在群作用下不变函数空间上的Frobenius互易性
Frobenius reciprocity on the space of functions invariant under a group action
摘要：
本文研究了群作用与其对应的向量空间之间的关系。给定一个作用的一组$G$上的一个非空集合$X$，我们研究了空间$L(X)$上的$X$和它的固定子空间：$$ L^G(X) = \{f\in L(X)\colon f(a\cdot x) = f(x) \textrm{ for all }a\in G, x\in X\}. $$特别是，我们表明，$L^G(X)$是一个不变量的作用$G$上的$X$。在作用量有限的情况下，利用$X$的不动点计算了$L^G(X)$的维数，并证明了$L^G(X)$的几个重要结果，包括Bessel不等式和Frobenius互易性.
英文摘要：
This article studies connections between group actions and their corresponding vector spaces. Given an action of a group $G$ on a nonempty set $X$, we examine the space $L(X)$ of scalar-valued functions on $X$ and its fixed subspace: $$ L^G(X) = \{f\in L(X)\colon f(a\cdot x) = f(x) \textrm{ for all }a\in G, x\in X\}. $$ In particular, we show that $L^G(X)$ is an invariant of the action of $G$ on $X$. In the case when the action is finite, we compute the dimension of $L^G(X)$ in terms of fixed points of $X$ and prove several prominent results for $L^G(X)$, including Bessel's inequality and Frobenius reciprocity.
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