Coagulation fragmentation processes and some variations
Picture $n$ particles arranged into "blobs" (a partition of $n$). Each time, a pair of particles is randomly chosen. If they are in the same blob, the blob breaks into two (uniformly). If they are in different blobs, the blobs merge. Natural questions arise: What is the stationary distribution? How does the process evolve (say, starting from all singletons)? What is the mixing time? All of these questions have nice answers because the process is a "lumping" of the well understood "random transpositions" process on the symmetric group.
Mackenzie Simper, Arun Ram and I study extensions to general groups lumped by double cosets. Our leading example is $Gl(n,q)$ and the Bruhat decomposition. The lumped process captures the "pivoting permutations" during Gaussian elimination. Analysis requires an excursion into the world of Hecke algebras and symmetric function theory. I will try to explain all of this in "mathematical English".