×

zbMATH — the first resource for mathematics

Classification of group subschemes in \(\text{GL}_n\), that contain a split maximal torus. (Russian, English) Zbl 1110.14040
Zap. Nauchn. Semin. POMI 321, 281-296 (2005); translation in J. Math. Sci., New York 136, No. 3, 3988-3995 (2006).
From the text: We describe group subschemes of \(\text{GL}_n\) over an arbitrary field that contain a split maximal torus. The main results are:
Theorem. There is a canonical bijection between the set of all connected group subschemes of \(\text{GL}_{l+1}\) containing a split maximal torus and the set of functions \(\phi: A_l\rightarrow \mathbb{N}\cup\{0,\infty\}\) satisfying the property \(\phi(\alpha+\beta)\geq\min(\phi(\alpha), \phi(\beta))\) for the root system \(A_l\).
Theorem. There is a canonical bijection between the set of all group subschemes of \(\text{GL}_{l+1}\) containing a split maximal torus and the set of pairs \((W, \phi)\), where \(\phi\) is a function \(\phi: A_l\rightarrow \mathbb{N}\cup\{0, \infty\}\) satisfying the property \(\phi(\alpha+\beta) \geq\min(\phi(\alpha), \phi(\beta))\), and \(W\) is a certain subgroup of the Weyl group \(W(A_l)\) containing all the reflections \(w_\alpha\) for \(\alpha\in A_l\) such that \(\phi(\alpha)=\phi(-\alpha)=\infty\) and normalizing the function \(\phi\).
This is a joint generalization of the papers by Z. I. Borevich, N. A. Vavilov [Tr. Mat. Inst. Steklova 148, 43–57 (1978; Zbl 0444.20039)] and others on the description of overgroups of a maximal torus and the works by Ch. Wenzel [Proc. Am. Math. Soc. 117, No. 4, 899–904 (1993; Zbl 0785.20023)] on parabolic subschemes.
MSC:
14L15 Group schemes
20G15 Linear algebraic groups over arbitrary fields
PDF BibTeX XML Cite
Full Text: DOI EuDML