表示论在投票理论中的应用

似乎主要是 Donald G. Saari 和 Zajj Daugherty 等人的工作...

大致思路:用 tabloid 来描述投票中的对象,然后将过程中的状态空间视为 \(\mathbb{Q}S_n\)-模,然后将其分解为不可约模的直和,并分析在投票过程中真正起作用的成分。

太懒了...直接把曹老师作报告用的 Slides 放上来吧...

参考文献

  1. Barcelo, H., Bernstein, M., Bockting-Conrad, S., Mcnicholas, E., Nyman, K., Viel, S. (2018). Algebraic voting theory and representations of \(S_m ≀ S_n\) . arXiv:1807.03743v1 [math.CO] 6 Jul 2018.
  2. Crisman, K. D., Orrison, M. E. (2017). Representation theory of the symmetric group in voting theory and game theory. Algebraic and Geometric Methods in Discrete Mathematics, 685, 97.
  3. Daugherty, Z., Eustis, A. K., Minton, G., Orrison, M. E. (2009).
  4. Voting, the symmetric group, and representation theory. The American Mathematical Monthly, 116(8), 667-687.
  5. Saari, D. G. (1999). Explaining all three-alternative voting outcomes. Journal of Economic Theory, 87(2), 313-355.
  6. Saari, D. G. (2000). Mathematical structure of voting paradoxes. Economic Theory, 15(1), 1-53; 55-102