On the fixed space induced by a group actio

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) for all a ∈ 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.