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How do I find eigenvalues for the adjacency matrix of Cayley graph $X(S_n,S)$ where $S_n$ is the symmetric group of order $n$ and $S$ is the set of transpositions $(i,i+1)$, if the eigenvalues of the graph $X(S_n,T)$ are given by $|T|\chi(1,2)/\chi(i,j)$ with $T$ being the set of transpositions $(i,j)$, $1<i<j<n$.

Thank you

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    $\begingroup$ The eigenvalues are much easier to compute for generating sets closed under conjugation like $T$. $\endgroup$
    – Benjamin Steinberg
    Aug 9, 2020 at 18:23
  • $\begingroup$ Thank you. But how do I find the eigenvalues of the reduced set S? $\endgroup$
    – user625452
    Aug 10, 2020 at 0:13
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    $\begingroup$ I don’t think there is any way to get the eigenvalues for the set S from T but I wouldn't be surprised if they are known. The set S of generators is the set of Coxeter or Coxeter-Moore generators. You can't immediately get the eigenvalues just using character theory because S is not in the center of the group algebra $\endgroup$
    – Benjamin Steinberg
    Aug 10, 2020 at 0:43
  • $\begingroup$ Many many thanks $\endgroup$
    – user625452
    Aug 10, 2020 at 7:13

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