Parametric decompositions into products of involutions
Abstract
It is a well-known result of Gustafson, Halmos and Radjavi, dating back to 1976, that any matrix A in SLn(ℂ) is a product of at most 4 involutions. We consider a natural continuous and holomorphic parameter dependence of this result in the spirit of Vaserstein and Gromov. Our main result shows that null-homotopy characterizes exactly those matrices in the special linear group over rings of continuous or holomorphic functions that are finite products of involutions. We give an upper bound $I(n, d)$, depending only on the size n of the matrices and the dimension d of the parameter space, for the number of involutions needed in the factorization. Our upper bound is not optimal, however we show that for fixed n the optimal upper bound tends to infinity as d→ ∞. In some cases we are able to determine the optimal bound.
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BibTeX
@misc{huang2026parametric,
title = {{Parametric decompositions into products of involutions}},
author = {Gaofeng Huang and Frank Kutzschebauch and Tran Nam Son and Truong Huu Dung},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26808},
eprint = {2609.26808},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26808},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Parametric decompositions into products of involutions
AU - Gaofeng Huang
AU - Frank Kutzschebauch
AU - Tran Nam Son
AU - Truong Huu Dung
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.26808
UR - https://arxiv.org/abs/2609.26808
AB - It is a well-known result of Gustafson, Halmos and Radjavi, dating back to 1976, that any matrix A in SL_n(ℂ) is a product of at most 4 involutions. We consider a natural continuous and holomorphic parameter dependence of this result in the spirit of Vaserstein and Gromov. Our main result shows that null-homotopy characterizes exactly those matrices in the special linear group over rings of continuous or holomorphic functions that are finite products of involutions. We give an upper bound $I(n, d)$, depending only on the size n of the matrices and the dimension d of the parameter space, for the number of involutions needed in the factorization. Our upper bound is not optimal, however we show that for fixed n the optimal upper bound tends to infinity as d→ ∞. In some cases we are able to determine the optimal bound.
ER -