The Tsetlin library is a random-to-top shuffle process on permutations.
A q-deformation using the Hecke algebra can also be implemented on
complete flags over the finite field F_q^n. The semigroup
underlying the random-to-top Markov chain on flags is a left regular
band, which provides a method to compute the stationary distribution and
eigenvalues. We consider a further generalisation by introducing
inhomogeneous rates. This is work in collaboration with Sarah Brauner,
Jan de Gier and Anne Schilling and is available at arXiv:2601.21195.