Spectral gap of the adjacent transposition shuffle ($\gamma =2(1-\cos (\pi /n))/n$)

For the random adjacent transposition shuffle with $n\ge 2$, the generator $P-I$ is the interchange process on a path graph with each edge rate $1/n$. The Aldous spectral gap theorem reduces its spectral gap to that of the single-label random walk. The eigenfunctions $k\mapsto \cos (\ell \pi (k-1/2)/n)$, for $0\le \ell <n$, have generator eigenvalues $-2(1-\cos (\ell \pi /n))/n$; direct substitution verifies both the interior and endpoint equations. Consequently $\gamma \sim {\pi }^2/n^3$.