On the Generating Rank and Embedding Rank of the Hexagonic Lie Incidence Geometries

IF 1 2区数学 Q1 MATHEMATICS

Combinatorica Pub Date : 2024-01-05 DOI:10.1007/s00493-023-00075-y

引用次数: 0

Abstract

Given a (thick) irreducible spherical building \(\Omega \) , we establish a bound on the difference between the generating rank and the embedding rank of its long root geometry and the dimension of the corresponding Weyl module, by showing that this difference does not grow when taking certain residues of \(\Omega \) (in particular the residue of a vertex corresponding to a point of the long root geometry, but also other types of vertices occur). We apply this to the finite case to obtain new results on the generating rank of mainly the exceptional long root geometries, answering an open question by Cooperstein about the generating ranks of the exceptional long root subgroup geometries. We completely settle the finite case for long root geometries of type \({{\textsf{A}}}_n\) , and the case of type \(\mathsf {F_{4,4}}\) over any field with characteristic distinct from 2 (which is not a long root subgroup geometry, but a hexagonic geometry).

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论六方列入射几何的生成秩和嵌入秩

摘要给定一个（厚的）不可还原的球面建筑（\Omega \），我们建立了一个关于其长根几何的生成秩和嵌入秩与相应的韦尔模子的维数之间的差异的约束，通过证明当取\(\Omega \)的某些残差（特别是长根几何的一个点对应的顶点的残差，但也有其他类型的顶点）时，这个差异不会增长。我们将其应用于有限情形，得到了主要是特殊长根几何的生成秩的新结果，回答了库珀斯坦关于特殊长根子群几何生成秩的一个开放问题。我们完全解决了类型为 \({{\textsf{A}}}_n\) 的长根几何的有限情形，以及任何特征与 2 不同的域上类型为 \(\mathsf {F_{4,4}}\) 的情形（这不是长根子群几何，而是六元几何）。

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

Combinatorica 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： COMBINATORICA publishes research papers in English in a variety of areas of combinatorics and the theory of computing, with particular emphasis on general techniques and unifying principles. Typical but not exclusive topics covered by COMBINATORICA are - Combinatorial structures (graphs, hypergraphs, matroids, designs, permutation groups). - Combinatorial optimization. - Combinatorial aspects of geometry and number theory. - Algorithms in combinatorics and related fields. - Computational complexity theory. - Randomization and explicit construction in combinatorics and algorithms.