| SL(3,3~n)和SU(3,3~n)的第一Cartan不变量 |
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| 引用本文： | 吴隋超,叶家琛.SL(3,3~n)和SU(3,3~n)的第一Cartan不变量[J].数学年刊A辑(中文版),2015,36(2):137-150. |
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| 作者姓名： | 吴隋超 叶家琛 |
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| 作者单位： | 上海工程技术大学基础教学学院, 上海 201620.,同济大学数学系, 上海 200092. |
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| 基金项目： | 本文受到国家自然科学基金 (No.11071187) 的资助. |
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| 摘 要： | 确定Cartan不变量是代数群与相关的李型有限群的模表示理论中的一个重要方面.作者利用代数群模表示理论中的一系列结果,计算了3~n个元素的有限域上特殊线性群SL(3,3~n)和特殊酉群SU(3,3~n)的第一Cartan不变量,得到如下结论:当G=SL(3,3~n)时,C_(00)~((n))=a~n+b~n+6~n-2·8~n;而当G=SU(3,3~n)时,C_(00)~((n))=a~n+b~n+6~n-2·8~n+2·(1+(-1)~n),其中a,b是多项式x~2-20x+48的两个根.另外,作者也得到了射影不可分解模U_n(0,0)的维数公式:dim U_n(0,0)=(12~n-6~n+∈)·3~(3n),其中,当G=SL(3,3~n)时,∈=1;而当G=SU(3,3~n)时,∈=-1.
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| 关 键 词： | 特殊线性群 特殊酉群 第一Cartan不变量 |
The First Cartan Invariant of SL(3, 3n) and SU(3, 3n) |
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| WU Suichao and YE Jiachen.The First Cartan Invariant of SL(3, 3n) and SU(3, 3n)[J].Chinese Annals of Mathematics,2015,36(2):137-150. |
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| Authors: | WU Suichao and YE Jiachen |
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| Affiliation: | School of Fundamental Studies, Shanghai University of Engineering Science,
Shanghai 201620, China. and Department of Mathematics, Tongji University, Shanghai 200092, China. |
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| Abstract: | The determination of Cartan invariants is an important aspect in the modular
representations of algebraic groups and related finite groups of Lie type. In this paper, the
first Cartan invariants for the groups SL(3, 3n) and SU(3, 3n) are calculated by using some
results from the representations of algebraic groups. Our main results are as follows:
dimUn(0, 0) = (12n ? 6n + ?) · 33n,
where ? = 1 when G = SL(3, 3n) and ? = ?1 when G = SU(3, 3n), and
C(n)
00 = an + bn + 6n ? 2 · 8n, when G = SL(3, 3n),
and
C(n)
00 = an + bn + 6n ? 2 · 8n + 2 · (1 + (?1)n) , when G = SU(3, 3n),
where a, b are the roots of the polynomial x2 ? 20x + 48. |
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| Keywords: | Special linear group Special unitary group First Cartan invariant |
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