Schedule for: 21w5514 - Dynamical Algebraic Combinatorics

In this talk, we systematize a technique for proving that various statistics of interest are homomesic by writing these statistics as linear combinations of “toggleability statistics” (originally introduced by Striker) plus a constant. This technique recaptures most of the known homomesies for the posets on which rowmotion has been most studied. We will give illustrations of this. Furthermore, we will introduce a q-analogue of rowmotion and show that this technique continues to work in this modified context and yields homomesies for “q-rowmotion”. This is joint work with Colin Defant, Sam Hopkins, and James Propp.

In this talk we introduce a class of lattice-posets called semidistributive lattices. Like the unique factorization domains of commutative algebra, each element of a finite semidistributive lattices has a unique ``factorization'' in terms of the join operation and a unique ``factorization'' in terms of the meet. In this talk, we discuss a map, which we call the kappa-map, that translates between these two unique factorizations. We relate the kappa-map to the Kreweras complement of noncrossing partitions and rowmotion. Finally, we discuss new results related to the representation theory of quivers. The talk will be self-contained, and all are welcome.

1. Jim Propp - q-Panyushev operation 2. Bruce Sagan - Rowmotion on fences 3. Sam Hopkins - Evacuation and the toggle description of RSK 4. Sylvester Zhang - A shifted octahedral recurrence 5. Colin Defant - Non-invertible rowmotion on meet-semidistributive lattices 6. Brian Hopkins - Cycles in Austrian solitaire (dynamics on labeled partitions)

A fence is a poset obtained from a sequence of chains by identifying maximal and minimal elements in an alternating fashion. We investigate rowmotion on antichains and ideals of fences. In particular, we show that orbits of antichains can be visualized using tilings. This permits us to prove various homomesy results for the number of elements of an antichain or ideal in an orbit. Rowmotion on fences also exhibits a new phenomenon, which we call orbomesy, where the value of a statistic is constant on orbits of the same size. Along the way, we prove a homomesy result for all self-dual posets and show that any two Coxeter elements in certain toggle groups behave similarly with respect to homomesies which are linear combinations of ideal indicator functions. We end with some conjectures and avenues for future research. This is joint work with Sergi Elizalde, Matthew Plante, and Tom Roby.

In order to give representation-theoretic proofs of cyclic sieving results of Reiner-Stanton-White and Pechenik, in 2014 the presenter used certain set partition skein relations to give a combinatorially defined action of $S_n$ on the vector space spanned by noncrossing partitions. In 2020 joint work with Jongwon Kim, the presenter determined the bigraded $S_n$-structure of the ring of fermionic diagonal coinvariants, proving a conjecture of Zabrocki. We explain how the skein action of the symmetric group arises naturally in the context of fermions. Joint with Jesse Kim.

The Specht modules $S^{\lambda}$, indexed by partitions, are the irreducible representations of the symmetric group. We can realize $S^{\lambda}$ as a certain graded piece of a ring of invariants, equivalently as global sections of a line bundle on a partial flag variety. There are many general ways to choose useful bases of this module. Particularly powerful are web bases, which make connections with cluster algebras and quantum link invariants, except that web bases are only available in very special cases; essentially, we only know web bases in the cases $\lambda=(m,m)$ and $\lambda=(m,m,m)$. Building on work of B. Rhoades, we find what appears to be a web basis of invariants for a special family of Specht modules with lambda of the form $(a,a,1^b)$. The planar diagrams that appear are noncrossing set partitions, and we thereby obtain geometric interpretations of earlier cyclic sieving phenomena. (For a different algebraic interpretation, see Rhoades' talk at this workshop.)

The purpose of this talk is to advertise noninvertible combinatorial dynamics, a largely unexplored area with several interesting directions. We will focus on the pop-stack sorting map, a specific noninvertible operator on the symmetric group, along with several of its recently-defined variants. We will discuss generalizations and extensions of the pop-stack sorting map to Coxeter groups and meet-semilattices. We will also see how to use pop-stack sorting to extend the definition of rowmotion to meet-semidistributive lattices; the resulting rowmotion operators are not always invertible. Finally, we will consider "dual" versions of the Coxeter group pop-stack sorting operators called Coxeter pop-tsack torsing operators. The discussions of rowmotion for meet-semidistributive lattices and Coxeter pop-tsack torsing are based on joint work with Nathan Williams.

