Darij Grinberg: The random-to-random shuffles and their q-deformations

Abstract: Consider a random shuffle acting on a deck of $n$ cards as follows: Uniformly at random, we select $k$ out of our $n$ cards, remove them from the deck, and then move them back to $k$ uniformly random positions.

This shuffle -- the so-called "$k$-random-to-random shuffle" -- is a Markov chain that is given by a certain element of the group algebra of the symmetric algebra. A celebrated result of Dieker, Saliola and Lafrenière says that this shuffle is diagonalizable with all eigenvalues rational. Earlier, it was observed by Reiner, Saliola and Welker that two such shuffles for different $k$'s always commute. Both results are deep and hard.

I will discuss a new approach to these shuffles that has resulted in simpler proofs as well as a $q$-deformation -- i.e., a generalization into the Hecke algebra of the symmetric group. Along the way, some properties of the Hecke algebras have been revealed, as well as some general results about integrality of eigenvalues.

Joint work with Sarah Brauner, Patricia Commins and Franco Saliola.

Preprint: https://www.cip.ifi.lmu.de/~grinberg/algebra/r2r2.pdf