto-Reutenauer Hopf代数的基本基础一致。最后，我们讨论了如何通过使用索引上的总订单来概括这些基础及其属性。
We introduce a new $P$ basis for the Hopf algebra of quasisymmetric functions that refine the symmetric powersum basis. Unlike the quasisymmetric power sums of types 1 and 2, our basis is defined combinatorially: its expansion in quasisymmetric monomial functions is given by fillings of matrices. This basis has a shuffle product, a deconcatenate coproduct, and has a change of basis rule to the quasisymmetric fundamental basis by using tuples of ribbons. We lift our quasisymmetric powersum $P$ basis to the Hopf algebra of quasisymmetric functions in non-commuting variables by introducing fillings with disjoint sets. This new basis has a shifted shuffle product and a standard deconcatenate coproduct, and certain basis elements agree with the fundamental basis of the Malvenuto-Reutenauer Hopf algebra of permutations. Finally we discuss how to generalize these bases and their properties by using total orders on indices.
