AI 中文总结
该研究引入双分次$\u210b_n$-复形，通过切割分解将其关联至经典带形复形，确定同调表示的弗罗贝尼乌斯特征，还细化多面体分解并建立微分分次霍普夫条形相关性质。
AI 中文摘要
我们引入双分次$\u210b_n$-复形，其有序集合分条形由普通置换和带根置换装饰。其希尔伯特-欧拉特征为$n!Z_\alpha(t,q)$，由非交换对称函数的双参数特征得到。当分拆$\alpha$至多有一个奇部时，总装饰的同时唯一分解给出典范分裂$C_\bullet^{t,q}(\alpha)\cong\bigoplus_{\theta\in\mathcal D_n}C_\bullet(\gamma_\alpha(\theta))$，分解为经典带形复形。由此确定每个双分次同调表示：其弗罗贝尼乌斯特征为带形正的，重数计数具有指定精确切割集的装饰。对于阶梯分拆$\delta_n$，特化$Z_{\delta_n}(t,-1)$是交替栅栏的序多项式。其猜想的循环符号模式在每条整数射线$t=m\\,u$上成立。其极端同调层实现贪婪记录模型的极端纤维，而贝蒂数阻碍直接莫尔斯压缩至每个置换一个单元。我们还通过显式分解细化富集链多面体的卦限分解，对交替栅栏，其给出扩展斐波那契递推和二次型精确希尔伯特-昆兹公式中两个埃尔哈特项的格罗滕迪克提升。最后，平均装饰乘积将所有分拆的复形置于微分分次霍普夫条形中，经典归一化匹配严格霍普夫兼容，而刚性定理排除任何进一步的非恒等归一化收缩，该收缩在约化偶块模型上保持相同的去连接。
英文摘要
Let $α\models n$. We define a two-variable specialization $Z_α(t,q)$ of the ribbon basis of noncommutative symmetric functions from cycle enumerators of an ordinary permutation and a rooted permutation, with $q$ recording reflection length. We realize $n!Z_α(t,q)$ as the shifted bigraded Euler characteristic of an $\mathfrak S_n$-equivariant ordered-set-partition complex. When $α$ has at most one odd part, each total decoration determines a set of simultaneous factorization cuts, and its fiber is the classical ribbon complex indexed by that cut set. This gives explicit nonnegative ribbon expansions for every bigraded homology representation. Organizing the complexes on labelled finite sets yields a counital differential graded comonoid in the Cauchy monoidal category of species, whose cut coproduct is compatible with the fiber decomposition. At a factorization cut, the induced map on top homology is injective; its cokernel has the near-concatenation ribbon character, and the kernel of the aggregate reduced cut coproduct on homology is $H_1$. For the zigzag compositions $δ_n$, $Z_{δ_n}(t,-1)$ is the order polynomial of the alternating fence. We prove \[ n!Z_{δ_n}(t,-u)\in\mathbb N[t,u] \] using coefficientwise nonnegative recurrences derived from a Riccati equation. The same cut data define a nonnegative factorization defect $d=\lceil n/2\rceil-k-j$, which controls the homological support and makes the Euler sign constant on each defect layer. We also determine the full defect-zero edge by explicit Frobenius-character formulas.
Comments22 pages
