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

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

Speaker: Darij Grinberg, Drexel
Date and time: Monday, April 10, 4–5 pm
Place: Rome 206

Abstract: Given a positive integer n, we define n elements t_1, t_2,
..., t_n in the group algebra of the symmetric group 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). Note that t_1 is
the famous "top-to-random shuffle" element studied by many.

These 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. Various surprises emerge in describing
these eigenvalues and their multiplicities; in particular, the Fibonacci
numbers appear prominently.

This talk will include an overview of other families (both well-known
and exotic) of elements of these group algebras. A card-shuffling
interpretation will be given and some tempting conjectures stated.

This is joint work with Nadia Lafrenière.