University of Melbourne
Time: 16:00-18:00 (GMT+8), Tuesday April 26th, 2022
Location: Tencent Meeting
Abstract: Aldous' spectral gap conjecture states that the second largest eigenvalue of any connected Cayley graph on the symmetric group \(S_n\) with respect to a set of transpositions is achieved by the standard representation of \(S_n\). This celebrated conjecture, which was proved in its general form in 2010, has inspired much interest in searching for other families of Cayley graphs on \(S_n\) with the "Aldous property" that the largest eigenvalue strictly smaller than the degree is attained by the standard representation of \(S_n\).
In this talk, I will first introduce the probabilistic background of Aldous' conjecture and different ways of viewing this conjecture from the perspective of algebraic graph theory. I will then review some basic results on representation theory of finite groups and symmetric groups. Finally, I will report three results on normal Cayley graphs on \(S_n\) possessing the Aldous property for sufficiently large \(n\), one of which can be viewed as a generalization of the "normal" case of Aldous' spectral gap conjecture. This talk is based on collaborative work with Binzhou Xia and Sanming Zhou.
Host: 黄弘毅 Hong Yi Huang