Study of Smith's Problem and Laitinen's Conjecture

对于有限群G，由球面上恰好有两个不动点的光滑G-作用在不动点处切线表示的所有差组成的实表示环的子集称为G的Smith集。在本研究中，我们计算了许多有限群G的Smith集。此外，我们在Smith集上找到了Laitinen猜想的反例以及对其成立的一族有限群族，这有助于Smith问题的研究的发展。

For a finite group G, the subset of real representation ring consisting of all differences of tangential representations at fixed points of smooth G-actions with exactly two fixed points on spheres is called the Smith set of G. In this research, we computed the Smith sets for many finite groups G.Furthermore, we found counterexamples to Laitinen's Conjecture on the Smith sets as well as a family of finite groups for which the conjecture is valid, which contributed to development of the study of Smith's problem.