Minimal eigenvalue of the Coxeter Laplacian for the symmetric group (Q1330041)

Let \(X\) denote a finite graph (with loops and multiple edges). Let \(X^ 0\) be its vertex set, \(X^ 1\) its edge set, and let \(l^ 2(X^ 0)\) denote the space of functions from \(X^ 0\) to the complex numbers, with an inner product that has the characteristic functions of the vertices as an orthonormal basis. Let \(a_{x,y}\) denote the number of edges joining the two vertices \(x\) and \(y\). The Laplacian of \(X\) is the linear operator \(\Delta\) of \(l^ 2(X^ 0)\) defined by \[ \Delta v(x)= \deg(x) v(x)- \sum_{y\in X^ 0} a_{x,y} v(y) \] for all \(v\in l^ 2(X^ 0)\). The eigenvalues of \(\Delta\) are necessarily nonnegative. The author proves that if \(X\) is the Cayley graph of the symmetric group \(S_ n\) relative to the generator set consisting of the transpositions \((1 2)\), \((2 3),\dots, (n-1 n)\), then the least positive eigenvalue of \(\Delta\) has value \(2- 2\cos{\pi\over n}\).