Orbits under dual symplectic transvections

IF 1 3区数学 Q1 MATHEMATICS

Linear Algebra and its Applications Pub Date : 2025-02-10 DOI:10.1016/j.laa.2025.02.010

Jonas Sjöstrand

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引用次数: 0

Abstract

Consider an arbitrary field K and a finite-dimensional vector space X over K equipped with a, possibly degenerate, symplectic form ω. Given a spanning subset S of X, for each k in K and each vector s in S, consider the symplectic transvection mapping a vector x to

x + k ω (x, s) s

. The group generated by these transvections has been extensively studied, and its orbit structure is known. In this paper, we obtain corresponding results for the orbits of the dual action on

X^{⁎}

. As for the non-dual case, the analysis gets harder when the field contains only two elements. For that field, the dual transvection group is equivalent to a game known as the lit-only sigma game, played on a graph. Our results provide a complete solution to the reachability problem of that game, previously solved only for some special cases.

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对偶辛横切下的轨道

考虑一个任意域K和一个有限维向量空间X / K，它具有一个可能退化的辛形式ω。给定X的生成子集S，对于k中的每个k和S中的每个向量S，考虑向量X到X +kω(X， S) S的辛横切映射。这些横切所产生的星系群已经被广泛研究，其轨道结构也已为人所知。在本文中，我们得到了X *上对偶作用的轨道的相应结果。对于非对偶的情况，当字段只包含两个元素时，分析变得更加困难。在这个领域中，对偶横切群相当于一个在图上玩的游戏，称为仅liti - sigma游戏。我们的研究结果为该游戏的可达性问题提供了一个完整的解决方案，以前只解决了一些特殊情况。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Linear Algebra and its Applications 数学-应用数学

CiteScore

2.20

自引率

9.10%

发文量

333

审稿时长

13.8 months

期刊介绍： Linear Algebra and its Applications publishes articles that contribute new information or new insights to matrix theory and finite dimensional linear algebra in their algebraic, arithmetic, combinatorial, geometric, or numerical aspects. It also publishes articles that give significant applications of matrix theory or linear algebra to other branches of mathematics and to other sciences. Articles that provide new information or perspectives on the historical development of matrix theory and linear algebra are also welcome. Expository articles which can serve as an introduction to a subject for workers in related areas and which bring one to the frontiers of research are encouraged. Reviews of books are published occasionally as are conference reports that provide an historical record of major meetings on matrix theory and linear algebra.