Algebra Seminar: The one-sided cycle shuffles in the symmetric group algebra

Welcome to join on site or via Zoom link (Meeting ID: 645 5572 6999, please contact the organizer for passcode)

Abstract: Elements in the group algebra of a symmetric group S_n are known to have an interpretation in terms of card shuffling. I will discuss a new family of such elements, recently constructed by Nadia Lafreniere: Given a positive integer n, we define n elements t_1, t_2, ..., t_n in the group algebra of S_n, where each t_i is the sum of the cycles (i, i+1, ..., j) over all j from i to n. (This includes the trivial cycle (i) = identity.) The first of them, t_1, is known as the top-to-random shuffle and has been studied by Diaconis, Fill, Pitman (among others). The n elements t_1, t_2, ..., t_n do not commute. However, we show that they can be simultaneously triangularized in an appropriate basis of the group algebra (the "descent-destroying basis"). As a consequence, any rational linear combination of these n elements has rational eigenvalues. The maximum number of possible distinct eigenvalues turns out to be the Fibonacci number f_{n+1}, and underlying this fact is a filtration of the group algebra connected to "lacunar subsets" (i.e., subsets containing no consecutive integers). This talk will include an overview of other families (both well-known and exotic) of elements of these group algebras. Some conjectures will be included, as well as the beginning of a new development (yet another family of elements with integer eigenvalues).

This is joint work with Nadia Lafreniere.

This is a seminar in our algebra seminar series.