发表机构
School of Mathematics, Southwest Jiaotong University（西南交通大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过分圆形式与Householder反射表示，证明了收缩代数反射群有限性的算术刚性判据，并给出BPS/Riemann-Hilbert实现。
AI 中文摘要
我们将有限块大小数据 \\[ \bm n=(n_d)_{d\ge1},\qquad n_1>0, \\] 关联到一个正定分圆形式和一个Householder反射表示。记 \\[ N=\operatorname{lcm}\{d:n_d\ne0\},\qquad \wt=\sum_d d^2n_d, \\] 则循环群 $C_N$ 上的形式为 \\[ B_{\bm n}(g,h)=\frac1{\wt}\sum_{(g^{-1}h)^d=1}d^2n_d. \\] 其标记的Fourier谱通过Möbius反演重构完整的重数序列。若 $n_d$ 是分裂半单代数 $S=\prod_d M_d(k)^{n_d}$ 的Wedderburn重数，则整除宽度是Wedderburn块的特征和的维数，该形式连同 $\dim_kS$ 一起恢复 $S$ 的同构类型。主定理是一个算术刚性结果。对于 $N\ge2$，反射群是有限的当且仅当它保持一个满秩格，等价地 \\[ \supp(\bm n)=\{1,N\},\qquad n_1=N^2n_N. \\] 此时 $B_{\bm n}=\frac12(I+J)$ 且群为 $W(A_N)\cong S_{N+1}$。在半单代数术语中，有限性等价于 \\[ S\cong k^{N^2m}\times M_N(k)^m \\] 对某个 $m\ge1$。对于 $N=2$，每个非有限情形都是 $D_\infty$ 在相应正交群中的稠密副本。对于三维flop的收缩代数，Toda的宽度公式将总尺度和原始Fourier模式识别为 $\dim A_{\con}$ 和 $\dim A_{\con}^{\ab}$。这得出所有长度至少为3的不可约flop的无穷性，并且在长度2时，\\[ \Gamma_f\text{ 有限}\quad\Longleftrightarrow\quad \dim A_{\con}=2\dim A_{\con}^{\ab}. \\] 最后部分给出了相同反射矩阵的互补BPS/Riemann–Hilbert实现。
英文摘要
We associate to finite block size data \[ \bm n=(n_d)_{d\ge1},\qquad n_1>0, \] a positive definite cyclotomic form and a Householder reflection representation. Writing \[ N=\operatorname{lcm}\{d:n_d\ne0\},\qquad \wt=\sum_d d^2n_d, \] the form on the cyclic group $C_N$ is \[ B_{\bm n}(g,h)=\frac1{\wt}\sum_{(g^{-1}h)^d=1}d^2n_d. \] Its labelled Fourier spectrum reconstructs the complete multiplicity sequence by Möbius inversion. If the $n_d$ are the Wedderburn multiplicities of a split semisimple algebra $S=\prod_d M_d(k)^{n_d}$, the divisibility widths are dimensions of characteristic sums of Wedderburn blocks, and the form together with $\dim_kS$ recovers the isomorphism type of $S$. The main theorem is an arithmetic rigidity result. For $N\ge2$, the reflection group is finite if and only if it preserves a full rank lattice, equivalently \[ \supp(\bm n)=\{1,N\},\qquad n_1=N^2n_N. \] In this case $B_{\bm n}=\frac12(I+J)$ and the group is $W(A_N)\cong S_{N+1}$. In semisimple algebra terms, finiteness is equivalent to \[ S\cong k^{N^2m}\times M_N(k)^m \] for some $m\ge1$. For $N=2$, every nonfinite case is a dense copy of $D_\infty$ in the corresponding orthogonal group. For contraction algebras of threefold flops, Toda's width formulas identify the total scale and primitive Fourier mode with $\dim A_{\con}$ and $\dim A_{\con}^{\ab}$. This yields infinitude for all irreducible flops of length at least three and, in length two, \[ Γ_f\text{ finite}\quad\Longleftrightarrow\quad \dim A_{\con}=2\dim A_{\con}^{\ab}. \] The final part gives a complementary BPS/Riemann--Hilbert realization of the same reflection matrices.
