I have a vague question, a less vague question and a lot of vaguer questions about permutation representations of a finite group $G$.
Vague question. Recall that if $G$ acts on a finite set $X$, we get a permutation representation $$G \to GL_{\lvert X \vert}(\mathbb C).$$ (Unless $X$ is very small,) this representation is never irreducible, for $\mathbb C (1, 1, \ldots, 1)^T$ splits off as a subrepresentation. What's left is the permutation representation $V_X$ associated to $G$. A classical fact is that $V_X$ is irreducible if and only if the original action was $2$-transitive. My question is: conversely, how do I know if some irreducible representation $V$ (whose character I know) is obtained by this construction? Clearly, a necessary condition is that $\chi_V$ should only take integer values $\geq -1$, but is it sufficient? If not, do we have another criterion?
Less vague question. (A special case of the first question). If I have not blundered, $\mathfrak S(6)$ has a degree-nine representation whose character only takes the values $-1$, $0$, $1$, $3$ and $9$. If you believe that $\mathfrak A(6) \simeq PSL_2(\mathbb F_9)$, you could get it by first considering the permutation representation $V_{\mathbb P^1(\mathbb F_9)}$ of $\mathfrak A(6)$, and inducing it to $\mathfrak S(6)$. This splits off as $$\mathrm{Ind}_{\mathfrak A(6)}^{\mathfrak S(6)} V_{\mathbb P^1(\mathbb F_9)} = V \oplus (V \otimes \epsilon),$$ where $\epsilon$ is the sign morphism. Is this $V$ a permutation representation? (Note that $\mathfrak S(6)$ is not isomorphic to $PGL_2(\mathbb F_9)$, so the action on the projective plane does not extend to $\mathfrak S(6)$ in a trivial way.) I believe this representation is not permutation, but I have no proof and not much confidence in my intuition (a brute-force examination of all index-nine subgroups of $\mathfrak S(6)$ would work, but I'd rather avoid hard work and I don't know any conceptual arguments or how to use modern computer tools to figure this out).
Vaguer question. The permutation representation machinery gives a morphism $$\mathbf{Burn}(G) \to \mathbf{R}(G)$$ from the Burnside ring of $G$ to its representation ring. Better still, it gives a morphism to all representation rings (whatever the field). What can we say about this morphism, beyond Wikipedia's example showing it can be noninjective and nonsurjective? Is there any textbook where this morphism is considered quite thoroughly? Do you know of any work on representation theory where the $\mathbf{Burn}(G)$-algebra structure on $\mathbf R(G)$ has been usefully exploited?
I realize a lot of my questions are very imprecise and I apologise unreservedly for that. I would nonetheless appreciate any comment, answer or reference recommendation about this topic.
