Applications are closed
for this workshop

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

This workshop, sponsored by AIM and the NSF, will focus on dynamical systems arising from algebraic combinatorics. Some well-known examples of actions on combinatorial objects are:

  • 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.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

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