#
Dynamical algebraic combinatorics

March 23 to March 27, 2015
at the

American Institute of Mathematics,
Palo Alto, 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.

Papers arising from the workshop: