Dynamical algebraic and probabilistic combinatorics

The first theme of this project is to study properties of random objects from algebraic combinatorics. This is a strong current trend. One such object is standard Young tableaux of a certain fixed shape.

For a particular shape that we want to study there is a bijection to reduced words of the longest element of the Weyl group B_n. Thus a random tableau also gives a random word. Thus a random tableau also gives a random word.

This would generalize recent discoveries, by us and other researchers, for type A (permutations) to so called sorting networks.

Studying random words of Weyl groups amounts to model and study certain Markov chains which exploits connections between seemingly disparate objects in representation theory, algebraic combinatorics, and statistical physics.The second theme is dynamical algebraic combinatorics.

There an operator acts on the objects and we study how certain statistics on the objects behave, in cyclic sieving (when inserting roots of unity into the generating polynomial gives the number of fixpoints) or homomessy (the average of a statistic is the same in all orbits). In previous work, homomessy was a tool to enumerate set-valued Young tableaux, with one more element than boxes.

We want to extend this to set-valued tableaux with k extra elements. To this end we plan to study toggling at several positions at the same time in a poset. We will study skew parts on random Young tableau with different distributions, this connects the two themes.