Linear Representations coming from Permutation Representations

Question

Let $G$ be a finite group, and consider a permutation representation of $G$ on some finite set $\Sigma$ with $|\Sigma| = n$. By considering the vector space $V$ over $\mathbf{C}$ of dimension $n$ generated by the elements of $\Sigma$, one obtains a representation $V$ of $G$ with character which we denote by $\chi_{\Sigma}$.

Problem: If one is only given a character $\chi$ of $G$, what is the easiest way to tell whether it is equal to $\chi_{\Sigma}$ for some permutation representation $\Sigma$ of $G$?

Some Remarks: It seems to be a quite stringent condition. The character $\chi$ must be valued in $\mathbf{Z}$. Indeed, $\chi$ must extend to the character of $S_n$ given by the trivial plus the standard representation.

A Caution : Suppose that $G = D_8$ is the dihedral group of order $8$, and $H = Q_8$ is the quaternion group of order $8$. Then the character tables of $G$ and $H$ are the same. In particular, the inclusion $G \rightarrow S_4$ gives rise to a character $\chi$ of $G$ which does come from a permutation representation, but the "corresponding" character of $H$ does not. So the answer must involve knowing more than the character table of $G$.

Tautologies : Clearly, given $\chi$, one can simply "compute" all the permutation representations of $G$, and see if $\chi = \chi_{\Sigma}$ for any such representations that arise, but this is not necessarily practical.

Answer 1

To begin with, let me dampen your hopes: there are no easy ways to do that. But there are less easy ones. I will just give you some key words to search for, marked in italics.

First, not only must the character be defined over $\mathbb{Z}$, it must belong to a representation that is defined over $\mathbb{Q}$ (or $\mathbb{Z}$ if you like - this is the same). As you note, there are characters that are $\mathbb{Q}$-valued, but whose representation cannot be defined over $\mathbb{Q}$. This defect is measured by the so-called Schur index. The second volume of Curtis and Reiner "Methods of representation theory with applications to finite groups and orders" contains a wonderful chapter on Schur indices (actually, the whole book is wonderful). Also, there are lots of papers out there, investigating Schur indices.

So now, let us suppose that $\chi$ is a character that belongs to a $\mathbb{Q}$-irreducible representation (start with a complex irreducible character, take the sum over its Galois conjugates, then multiply by the Schur index to get such a $\chi$), and we want to know whether it is a virtual permutation character, i.e. whether it can be written as a difference of permutation characters. Artin's induction theorem (see e.g. Isaacs, or Curtis and Reiner) implies that $|G|\cdot \chi$ is a virtual permutation character. So you want to know the smallest integer $n$ such that $n\cdot \chi$ is a virtual permutation character.

The way people study this question is by defining two rings attached to $G$: the Burnside ring of $G$ is spanned over $\mathbb{Z}$ by symbols $[T]$ for each transitive $G$-set $T$ (up to isomorphism of course) with addition $[T\cup S] = [T] + [S]$ and with multiplication given by Cartesian product: $[T]\cdot[S] = [T\times S]$. The rational representation ring of $G$ is spanned by $\mathbb{Q}$-irreducible representations of $G$ with addition given by direct sum, and multiplication given by tensor product. There is a natural map from the former to the latter, taking a set and turning it into a permutation representation in the familiar way. Artin's induction theorem says that the cokernel of this map is finite. If it is 1 (and it often is), then every character of a rational representation is a virtual permutation character. In general, this index can be greater than 1, the smallest example is $Q_8\times C_3$ and is due to Serre (see Serre's Representation Theory book). The index is often called the Artin index and has also been extensively studied. The first page of this paper gives several references.

Ok, this should get you started.