- 1、本文档共6页,可阅读全部内容。
- 2、原创力文档(book118)网站文档一经付费(服务费),不意味着购买了该文档的版权,仅供个人/单位学习、研究之用,不得用于商业用途,未经授权,严禁复制、发行、汇编、翻译或者网络传播等,侵权必究。
- 3、本站所有内容均由合作方或网友上传,本站不对文档的完整性、权威性及其观点立场正确性做任何保证或承诺!文档内容仅供研究参考,付费前请自行鉴别。如您付费,意味着您自己接受本站规则且自行承担风险,本站不退款、不进行额外附加服务;查看《如何避免下载的几个坑》。如果您已付费下载过本站文档,您可以点击 这里二次下载。
- 4、如文档侵犯商业秘密、侵犯著作权、侵犯人身权等,请点击“版权申诉”(推荐),也可以打举报电话:400-050-0827(电话支持时间:9:00-18:30)。
查看更多
Extending $pi$-systems to bases of root systems
a
r
X
i
v
:
m
a
t
h
/
0
4
1
0
3
5
7
v
3
[
m
a
t
h
.R
T
]
4
O
c
t
2
0
0
5
Extending π-systems to bases of root systems
Helmer ASLAKSEN
Department of Mathematics
National University of Singapore
Singapore 117543
Singapore
aslaksen@.sg
.sg/aslaksen/
Mong Lung LANG
Department of Mathematics
National University of Singapore
Singapore 117543
Singapore
matlml@.sg
Abstract
Let R be an indecomposable root system. It is well known that any root
is part of a basis B of R. But when can you extend a set of two or more
roots to a basis B of R? A π-system is a linearly independent set of roots,
C , such that if α and β are in C , then α ? β is not a root. We will use
results of Dynkin and Bourbaki to show that with two exceptions, A3 ? Bn
and A7 ? E8, an indecomposable π-system whose Dynkin diagram is a
subdiagram of the Dynkin diagram of R can always be extended to a basis
of R.
1 Introduction
Let R be an indecomposable root system in a Euclidean space V . A subset B
of R is called a basis of R if B is a vector space basis of V and each root of R
can be written as a linear combination of roots in B with integral coefficients that
1
are all nonnegative or all nonpositive. It is well known that any root is part of
a basis B of R. But when can you extend a set of two or more roots to a basis
B of R? A π-system ([Dyn, Oni-Vin]) is a linearly independent set of roots, C,
such that if α and β are in C, then α ? β is not a root. (It is not assumed to be
linearly independent in [Oni-Vin].) A subset of a basis will be a π-system, and a
π-system will be a basis of a root subsystem. We can associate a Dynkin diagram
to a π-system, and in order to extend to a basis of R, the Dynkin diagram of the
π-system must be a subdiagram of the Dynkin diagram of R. By a subdiagram
we mean a diagram obtained by deleting some nodes and their corresponding
links. We will assume that the nodes corresponding to short roots are marked, and
that the subdiagram preserves the marking. Hence two orthogona
您可能关注的文档
- Discovery_V12_中文操作说明.pdf
- DISCOVERY心得_GEOALTLAS等值线图直接导入Geomap之一.pdf
- Distribution of bacteria involved in nitrogen and sulphur cycles in shrimp culture systems of Wes.pdf
- Distribution of the extensive Doppler redshift of quasars.pdf
- Division and Modulus for Computer Scientists.pdf
- DM00039084【Discovery kit for STM32F407417 lines】.pdf
- DM-Intro_14.5_L01_Introduction_to_ANSYS.pdf
- DMN3009SFG替代型号AON7426.docx
- DNA编码化合物库应用与新药发现的效率分析-roche.pdf
- Do extragalactic cosmic rays induce cycles in fossil diversity.pdf
- 2024年江西省寻乌县九上数学开学复习检测模拟试题【含答案】.doc
- 2024年江西省省宜春市袁州区数学九上开学学业水平测试模拟试题【含答案】.doc
- 《GB/T 44275.2-2024工业自动化系统与集成 开放技术字典及其在主数据中的应用 第2部分:术语》.pdf
- 中国国家标准 GB/T 44275.2-2024工业自动化系统与集成 开放技术字典及其在主数据中的应用 第2部分:术语.pdf
- GB/T 44285.1-2024卡及身份识别安全设备 通过移动设备进行身份管理的构件 第1部分:移动电子身份系统的通用系统架构.pdf
- 《GB/T 44285.1-2024卡及身份识别安全设备 通过移动设备进行身份管理的构件 第1部分:移动电子身份系统的通用系统架构》.pdf
- 中国国家标准 GB/T 44285.1-2024卡及身份识别安全设备 通过移动设备进行身份管理的构件 第1部分:移动电子身份系统的通用系统架构.pdf
- GB/T 44275.11-2024工业自动化系统与集成 开放技术字典及其在主数据中的应用 第11部分:术语制定指南.pdf
- 中国国家标准 GB/T 44275.11-2024工业自动化系统与集成 开放技术字典及其在主数据中的应用 第11部分:术语制定指南.pdf
- 《GB/T 44275.11-2024工业自动化系统与集成 开放技术字典及其在主数据中的应用 第11部分:术语制定指南》.pdf
文档评论(0)