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Cokernels of random p-adic orthogonal matrices and their linearizations
Abstract
We study the linearization of the random p-adic orthogonal matrix model. Let An and Bn be Haar-random n× n special orthogonal and alternating matrices over ℤp, respectively. For an odd prime p, we prove that cok(An−In) and cok(Bn) have the same limiting distribution as n → ∞ through integers of the same parity. The two parity-dependent limits are interchanged for orthogonal matrices of determinant −1, while the full orthogonal group gives their equal mixture. For p=2, linearization does not preserve the limiting cokernel distribution, and we explicitly determine the limiting distributions for the special orthogonal model in terms of the spinor norm, which controls the parity of the $2$-adic valuation of the order of the torsion subgroup.
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BibTeX
@misc{lee2026cokernels,
title = {{Cokernels of random $p$-adic orthogonal matrices and their linearizations}},
author = {Jungin Lee and Myungjun Yu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26404},
eprint = {2609.26404},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26404},
}
RIS
TY - UNPB
TI - Cokernels of random p-adic orthogonal matrices and their linearizations
AU - Jungin Lee
AU - Myungjun Yu
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26404
UR - https://arxiv.org/abs/2609.26404
AB - We study the linearization of the random p-adic orthogonal matrix model. Let A_n and B_n be Haar-random n× n special orthogonal and alternating matrices over ℤ_p, respectively. For an odd prime p, we prove that cok(A_n−I_n) and cok(B_n) have the same limiting distribution as n → ∞ through integers of the same parity. The two parity-dependent limits are interchanged for orthogonal matrices of determinant −1, while the full orthogonal group gives their equal mixture. For p=2, linearization does not preserve the limiting cokernel distribution, and we explicitly determine the limiting distributions for the special orthogonal model in terms of the spinor norm, which controls the parity of the $2$-adic valuation of the order of the torsion subgroup.
ER -