典型Renner幺半群的共轭类-基础数学专业论文.docxVIP

I I 摘 要 J-不可约代数幺半群有Am, Bm, Cm, Dm, E6, E7, E8, F4, G2九种形式, 它们的Renner幺 半群也有相应的九种形式, 其中前面四种Renner幺半群叫做典型幺半群, 特别地, Am型Renner幺半群叫做rook幺半群, Cm型Renner幺半群叫做辛幺半群, Bm, Dm型Renner 幺半群叫做正交Renner幺半群. 人们已经确定了Am, Cm型Renner幺半群共轭类，其余 类型的Renner幺半群共轭类尚无人研究. 本文将详细描述典型Renner幺半群以及它们 的单位群(典型Weyl群)的结构和实现，全面系统地综述Renner幺半群共轭类的已有结 果并确定正交Renner幺半群的共轭类. 关键词：Renner幺半群; Weyl群; 共轭类. II II Abstract There are nine types of J-irreducible algebraic monoids: Am, Bm, Cm, Dm, E6, E7, E8, F4, G2. The Renner monoids of these algebraic monoids have nine types. The Renner monoids of types Am, Bm, Cm, Dm are called classical Renner monoids. In particular, the Renner monoid of type Am is called the rook monoid, the Renner monoid of types Cm is called the symplectic monoid and the Renner monoids of types Bm, Dm are called orthogonal Renner monoids. Conjugacy classes of the Renner monoids of types Am and Cm have been determined. And conjugacy classes of the Renner monoids of the other types have not been investigated so far. In this paper we will give an explicit description for the structure and realizations of the classical Renner monoids and their unit groups (the classical Weyl groups). Also, we will comprehensively and systematically summarize the known results for conjugacy of the classical Renner monoids and will determine conjugacy classes of the orthogonal Renner monoids. Key words：Renner monoid; Weyl group; Conjugacy class. 记 号 所列记号按照在正文中出现的先后排序 n 集合{1, 2, . . . , n} m 集合{1, 2, . . . , m} Sn n级对称群 sgn(σ) 置换σ ∈ Sn的符号 An n级交错群 ˉi i ∈ n在n的对合之下的象 |i| i ∈ n的绝对值，当1 ≤ i ≤ n 时|i| = i，当 n i ≤ 2m时|i| = ˉi， 2 2 ∥I∥ ∥I∥ = {|i1|, |i2|, . . . , |is|}, 若I = {i1, i2, . . . , is} Sn σ ∈ Sn | ˉiσ = iσ, 1 ≤ i ≤ n , 辛Weyl An Sn ∩ An(n = 2m是偶数), 偶正交Weyl群的一种实现 σ) 集合{i | 1 ≤ i ≤ m 且 m + 1 ≤ iσ ≤ n}所含元素的个数. q( K 域 K? 域K的非零元素集合 Mn(K) 域K上所有n × n矩阵构成的幺半群 GLn(K) Mn(K)的单位群, 即域K上的n级一般线性群 S Ln(K) {x ∈ GLn(K) | det x = 1}, 即域K上的n级特殊线性群 D = Dn(K) Mn(K)的全体对角矩阵构成的幺半群 T = Tn(K) D中全体可逆矩阵构成的单位群, 即GLn(K)的极大环面 Ei j Mn(K)中的矩阵单位