  We generalize random-to-random shuffling from a Markov chain on the symmetric
group to one on the Type A Iwahori Hecke algebra, and show that its eigenvalues
are polynomials in q with non-negative integer coefficients. Setting q=1
recovers results of Dieker and Saliola, whose computation of the spectrum of
random-to-random in the symmetric group resolved a nearly 20 year old
conjecture by Uyemura-Reyes. Our methods simplify their proofs by drawing novel
connections to the Jucys-Murphy elements of the Hecke algebra, Young seminormal
forms, and the Okounkov-Vershik approach to representation theory.
