Spectra of symmetrized shuffling operators

In 1999, Bidigare, Hanlon, and Rockmore (BHR) introduced an interesting family of shuffling operators, or random walks on the symmetric group S_n, and completely determined the spectra of their transition matrices. A favorite example is the random-to-top shuffling operator.

Other interesting shuffling operators occur by symmetrizing these BHR operators, in the sense that one takes the transition matrix times its transpose. An example is the random-to-random shuffling operator, discussed in a 2002 Stanford PhD thesis by Uyemura-Reyes.

This talk will discuss some results and conjectures about some of these symmetrized operators, e.g. when they commute and what their eigenvalues look like. Representations of the symmetric group play a helpful role.