Let $G$ be a group acting doubly-transitively on a set $X$. Then the vector space $V\_X$ of functions $f\colon X\to\mathbb C$ with finite support such that $\sum\_{x\in X}f(x)=0$ carries an action of $G$.



> 
> Is $V\_X$ necessarily irreducible? Is it indecomposable?
> 
> 
> 


When $G$ is finite, then there is an easy [character-theoretic argument](https://math.stackexchange.com/a/3471289). When $\mathbb F$ is any field and $G=\mathrm{GL}\_2(\mathbb F)$ acts on $X=\mathbb P\_{\mathbb F}^1$, this is answered [here](https://mathoverflow.net/q/228359). What can be said in general?




---


**EDIT (6/15):** Although Noam gave a great example of a reducible $V\_X$, the question still remains of whether $V\_X$ can be decomposable. I suspect this can be proven by proving the only $G$-homomorphisms $V\_X\to V\_X$ are scalars, and such homomorphisms are uniquely determined by where $[y]-[x]$ is mapped to, where $x\ne y\in X$ and $[x]$ denotes the characteristic function of $\{x\}$. Then, the condition for such a map to be a homomorphism reduces to a complicated cocycle condition.


**Further update:** Per YCor's suggestion, I have posted a follow-up [here](https://mathoverflow.net/questions/448953/do-doubly-transitive-actions-give-rise-to-indecomposable-representations-for-inf) regarding indecomposability.



Do doubly-transitive actions give rise to irreducible representations for infinite groups?
$V\_X$ can fail to be irreducible. If $X \subseteq {\bf C}$ is a field and $G$ is the group of affine linear transformations $x \mapsto ax+b$ (any $a,b \in X$ with $a \neq 0$) then $V\_X$ has the proper subrepresentation consisting of functions $f : X \to {\bf C}$ of finite support such that $\sum\_x f(x) = \sum\_x x f(x) = 0$.


