有限个奇特征典型群的Character度和与实表示
Character degree sums and real representations of finite classical groups of odd characteristic
摘要：
设$\mathbb{F}_q$是具有$q$元的有限域，其中$q$是奇素数的幂，$\mathrm{GSp}(2n, \mathbb{F}_q)$和$\mathrm{GO}^{\pm}(2n, \mathbb{F}_q)$分别表示$\mathbb{F}_q$上的辛相似群和正交相似群.我们证明了$\mathrm{GSp}(2n, \mathbb{F}_q)$或$\mathrm{GO}^{\pm}(2n, \mathbb{F}_q)$的每一个实值不可约特征标都是一个实表示的特征标，并且我们找到了每个群的实表示的维数之和。我们还证明了，如果$\boldsymbol{G}$是一个定义在$\mathbb{F}_q$上的经典连通群，且中心是连通的，且维数为$d$，秩为$r$，则$\boldsymbol{G}(\mathbb{F}_q)$的不可约特征标的度之和上有界于$(q+1)^{(d+r)/2}$ .最后，证明了若$\boldsymbol{G}$是定义在$\mathbb{F}_q$上的任意连通约化群，则对任意$q$，$\boldsymbol{G}(\FF_q)$的不可约特征标的度之和下有界于$q^{(d-r)/2}(q-1)^r$ .我们猜想这个和总是可以被$q^{(d-r)/2}(q+1)^r$上界。
英文摘要：
Let $\mathbb{F}_q$ be a finite field with $q$ elements, where $q$ is the power of an odd prime, and let $\mathrm{GSp}(2n, \mathbb{F}_q)$ and $\mathrm{GO}^{\pm}(2n, \mathbb{F}_q)$ denote the symplectic and orthogonal groups of similitudes over $\mathbb{F}_q$, respectively. We prove that every real-valued irreducible character of $\mathrm{GSp}(2n, \mathbb{F}_q)$ or $\mathrm{GO}^{\pm}(2n, \mathbb{F}_q)$ is the character of a real representation, and we find the sum of the dimensions of the real representations of each of these groups. We also show that if $\boldsymbol{G}$ is a classical connected group defined over $\mathbb{F}_q$ with connected center, with dimension $d$ and rank $r$, then the sum of the degrees of the irreducible characters of $\boldsymbol{G}(\mathbb{F}_q)$ is bounded above by $(q+1)^{(d+r)/2}$. Finally, we show that if $\boldsymbol{G}$ is any connected reductive group defined over $\mathbb{F}_q$, for any $q$, the sum of the degrees of the irreducible characters of $\boldsymbol{G}(\FF_q)$ is bounded below by $q^{(d-r)/2}(q-1)^r$. We conjecture that this sum can always be bounded above by $q^{(d-r)/2}(q+1)^r$.
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