发表机构
Rutgers University; Princeton University（罗格斯大学; 普林斯顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明格拉斯曼流形中环面理查森簇的上同调环为拟对称函数环的有限截断，构造仿射铺砌，并利用直接极限得到具有弱H-群结构的ind-簇，其上同调为拟对称函数的Hopf代数，同时推出碎片多面体的f-向量对数凹性。
AI 中文摘要
我们证明格拉斯曼流形中环面理查森簇的上同调环是拟对称函数环的有限截断。我们展示了每个此类簇的仿射铺砌，其胞腔闭包给出基本拟对称函数基。我们类似地用非交换对称函数环解释这些簇的同调，并表明任何环面不变子簇的同调类在仿射铺砌基中的展开与相应的广义非交换带状函数在带状基中的展开一致。通过取所有环面理查森簇的直接极限，我们得到一个配备弱$H$-群结构的ind-簇，其上同调是拟对称函数的Hopf代数。我们猜想它与Baker--Richter构造的类似$H$-群同构。作为副产品，我们推导出碎片多面体的$f$-向量是对数凹的，这在Ferroni--Schröter关于拟阵基多面体的问题上取得了首次进展。
英文摘要
We show that the cohomology rings of toric Richardson varieties in the Grassmannian are finite truncations of the ring of quasisymmetric functions. We exhibit an affine paving of each such variety whose cell closures give rise to the basis of fundamental quasisymmetric functions. We similarly interpret the homology of these varieties in terms of the ring of noncommutative symmetric functions and show that the expansion of the homological class of any torus-invariant subvariety into the affine paving basis agrees with the expansion of a corresponding generalized noncommutative ribbon function into the ribbon basis. By taking the direct limit of all toric Richardson varieties, we obtain an ind-variety equipped with a weak $H$-group structure whose cohomology is the Hopf algebra of quasisymmetric functions. We conjecture that it is isomorphic to a similar $H$-group constructed by Baker--Richter. As a byproduct, we deduce that the $f$-vectors of shard polytopes are log-concave, making the first progress on a question of Ferroni--Schröter for matroid base polytopes.
Comments48 pages, 16 figures, comments are welcome
