论文标题
在小组动作下，功能空间不变的Frobenius互惠
Frobenius reciprocity on the space of functions invariant under a group action
论文作者
论文摘要
本文研究了小组动作与其相应向量空间之间的联系。给定在非空置集合$ x $的组$ g $的操作，我们检查了$ x $及其固定子空间的标量值函数的空间$ l（x）$：$$ l^g（x）= \ {f \ in l（x）\ in L（x）\ colon f（a \ cdot x）= f（x）= f（x）= f（x）= f（x）\ textrm {尤其是$$，我们表明$ l^g（x）$是$ x $的$ g $动作的不变性。在操作是有限的情况下，我们根据$ x $的固定点计算$ l^g（x）$的尺寸，并证明了$ l^g（x）$的几个突出结果，包括贝塞尔的不平等和弗罗贝尼乌斯的互惠。
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.
