# 群与图讨论班 Seminars on Groups and Graphs

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# Aldous' spectral gap conjecture and generalizations

## 李雨轩 Yuxuan Li

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

Slides