Sarah Brauner: Spectra of shuffling operators via combinatorial representation theory

Sarah Brauner (Brown)

In algebraic combinatorics, we are often interested in naturally arising sequences of positive integers because they detect the presence of deep underlying algebraic structure. Proving properties of these sequences can in turn forge connections between combinatorics, representation theory, geometry, topology, probability, and more. In this talk, I will discuss a surprising source of positive integers: the eigenvalues of certain Markov chains modeling card shuffling. My work generalizes these processes to the Type A Iwahori Hecke algebra, where they become richer from a combinatorial, algebraic, and probabilistic perspective. We apply tools from combinatorial representation theory to illuminate long-standing mysteries about these operators, including that their eigenvalues are polynomials with non-negative integer coefficients.