有限辛正交群的Sylow p-子群的超特征标理论
A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups
摘要：
我们定义了$B_{n}(q)$，$C_{n}(q)$，或$D_{n}(q)$型的经典有限幂幺群$U$的超类，并证明了它们与在以前的论文中定义的超特征标一起构成了超特征标理论.特别是，我们证明了超字符在每个超类上取一个常数值，并评估这个值。作为结果，我们得到一个因式分解的任何超类作为产品的基本超类。此外，我们还定义了超类函数空间，并证明了它是由超特征标所张成的。作为结果，我们（重新）获得的分解的正规字符作为一个正交的线性组合的超字符。最后，我们定义了$U$的超特征标表，并证明了超特征标的各种正交关系（类似于著名的不可约特征标的正交关系）。
英文摘要：
We define the superclasses for a classical finite unipotent group $U$ of type $B_{n}(q)$, $C_{n}(q)$, or $D_{n}(q)$, and show that, together with the supercharacters defined in a previous paper, they form a supercharacter theory. In particular, we prove that the supercharacters take a constant value on each superclass, and evaluate this value. As a consequence, we obtain a factorization of any superclass as a product of elementary superclasses. In addition, we also define the space of superclass functions, and prove that it is spanned by the supercharacters. As as consequence, we (re)obtain the decomposition of the regular character as an orthogonal linear combination of supercharacters. Finally, we define the supercharacter table of $U$, and prove various orthogonality relations for supercharacters (similar to the well-known orthogonality relations for irreducible characters).
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