Contraction Algebras, Cyclotomic Forms, and Reflection Rigidity
Abstract
We associate to finite block size data n=(nd)d≥1, n1>0, a positive definite cyclotomic form and a Householder reflection representation. Writing N=lcm{d:nd≠0}, wt=∑d d2nd, the form on the cyclic group CN is Bn(g,h)=1/wt∑(g−1h)d=1d2nd. Its labelled Fourier spectrum reconstructs the complete multiplicity sequence by Möbius inversion. If the nd are the Wedderburn multiplicities of a split semisimple algebra S=∏d Md(k)nd, the divisibility widths are dimensions of characteristic sums of Wedderburn blocks, and the form together with dimkS recovers the isomorphism type of S. The main theorem is an arithmetic rigidity result. For N≥2, the reflection group is finite if and only if it preserves a full rank lattice, equivalently supp(n)={1,N}, n1=N2nN. In this case Bn=1/2(I+J) and the group is W(AN)≅ SN+1. In semisimple algebra terms, finiteness is equivalent to S≅ kN2m× MN(k)m for some m≥1. For N=2, every nonfinite case is a dense copy of D∞ 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 Acon and dim Aconab. This yields infinitude for all irreducible flops of length at least three and, in length two, Γf finite Longleftrightarrow dim Acon=2dim Aconab. The final part gives a complementary BPS/Riemann--Hilbert realization of the same reflection matrices.
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BibTeX
@misc{li2026contraction,
title = {{Contraction Algebras, Cyclotomic Forms, and Reflection Rigidity}},
author = {Xiaobin Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00263},
eprint = {2610.00263},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00263},
}
RIS
TY - UNPB
TI - Contraction Algebras, Cyclotomic Forms, and Reflection Rigidity
AU - Xiaobin Li
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.00263
UR - https://arxiv.org/abs/2610.00263
AB - We associate to finite block size data n=(n_d)_d≥1, n_1>0, a positive definite cyclotomic form and a Householder reflection representation. Writing N=lcm{d:n_d≠0}, wt=∑_d d^2n_d, the form on the cyclic group C_N is B_n(g,h)=1/wt∑_(g^−1h)^d=1d^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=∏_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≥2, the reflection group is finite if and only if it preserves a full rank lattice, equivalently supp(n)={1,N}, n_1=N^2n_N. In this case B_n=1/2(I+J) and the group is W(A_N)≅ S_N+1. In semisimple algebra terms, finiteness is equivalent to S≅ k^N^2m× M_N(k)^m for some m≥1. For N=2, every nonfinite case is a dense copy of D_∞ 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 finite Longleftrightarrow dim A_con=2dim A_con^ab. The final part gives a complementary BPS/Riemann--Hilbert realization of the same reflection matrices.
ER -