Card Shuffling, Hyperplane Arrangements, and Mysterious Representations of $S_n$

Last semester, Trevor showed us how tableau combinatorics is useful in solving problems in card shuffling. In this talk, we will examine another, surprising way in which combinatorics pervades this theory: hyperplane arrangements! We will focus on two examples of Markov chains and explain how arrangements help us compute their eigenvalues. Time permitting, we will also explore the eigenspaces of "symmetric" shuffling operators as representations of the symmetric group. These representations are the famous yet mysterious "higher Lie representations", whose decompositions into irreducible representations remains an open problem. No prior knowledge of hyperplane arrangements will be assumed. This will be a pretalk for tomorrow's combinatorics seminar.