非交换变量中峰值代数与Schur Q函数

非交换变量中的对称函数理论是代数组合学中一个活跃的研究领域。本项目旨在将经典对称函数理论中的Schur Q函数推广至非交换变量Schur Q代数。经典Schur Q函数理论在表示论、Macdonald多项式等领域都有重要应用。Schur Q函数于1997年被Stembridge以组合的方法推广为拟对称函数中的峰值代数。过去20余年中，虽然对峰值代数的研究已经十分深入，但峰值代数仅被推广到了少数有特殊组合结构的Hopf代数。为了解决这些问题，我们计划提出一套系统的广义峰值代数理论，找出其统一的构造方式，使经典的Schur Q函数和峰值代数作为其特例存在。我们的方法是构造一种非交换的P-分拆，从而将峰值代数推广为非交换变量中的峰值代数。我们构造的空间隶属于非交换变量拟对称函数空间。通过限制到子空间或投影到商空间，我们即可得到非交换变量Schur Q函数以及其它广义峰值代数。

The theory of symmetric functions in non-commuting variables is an active and fruitful research area in algebraic combinatorics. This project aims at extending classical theory of Schur Q functions to non-commuting variables. Schur Q functions is an important class of symmetric functions that has numerous applications in representation theory and MacDonald polynomials. Classical theory of Schur Q functions has been extended combinatorially by Stembridge to the peak algebra in quasi-symmetric functions since 1997. The theory of peak algebra has been deeply studied in the past 20 years. However, though there have been many trials, the peak algebra is only generalized to a few Hopf algebras that have special underlying combinatorial structures. To resolve these problems, we propose a unified way of constructing generalizations of peak algebras that contain both the Schur Q functions in non-commuting variables and existing generalized peak algebras as special cases. Our approach is to construct peak algebra in non-commuting variables through a non-commutative analogue of P-partitions. Our construction lies in the space of quasi-symmetric functions in non-commuting variables. By restrictions or projections, we obtain not only Schur Q functions in non-commuting variables but also other peak objects.