We give a natural definition of rowmotion for 321-avoiding permutations, by translating, through bijections involving Dyck paths and the Lalanne-Kreweras involution, the analogous notion for antichains of the positive root poset of type A. We prove that some permutation statistics, including the number of fixed points, are homomesic under rowmotion, meaning that they have a constant average over its orbits. We also show that the Armstrong-Stump-Thomas equivariant bijection between order ideals in type A and non-crossing matchings can be described in terms of the Robinson–Schensted–Knuth correspondence on permutations. This is joint work with Ben Adenbaum.

(1) Emily Gunawan - Box-ball systems and Robinson--Schensted--Knuth tableaux (2) Alexander Lazar (joint with Sam Hopkins and Svante Linusson) - Refined Enumeration of Barely Set-Valued Tableaux and Reverse Plane Partitions (3) Jaeseong Oh - Macdonald polynomials and cyclic sieving, (4) Joseph Pappe and Anne Schilling - An area-depth symmetric q,t-Catalan polynomial (5) Matthew Plante - Periodicity and Homomesy for Whirling Proper 3-Colorings of a Graph (6) Stefan Trandafir - Littlewood-Richardson coefficients from the vector partition perspective

In this talk we consider a generalization of the results of Joseph and Roby on a toggle dynamical system whose state space consists of independent sets on a path graph. Associated with this dynamical system, a simple generalization of the rotation (or circular shift) of the binary sequences arises. We show each orbit of this generalized rotation has a certain statistical symmetry.

We will discuss recent developments around famous inequalities concerning linear extensions of general posets. We will sketch some combinatorial proofs using lattice path bijections. We will also discuss the algebraic framework on restricted linear extensions, and applications of promotion. Joint work with Swee Hong Chan and Igor Pak.

Promotion is an important cyclic action on the set of standard Young tableaux of a fixed shape (or more generally, the set of linear extensions of a poset). Following the work of Khovanov-Kuperberg, Petersen-Pylyavskyy-Rhoades, and Tymoczko, it is known that promotion of three row rectangular SYTs is in equivariant bijection with rotation of (sl(3)-)webs. Here webs are certain bipartite, planar graphs embedded in a disc introduced by Kuperberg to study tensor invariants. In this talk we will advertise a new observation about how Postnikov's theory of plabic graphs also fits nicely into this story. Namely, any web can directly be viewed as a plabic graph, and hence has an associated trip permutation. But also, each SYT determines a permutation via the growth diagram of its promotion orbit. The observation is that, under the bijection between SYTs and webs, these two permutations coincide. This talk is based on joint work with Martin Rubey (arXiv:2005.14031), although that paper discusses a slightly different and more exotic context: promotion of linear extensions of the poset V x [n]. The aims of this talk are to ask the audience what combinatorial consequences this connection between promotion and plabic graphs might entail, and whether the connection has any algebraic meaning.

The operation of birational rowmotion on a finite poset has been a mainstay in dynamical algebraic combinatorics for the last 8 years. Since 2015, it is known that for rectangular posets of the form $[p] \times [q]$ this operation is periodic with period $p+q$. (This result, as has been observed by Max Glick, is equivalent to Zamolodchikov's periodicity conjecture in type AA, proved by Volkov.) In this talk, I will outline a proof (joint work with Tom Roby) of a noncommutative generalization of this result. The generalization does not quite extend to the full generality one could hope for -- it covers noncommutative rings, but not semirings; however, the proof is novel and simpler than the original commutative one. Extending this to other posets is work in progress.

We introduce semidistrim lattices, a simultaneous generalization of semidistributive lattices and trim lattices that shares many of their properties—for example, intervals of semidistrim lattices are again semidistrim. We define a rowmotion operator on semidistrim lattices that generalizes the definition Barnard gave for semidistributive lattices and the definition Thomas and the second author gave for trim lattices. This is joint work with Colin Defant.