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
- 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.
- 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.
- Daugherty, Z., Eustis, A. K., Minton, G., Orrison, M. E. (2009).
- Voting, the symmetric group, and representation theory. The American Mathematical Monthly, 116(8), 667-687.
- Saari, D. G. (1999). Explaining all three-alternative voting outcomes. Journal of Economic Theory, 87(2), 313-355.
- Saari, D. G. (2000). Mathematical structure of voting paradoxes. Economic Theory, 15(1), 1-53; 55-102