拟对称函数的拟阵友好基
A matroid-friendly basis for the quasisymmetric functions
摘要：
提出了拟对称函数空间的一种新的Z-基（简称QSym）。它具有非负的结构常数，和几个有趣的性质，相对于空间的拟对称函数相关联的拟阵的霍普夫代数态射（F）的Billera，贾，Reiner。特别是，对于无环拟阵，这个基础反映了拟阵秩的分级，以及基集的大小。证明了态射F在二阶拟阵的集合上是内射的，二阶拟阵的拟对称函数的可分解性反映了它的基多面体的可分解性。对Billera、Jia和Reiner提出的Hilbert基问题给出了肯定的回答。
英文摘要：
A new Z-basis for the space of quasisymmetric functions (QSym, for short) is presented. It is shown to have nonnegative structure constants, and several interesting properties relative to the space of quasisymmetric functions associated to matroids by the Hopf algebra morphism (F) of Billera, Jia, and Reiner. In particular, for loopless matroids, this basis reflects the grading by matroid rank, as well as by the size of the ground set. It is shown that the morphism F is injective on the set of rank two matroids, and that decomposability of the quasisymmetric function of a rank two matroid mirrors the decomposability of its base polytope. An affirmative answer is given to the Hilbert basis question raised by Billera, Jia, and Reiner.
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