Haar graphical representations of finite groups and an application to poset representations

Let $R$ be a group and let $S$ be a subset of $R$. The Haar graph $\mathrm{Haar}(R,S)$ of $R$ with connection set $S$ is the graph having vertex set $R\times\{-1,1\}$, where two distinct vertices $(x,-1)$ and $(y,1)$ are declared to be adjacent if and only if $yx^{-1}\in S$. The name Haar graph was coined by Toma\v{z} Pisanski in one of the first investigations on this class of graphs. For every $g\in R$, the mapping $\rho_g:(x,\varepsilon)\mapsto (xg,\varepsilon)$, $\forall (x,\varepsilon)\in R\times\{-1,1\}$, is an automorphism of $\mathrm{Haar}(R,S)$. In particular, the set $\hat{R}:=\{\rho_g\mid g\in R\}$ is a subgroup of the automorphism group of $\mathrm{Haar}(R,S)$ isomorphic to $R$. In the case that the automorphism group of $\mathrm{Haar}(R,S)$ equals $\hat{R}$, the Haar graph $\mathrm{Haar}(R,S)$ is said to be a Haar graphical representation of the group $R$. Answering a question of Feng, Kov\'{a}cs, Wang, and Yang, we classify the finite groups admitting a Haar graphical representation. Specifically, we show that every finite group admits a Haar graphical representation, with abelian groups and ten other small groups as the only exceptions. Our work on Haar graphs allows us to improve a 1980 result of Babai concerning representations of groups on posets, achieving the best possible result in this direction. An improvement to Babai's related result on representations of groups on distributive lattices follows.