# Dynamical algebraic combinatorics

March 23 to March 27, 2015

at the

American Institute of Mathematics, San Jose, California

organized by

James Propp, Tom Roby, Jessica Striker, and Nathan Williams

## Original Announcement

This workshop will focus on dynamical systems arising from algebraic combinatorics. Some well-known examples of actions on combinatorial objects are the following:
• promotion and evacuation for Young tableaux;
• the action of a Coxeter element on a parabolic quotient of a Coxeter group; and
• crystal operators on highest-weight representations.
Of particular relevance to this workshop are the actions and dynamical systems arising from:
• promotion and rowmotion for order ideals and antichains in posets; and
• their piecewise-linear and birational liftings.
A unifying theme is the central role played by involutions, such as the Bender-Knuth involutions whose composition gives promotion of Young tableaux and the toggle operations whose composition gives rowmotion of order ideals. Typical questions we ask in various contexts are: Why does this product of involutions --- a permutation on a large set --- have such small order? (Or, if it has large order, why does the action nevertheless resonate with a small integer $p$ as a pseudo-period, in the sense that most orbit-sizes are multiples of $p$?) Why do certain combinatorially significant numerical functions (statistics) on the set have the property that the average value of the function on each orbit is the same for all orbits (the homomesy phenomenon)?

Some of the properties of these cyclic actions can be explained by the importation of combinatorial or algebraic models that explain why the action exists. When the cyclic action has predictable orbit structure, this program has been very successful (as seen in the recent flurry of work on the cyclic sieving phenomenon). The encoding of alternating sign matrices under gyration by fully packed loops and their associated link-patterns shows that such models can exist even when the orbits of the cyclic action display resonance and some are quite large. We hope to study further actions of this last sort, such as rowmotion on plane partitions of height greater than two.

Some examples of problems we are interested in are:

• Develop a combinatorial model of alternating sign matrices of size n that explains the existence of the cyclic action, superpromotion, of pseudo-period $3n-2$ with properties similar to gyration.
• Uniformly prove that birational promotion and rowmotion have finite order on all minuscule posets.
• Express known combinatorial actions as compositions of piecewise-linear involutions and investigate their birational analogues.
• Uniformly prove a bijection between nonnesting partitions and clusters related to Panyushev's homomesy conjectures.
The main goals of the workshop are:
• To produce new combinatorial models that explain the existence of known cyclic actions and homomesies.
• To use data provided by cyclic actions, invariants, and homomesies to produce new bijections between combinatorial objects.
• To coordinate work on homomesy and generalized toggle group actions.
• To suggest directions for future research.

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

A list of workshop notes and open problems, by Sam Hopkins.

A pre-workshop problem list prepared by the organizers.

One of the outcomes of the workshop was the creation of a Dynamical Algebraic Combinatorics ("DAC") listserv. To join, send email to James Propp.