@article{Suksumran2022, 
author = {Teerapong Suksumran},
title = {On the fixed space induced by a group action},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {12},
pages = {20615-20626},
keywords = {function space, Bessel's inequality, free action, Frobenius reciprocity, group action},
url = {https://www.sciopen.com/article/10.3934/math.20221130},
doi = {10.3934/math.20221130},
abstract = {This article studies connections between group actions and their corresponding vector spaces. Given an action of a group  G on a non-empty set  X, we examine the space  L(X) of scalar-valued functions on  X and its fixed subspace:  LG(X)={f∈L(X):f(a⋅x)=f(x)foralla∈G,x∈X}. In particular, we show that  LG(X) is an invariant of the action of  G on  X. In the case when the action is finite, we compute the dimension of  LG(X) in terms of fixed points of  X and prove several prominent results for  LG(X), including Bessel's inequality and Frobenius reciprocity.}
}