发表机构
University of Basel; Université du Québec à Montréal（巴塞尔大学; 蒙特利尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究拟阵的色词拟对称函数，证明其可估值性，并利用特殊拟阵类给出映射秩的下界$2^d-d$，猜想上界$d!$紧。核心贡献是引入该映射并探讨其区分能力。
AI 中文摘要
Billera、Jia和Reiner（2009）引入了拟阵的拟对称函数，并证明了这定义了一个Hopf代数同态，该同态是一个可估值不变量，即同构的拟阵具有相同的拟对称函数，且拟阵基多面体的多面体细分定义了相应拟对称函数之间的关系。在本项目中，我们研究非交换变量中的类似物，即词拟对称函数。对每个拟阵$M$，我们关联一个词拟对称函数$\u03c8(M)$，并称之为拟阵的色词拟对称函数。拟阵和词拟对称函数构成Hopf代数，而我们的映射$\u03c8$是它们之间的同态。我们想要研究从拟阵到词拟对称函数的映射$\u03c8$的核，等价地，其像，即我们想要理解哪些拟阵无法通过色词拟对称函数区分。映射$\u03c8$不是不变量，但我们可以证明它是可估值的。利用舒伯特拟阵和嵌套拟阵这两类特殊的拟阵，我们证明了在次数$d$下，从拟阵到色词拟对称函数的映射$\u03c8$的秩的下界为$2^d-d$，并猜想上界$d!$是紧的。
英文摘要
Billera, Jia, and Reiner (2009) introduced the quasisymmetric functions of matroids and showed that this defines a Hopf algebra homomorphism which is a valuative invariant, i.e., isomorphic matroids have the same quasisymmetric function and polytopal subdivisions of matroid base polytopes define relations among the corresponding quasisymmetric functions. In this project we study an analogue in non-commuting variables, the word-quasisymmetric functions. To every matroid $M$ we associate a word-quasisymmetric function $ψ(M)$ and call this the chromatic word-quasisymmetric functions of a matroid. Matroids and word-quasisymmetric functions form Hopf algebras, and our map $ψ$ between them is a homomorphism. We want to study the kernel, equivalently the image, of the map $ψ$ from matroids to word-quasisymmetric functions, that is, we would like to understand which matroids are indistinguishable by the chromatic word-quasisymmetric functions. The map $ψ$ is not an invariant, but we can show that it is valuative. Using Schubert matroids and nested matroids, special classes of matroids, we prove a lower bound of $2^d-d$ for the rank of the map $ψ$ from matroids to the chromatic word-quasisymmetric functions in degree $d$ and conjecture the upper bound of $d!$ is tight.
Comments53 pages, 10 figures, comments welcome!
