Darij Grinberg : Shuffles in the symmetric group algebra

Ever since the famous 1992 work of Bayer and Diaconis, it has been known that random shuffles of a deck of cards (with the back side up) can be modelled as elements of the group algebra $\mathbb{R}[S_n]$ of the symmetric group $S_n$. This viewpoint has spawned progress in both card shuffling and the representation theory of the symmetric group. In this talk, I will focus on two projects in the latter: one focusing
on the "somewhere-to-below shuffles" 
$$t_i := (i) + (i,i+1) + (i,i+1,i+2) + \cdots + (i,i+1,\ldots,n) \in \mathbb{R}[S_n]$$
for $1 \leq i \leq n$ (where the parenthesized expressions mean cycles; the $1$-cycle $(i)$ is the identity), and one focusing on the "$k$-random-to-random shuffles" 
$$R_k := \sum_{1 \leq i_1 < i_2 < \cdots < i_k \leq n} \sum_{w \in S_n \text{ such that } w(i_1) < w(i_2) < \cdots < w(i_k)} w \in \mathbb{R}[S_n]$$
for $0 \leq k \leq n$. Both families have revealed a variety of unexpected properties. For instance, the $R_0, R_1, \ldots, R_n$ commute, whereas the $t_1, t_2, \ldots, t_n$ are simultaneously triangularizable (i.e., there is an -- explicitly describable -- basis of $\mathbb{R}[S_n]$ on which right multiplication by each $t_i$ acts as a triangular matrix). In both cases, all eigenvalues are integers and can be explicitly described and assigned to Specht modules (irreducible representations of $S_n$). Many of these properties furthermore generalize to the (type-A) Iwahori-Hecke algebra.

Due to the amount of results, this talk will be an overview with no proofs.

Some of the above is joint work with Nadia Lafrenière, Sarah Brauner, Patricia Commins and Franco Saliola.