Limit profile for the transpose top-2 with random shuffle

The transpose top-$2$ with random shuffle (J. Theoret. Probab., 2020) is a lazy random walk on the alternating group Aₙ generated by $3$-cycles of the form (,n-1,n) and (,n,n-1). We obtain the limit profile of this random walk by comparing it with the random walk on Aₙ generated by all $3$-cycles. Our method employs a non-commutative Fourier analysis analogue of the comparison method introduced by Nestoridi (Electron. J. Probab., 2024). We also give the complete spectrum of the alternating group graph, thus answering a question of Huang and Huang (J. Algebraic Combin., 2019).