Kromatic对称函数展开的新解释
New interpretations for Kromatic symmetric function expansions
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中文总结 AI 辅助
本文研究Kromatic对称函数在两种基下的展开，提出递归计算和包含-排除新方法，并给出图Hopf代数的$K$-类比余积公式。
中文摘要 AI 辅助
Kromatic对称函数（KSF）$\overline{X}_{G}$，由Crew、Pechenik和Spirkl（2026）引入，是色对称函数（CSF）$X_G$的$K$-理论类比。我们研究了KSF在两种不同基下的展开公式。首先，我们探索了在$K$-理论单项式对称函数基$\overline{\widetilde{m}}_\lambda$中计算KSF展开的递归方法，推广了第二作者和Samanta用来证明KSF能区分某些CSF无法区分的图族时所使用的公式。其次，我们给出了KSF在$K$-理论幂和基$\overline{p}_\lambda$中展开的新解释，采用了类似于Stanley（1995）的包含-排除方法，而非无环定向方法。最后，我们给出了Schmitt以及Humpert和Martin关于图Hopf代数的余积公式的$K$-类比，并给出了KSF的$\overline{p}$-展开和$\overline{\widetilde{m}}$-展开的Hopf代数解释。
英文摘要
The Kromatic symmetric function (KSF) $\overline{X}_{G}$, introduced by Crew, Pechenik, and Spirkl (2026), is a $K$-theoretic analogue of the chromatic symmetric function (CSF) $X_G$. We study expansion formulas for the KSF in two different bases. First, we explore recursive ways to compute the KSF's expansion in the $K$-theoretic monomial symmetric function basis $\overline{\widetilde{m}}_λ$, generalizing formulas that were used by the second author and Samanta to show that the KSF distinguishes certain families of graphs that are not distinguished by the CSF. Second, we give new interpretations for the KSF's expansion in the $K$-theoretic power sum basis $\overline{p}_λ$, using an inclusion-exclusion approach like the one from Stanley (1995) instead of an acyclic orientation approach. Finally, we give $K$-analogues of Schmitt's and of Humpert and Martin's antipode formulas for the Hopf algebra of graphs, along with a Hopf algebra interpretation for both the $\overline{p}$-expansion and $\overline{\widetilde{m}}$-expansion of the KSF.
发表机构
- Acton-Boxborough Regional High School（阿克顿-博克斯伯勒地区高中）
- Department of Mathematics, Harvard University（哈佛大学数学系）
机构由 AI 辅助整理，请以论文原文为准。
