AI 中文总结
该研究证实了Reiner和Rhoades关于立方体超平面切片q-埃哈特级数的猜想，通过找到轨道调和理想生成集等方法，证明超单形的调和代数有限生成，给出其有理性的第二个证明。
AI 中文摘要
我们证明，立方体的超平面切片的q-埃哈特级数是具有显式分母的有理函数，满足q-互反性，从而证实了Reiner和Rhoades针对这些多面体提出的猜想。为实现这一点，我们找到了轨道调和理想的生成集，这也得到了相关商空间的希尔伯特级数和分次弗罗贝尼乌斯特征。我们进一步利用正则多重图的2-因子的结构结果，证明超单形Δ的调和代数作为代数由Δ的调和空间生成。特别地，该调和代数是有限生成的，由此给出了有理性的第二个证明。
英文摘要
We prove that the $q$-Ehrhart series of a hyperplane slice of a cube is a rational function with an explicit denominator that satisfies $q$-reciprocity, confirming a conjecture of Reiner and Rhoades for these polytopes. To do this, we find a generating set for the orbit harmonics ideal, which also yields the Hilbert series and graded Frobenius characteristic of the associated quotient. We further show that the harmonic algebra of a hypersimplex $Δ$ is generated as an algebra by the harmonic space of $Δ$ using structural results on 2-factors of regular multigraphs. In particular, the harmonic algebra is finitely generated, giving a second proof of rationality.
