Algebraic Voting Theory & Representations of $S_m\wr S_n$

We consider the problem of selecting an $n$-member committee made up of one of $m$ candidates from each of $n$ distinct departments. Using an algebraic approach, we analyze positional voting procedures, including the Borda count, as $\mathbb{Q}{S}_m\wr S_n$-module homomorphisms. In particular, we decompose the spaces of voter preferences and election results into simple $$\mathbb{Q}$S_m \wr S_n$-submodules and apply Schur’s Lemma to determine the structure of the information lost in the voting process. We conclude with a voting paradox result, showing that for sufficiently different weighting vectors, applying the associated positional voting procedures to the same set of votes can yield arbitrarily different election outcomes.