Abstract 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 Q S m ≀ S n -module homomorphisms. In particular, we decompose the spaces of voter preferences and election results into simple Q S m ≀ 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 vastly different election outcomes.

中文翻译：

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S≀S的代数投票理论和表示

摘要 我们考虑选择一个由来自 n 个不同部门中的每个部门的 m 名候选人之一组成的 n 成员委员会的问题。使用代数方法，我们将位置投票程序（包括 Borda 计数）分析为 QS m ≀ S n 模同态。特别是，我们将选民偏好和选举结果的空间分解为简单的 QS m ≀ S n -子模块，并应用 Schur 引理来确定在投票过程中丢失的信息的结构。我们得出一个投票悖论结果，表明对于足够不同的权重向量，将相关的位置投票程序应用于同一组投票可以产生截然不同的选举结果。