| 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,2015,36(2):137~150 |
| SL(3,3^n) 和 SU(3,3^n)的第一 Cartan 不变量* |
| The First Cartan Invariant of SL(3, 3n) and SU(3, 3n) |
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| DOI： |
| 中文关键词: 特殊线性群, 特殊酉群, 第一Cartan不变量 |
| 英文关键词:Special linear group, Special unitary group, First Cartan invariant |
| 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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| 中文摘要: |
| 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\cdot 8^{n};而当 G=SU(3, 3^n) 时,
C_{00}^{(n)}= a^{n}+b^{n}+6^{n}-2\cdot 8^{n}+2\cdot\left(1+(-1)^{n}\right),$$
其中 $a,b$ 是多项式 $x^{2}-20x+48$ 的两个根. 另外, 作者也得到了射影不可分解模 $U_n(0,0)$ 的维数公式:
$$ \dim U_n(0,0)=(12^n-6^n+\epsilon)\cdot3^{3n},$$
其中, 当 $G=SL(3, 3^n)$ 时, $\epsilon=1$; 而当 $G=SU(3, 3^n)$ 时,$\epsilon=-1$. |
| 英文摘要: |
| 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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