Frobenius reciprocity on the space of functions invariant under a group action

TL;DR

This paper explores the structure of functions invariant under a group action, establishing their properties, computing their dimension in finite cases, and proving key results like Frobenius reciprocity and Bessel's inequality.

Contribution

It extends Frobenius reciprocity to the space of invariant functions and provides explicit dimension formulas for finite group actions.

Findings

Dimension of invariant function space related to fixed points

Proof of Frobenius reciprocity for invariant functions

Establishment of Bessel's inequality in this context

Abstract

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.