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1. The Sn -equivariant top weight Euler characteristic of Mg,n

   https://doi.org/10.1353/ajm.2023.a907705  

   Chan, Melody; Faber, Carel; Galatius, Søren; Payne, Sam (October 2023, American Journal of Mathematics)

   abstract: We prove a formula, conjectured by Zagier, for the $$S_n$$-equivariant Euler characteristic of the top weight cohomology of $$\scr{M}_{g,n}$$. 

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2. The second eigenvalue of some normal Cayley graphs of highly transitive groups

   Huang, Xueyi; Huang, Qiongxiang; Cioaba, Sebastian M (June 2019, The Electronic journal of combinatorics)

   Let $$\Gamma$$ be a finite group acting transitively on $$[n]=\{1,2,\ldots,n\}$$, and let $$G=\mathrm{Cay}(\Gamma,T)$$ be a Cayley graph of $$\Gamma$$. The graph $$G$$ is called normal if $$T$$ is closed under conjugation. In this paper, we obtain an upper bound for \textcolor[rgb]{0,0,1}{the second (largest) eigenvalue} of the adjacency matrix of the graph $$G$$ in terms of the second eigenvalues of certain subgraphs of $$G$$ (see Theorem 2.6). Using this result, we develop a recursive method to determine the second eigenvalues of certain Cayley graphs of $$S_n$$ and we determine the second eigenvalues of a majority of the connected normal Cayley graphs (and some of their subgraphs) of $$S_n$$ with $$\max_{\tau\in T}|\mathrm{supp}(\tau)|\leq 5$$, where $$\mathrm{supp}(\tau)$$ is the set of points in $[n]$ non-fixed by $$\tau$$. 

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3. The second eigenvalue of some normal Cayley graphs of highly transitive groups

   Huang, Xueyi; Huang, Qiongxiang; Cioaba, Sebastian M (June 2019, The Electronic journal of combinatorics)

   Let $$\Gamma$$ be a finite group acting transitively on $$[n]=\{1,2,\ldots,n\}$$, and let $$G=\mathrm{Cay}(\Gamma,T)$$ be a Cayley graph of $$\Gamma$$. The graph $$G$$ is called normal if $$T$$ is closed under conjugation. In this paper, we obtain an upper bound for \textcolor[rgb]{0,0,1}{the second (largest) eigenvalue} of the adjacency matrix of the graph $$G$$ in terms of the second eigenvalues of certain subgraphs of $$G$$ (see Theorem 2.6). Using this result, we develop a recursive method to determine the second eigenvalues of certain Cayley graphs of $$S_n$$ and we determine the second eigenvalues of a majority of the connected normal Cayley graphs (and some of their subgraphs) of $$S_n$$ with $$\max_{\tau\in T}|\mathrm{supp}(\tau)|\leq 5$$, where $$\mathrm{supp}(\tau)$$ is the set of points in $[n]$ non-fixed by $$\tau$$. 

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4. Double coset Markov chains

   https://doi.org/10.1017/fms.2022.106  

   Diaconis, Persi; Ram, Arun; Simper, Mackenzie (January 2023, Forum of Mathematics, Sigma)

   Abstract Let G be a finite group. Let $H, K$ be subgroups of G and $$H \backslash G / K$$ the double coset space. If Q is a probability on G which is constant on conjugacy classes ( $$Q(s^{-1} t s) = Q(t)$$ ), then the random walk driven by Q on G projects to a Markov chain on $$H \backslash G /K$$ . This allows analysis of the lumped chain using the representation theory of G . Examples include coagulation-fragmentation processes and natural Markov chains on contingency tables. Our main example projects the random transvections walk on $$GL_n(q)$$ onto a Markov chain on $$S_n$$ via the Bruhat decomposition. The chain on $$S_n$$ has a Mallows stationary distribution and interesting mixing time behavior. The projection illuminates the combinatorics of Gaussian elimination. Along the way, we give a representation of the sum of transvections in the Hecke algebra of double cosets, which describes the Markov chain as a mixture of Metropolis chains. Some extensions and examples of double coset Markov chains with G a compact group are discussed. 

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5. The Second Eigenvalue of some Normal Cayley Graphs of Highly Transitive Groups

   https://doi.org/10.37236/8054  

   Huang, Xueyi; Huang, Qiongxiang; Cioabă, Sebastian M. (April 2019, The Electronic Journal of Combinatorics)

   Let $$G$$ be a finite group acting transitively on $$[n]=\{1,2,\ldots,n\}$$, and  let $$\Gamma=\mathrm{Cay}(G,T)$$ be a Cayley graph of $$G$$. The graph $$\Gamma$$ is called  normal if $$T$$ is closed under conjugation. In this paper, we obtain an upper bound for the second (largest) eigenvalue of the adjacency matrix of the graph $$\Gamma$$ in terms of the second eigenvalues of certain subgraphs of $$\Gamma$$. Using this result, we develop a recursive method to  determine the second eigenvalues of certain  Cayley graphs of $$S_n$$, and we determine the second eigenvalues  of a majority of the connected normal Cayley graphs (and some of their subgraphs) of $$S_n$$  with  $$\max_{\tau\in T}|\mathrm{supp}(\tau)|\leqslant 5$$, where $$\mathrm{supp}(\tau)$$ is the set of points in $[n]$ non-fixed by $$\tau$$. 

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