辫群排列的循环相容形变

中文摘要 AI 辅助

我们证明了辫群排列的循环相容形变的双侧扩张的特征多项式移位公式。对于对角线为零的非负整数矩阵$M=(m_{ij})$，令$\boldsymbol{\textit{Arrange}} \boldsymbol{\textit{ment}} \boldsymbol{\textit{A}}_M$为满足$1 \boldsymbol{\textit{≤}} i \boldsymbol{<} j \boldsymbol{\textit{≤}} n$且$s \boldsymbol{\textit{∈}} [-m_{ij}, m_{ji}]_{\boldsymbol{\textit{Z}}}$的排列$x_i - x_j = s$。给定$\boldsymbol{\textit{α}}, \boldsymbol{\textit{β}} \boldsymbol{\textit{∈}} \boldsymbol{\textit{N}}^n$，其双侧扩张$\boldsymbol{\textit{Arrange}} \boldsymbol{\textit{ment}} \boldsymbol{\textit{A}}_M(\boldsymbol{\textit{α}}, \boldsymbol{\textit{β}})$通过将上述区间替换为$[-m_{ij} - \boldsymbol{\textit{α}}_i - \boldsymbol{\textit{β}}_j, m_{ji} + \boldsymbol{\textit{α}}_j + \boldsymbol{\textit{β}}_i]_{\boldsymbol{\textit{Z}}}$得到。若$M$为循环相容矩阵，即所有两两不同的$a,b,c$（满足$1 \boldsymbol{\textit{≤}} a,b,c \boldsymbol{\textit{≤}} n$）均满足$m_{ac} \boldsymbol{\textit{≤}} m_{ab} + m_{bc} + 1$，则对于简约特征多项式$\boldsymbol{\textit{χ}}(\boldsymbol{\textit{A}}, t) = \boldsymbol{\textit{χ}}(\boldsymbol{\textit{A}}, t)/t$，有$\boldsymbol{\textit{χ}}(\boldsymbol{\textit{A}}_M(\boldsymbol{\textit{α}}, \boldsymbol{\textit{β}}), t) = \boldsymbol{\textit{χ}}(\boldsymbol{\textit{A}}_M, t - |\boldsymbol{\textit{α}}| - |\boldsymbol{\textit{β}}|)$。证明采用有限域方法和循环间隙计数公式，还建立了重分布不变性，研究了弱和扰动，并给出了在Shi、均匀区间、图型和Ferrers型形变中的应用。

英文摘要

We study integer-interval deformations of Coxeter arrangements of types $A$, $B$, and $D$, assigning an integer interval to each root direction through an integer-valued root function. We introduce a local compatibility condition on this root function. For compatible root functions, subject to additional local bounds in types $B$ and $D$, we derive characteristic-polynomial formulas as binomial sums over cyclic orders in type $A$ and signed permutations in types $B$ and $D$. The proofs use the finite-field method and reduce hyperplane avoidance to consecutive-gap inequalities. These formulas recover, in particular, the classical Catalan and Shi characteristic polynomials.

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14 pages