Adjoint Transfer and Order-Lifting Dichotomies for 2 x 2 Matrices over Prime-Power Rings
Abstract
For an odd prime p, we study the change in matrix order under lifting from Z/p^k Z to Z/p^(k+1) Z. The first variation of the order defect is an affine translate of a cyclic adjoint transfer on sl_2(F_p). We classify the full-order transfer for every q in GL_2(F_p): its rank is 3 for scalar elements, 1 for nonscalar semisimple elements, 1 for nonscalar Jordan elements when p = 3, and 0 for nonscalar Jordan elements when p > 3. For a Jordan element q = lambda(I + U), U^2 = 0 and U != 0, the vanishing first-order transfer for p > 3 admits an exact higher-order refinement. Writing d = (p - 1)/2, the full-order perturbation has depth d when tr(UX) != 0, depth p - 1 when tr(UX) = 0 and X is not in the span of U, and vanishes identically for X in the span of U. For p = 3, among the 27 lifts of a nonscalar order-preserving parent with determinant fixed to +1 or -1, exactly 9 preserve its order and 18 triple it; after the first increase, every further lift triples the order. We also give a scalar obstruction criterion for order preservation among unrestricted Jordan lifts.
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BibTeX
@misc{vajc2026adjoint,
title = {{Adjoint Transfer and Order-Lifting Dichotomies for 2 x 2 Matrices over Prime-Power Rings}},
author = {Ivo Vajc},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18900},
eprint = {2610.18900},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18900},
}
RIS
TY - UNPB
TI - Adjoint Transfer and Order-Lifting Dichotomies for 2 x 2 Matrices over Prime-Power Rings
AU - Ivo Vajc
PY - 2026
DA - 2026/10/11
PB - arXiv
DO - 10.48550/arxiv.2610.18900
UR - https://arxiv.org/abs/2610.18900
AB - For an odd prime p, we study the change in matrix order under lifting from Z/p^k Z to Z/p^(k+1) Z. The first variation of the order defect is an affine translate of a cyclic adjoint transfer on sl_2(F_p). We classify the full-order transfer for every q in GL_2(F_p): its rank is 3 for scalar elements, 1 for nonscalar semisimple elements, 1 for nonscalar Jordan elements when p = 3, and 0 for nonscalar Jordan elements when p > 3. For a Jordan element q = lambda(I + U), U^2 = 0 and U != 0, the vanishing first-order transfer for p > 3 admits an exact higher-order refinement. Writing d = (p - 1)/2, the full-order perturbation has depth d when tr(UX) != 0, depth p - 1 when tr(UX) = 0 and X is not in the span of U, and vanishes identically for X in the span of U. For p = 3, among the 27 lifts of a nonscalar order-preserving parent with determinant fixed to +1 or -1, exactly 9 preserve its order and 18 triple it; after the first increase, every further lift triples the order. We also give a scalar obstruction criterion for order preservation among unrestricted Jordan lifts.
ER -