A central theme in social choice theory is that of impossibility theorems, such as Arrow's theorem [Arr63] and the Gibbard-Satterthwaite theorem [Gib73, Sat75], which state that under certain natural constraints, social choice mechanisms are impossible to construct. In recent years, beginning in Kalai [Kal01], much work has been done in finding robust versions of these theorems, showing "approximate" impossibility remains even when most, but not all, of the constraints are satisfied. We study a spectrum of settings between the case where society chooses a single outcome (a-laGibbard-Satterthwaite) and the choice of a complete order (as in Arrow's theorem). We use algebraic techniques, specifically representation theory of the symmetric group, and also prove robust versions of the theorems that we state. Our relaxations of the constraints involve relaxing of a version of "independence of irrelevant alternatives", rather than relaxing the demand of a transitive outcome, as is done in most other robustness results.