Line up a deck of cards on a table and repeatedly switch pairs of them at random. One might ask how many switches are necessary and/or sufficient to produce a deck which is well-shuffled. Answering this question and others like it motivates the study of random walks on groups, which has surprising connections to representation theory. I will provide a gentle introduction to this subject. If time permits, I will also discuss the result of Diaconis and Shahshahani (1981) that for the above shuffling process on an n card deck, a rapid shift from low to high entropy occurs after about (n/2)*log(n) switches.
From Card Shuffling to Representation Theory and Back