[math] \newcommand{\mathds}{\mathbb}[/math]

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Consider the symmetric group [math]S_N[/math], regarded as symmetry group of the [math]N[/math] coordinate axes of [math]\mathbb R^N[/math], and so as group of orthogonal matrices:

[[math]] S_N\subset O_N [[/math]]

Compute the main character for this group, then the law of this main character, and work out the [math]N\to\infty[/math] asymptotics.