AI 中文总结
本文针对Doignon 2023年提出的最佳-最差选择多面体维数问题，以对称群表示论解释为启发，用初等线性代数方法给出该维数的显式公式并解答该问题。
AI 中文摘要
对于包含$n$个备选方案的集合，最佳-最差选择多面体是由所有备选方案的线性排序所诱导的确定性最佳-最差选择向量的凸包。确定该多面体作为$n$的函数的维数这一问题由Doignon于2023年提出。本文通过找到该维数的显式公式来回答这一问题。我们的证明受对称群相关秩问题的表示论解释启发，但完全以初等线性代数术语呈现。与表示论的联系将在其他地方探讨。
英文摘要
For a set of $n$ alternatives, the best--worst choice polytope is the convex hull of the deterministic best--worst choice vectors induced by all linear rankings of the alternatives. The question of determining the dimension of this polytope as a function of $n$ has been raised by Doignon \cite{Doignon2023}. In this paper we answer this question by finding an explicit formula for this dimension. Our proof is motivated by a representation-theoretic interpretation of the relevant rank problem for the symmetric group, but is presented entirely in elementary linear-algebraic terms. The connection with representation theory will be explored elsewhere.
