Title: The one-sided cycle shuffles in the symmetric group algebra
| Speaker: | Darij Grinberg |
| Affiliation: | Drexel University |
| Zoom: | Contact Logan Crew or Olya Mandelshtam |
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 Lafrenière: Given a positive integer n, we define n elements t_1, t_2, ..., t_n in
the group algebra of S_n by
t_i = the sum of the cycles (i), (i, i+1), (i, i+1, i+2), ..., (i, i+1, ..., n),
where the cycle (i) is the identity permutation.
The first of them, t_1, is known as the top-to-random shuffle and has been studied by Diaconis, Fill, and 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. I will also briefly discuss the probabilistic meaning of these elements as well as some tempting conjectures.
This is joint work with Nadia Lafrenière.