Introduction

When studying a linear space, when some subspaces are known, we are interested in the contribution of these subspaces, by studying their sum or (inner) direct sum if possible. This philosophy can be applied to many other fields.

In the context of representation theory, say, we are given a finite group $G$, with a subgroup $H$, we want to know how a character of $H$ is related to a character of $G$, through induction if anything. Next we state the content of this post more formally.

Let $G$ be a finite group with distinct irreducible characters $\chi_1,\dots,\chi_h$. A class function $f$ on $G$ is a character if and only if it is a linear combination of the $\chi_i$’s with non-negative integer coefficients. We denote the space of characters by $R^+(G)$. However, $R^+(G)$ lacks a satisfying algebraic structure, for example, one is not even allowed to freely do subtraction. For this reason, we extend the coefficients to all of integers, by defining

$$R(G)=\mathbb{Z}\chi_1 \oplus \cdots \oplus \mathbb{Z}\chi_h.$$

An element of $R(G)$ is called a virtual character because when one coefficient of some $\chi_i$ is negative, it cannot be a character in the usual sense. Note that $R(G)$ is a finitely...