Triangular and Unitriangular Factorization of Twisted Chevalley Groups
Shripad M. Garge, Deep H. Makadiya
TL;DR
This work advances the theory of LU-type decompositions for twisted Chevalley groups by addressing the previously unresolved ${^2A_{2n}}$ case over general commutative rings. By introducing the special stable range one condition $(SSR_1)$ and the concept of $\theta$-complete rings, the authors establish triangular factorizations for elementary twisted groups $E'_{\sigma}(\Phi,R)$ under precise ring hypotheses, and unitriangular factorizations when $R$ is $\theta$-complete with finite $\mathcal{C}$-length. The approach blends explicit SU(3,$R$) machinery with Tavgen-type rank reduction on twisted root systems to reduce to manageable subsystems, yielding both existence and length bounds for factorizations. The results extend classical Gauss-type decompositions to a broad class of rings, providing structural insight and potential computational applications for twisted Chevalley groups. The paper also clarifies when torus and Levi components coincide in twisted settings, and develops a robust framework for further factorization questions in the twisted, non-simply connected regime.
Abstract
The existence of triangular and unitriangular factorizations has been extensively studied for untwisted Chevalley groups, as well as for twisted Chevalley groups of types other than ${}^2A_{2n} \ (n \geq 1)$. However, the case of twisted Chevalley groups of type ${}^2A_{2n} \ (n \geq 1)$, has remained unresolved in the general setting of commutative rings. Prior work by A. Smolensky addressed this case only over certain fields, including finite fields and the field of complex numbers. These results indicate that, even over fields, the ${}^2A_{2n}$ case demands more refined techniques, reflecting the difficulty of extending such factorizations to the broader class of commutative rings. In this paper, we introduce two new classes of commutative rings: those satisfying the \emph{special stable range one condition} and those that are \emph{$θ$-complete}. We discuss their basic properties and provide illustrative examples. Our main result establishes the existence of triangular and unitriangular factorizations for twisted Chevalley groups of type ${}^2A_{2n}$ over a certain class of commutative rings, which includes all fields, all local rings (with mild restrictions), and several other important classes of rings.