On the fixed space induced by a group action

Abstract

<abstract><p>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: $ L^G(X) = \{f\in L(X): f(a\cdot x) = f(x) \; {\rm{ 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.</p></abstract>