**Nadia Lafrenière (Dartmouth College) - Eigenvalues of the symmetrized shuffling operators**

**February 10th, 4:00 PM, Innovation E432**

Abstract: When looking at a procedure for shuffling a deck of cards, one can ask for the necessary and sufficient amount of time one needs to repeat it so that the deck is well shuffled. An important statistics in the study of that question is the eigenvalues of a Markov chain. For a given shuffle, that Markov chain is build on the state space of arrangements of the deck (the permutations).

The random-to-random shuffle corresponds to removing one card (randomly) and placing it back into the deck at a random place. In this talk, I will build the Markov chains for shuffling techniques similar to random-to-random called the symmetrized shuffling operators, and introduced by Victor Reiner, Franco Saliola and Volkmar Welker in 2014. I will show that their eigenvalues are all non-negative integers by writing them as statistics on combinatorial objects called standard tableaux. I will present some conjectures on the mixing time, that are likely to be proven using the eigenvalues.