## Combinatorics and beyond: the many facets of Sergey Fomin's mathematics, November 8-11, 2018 University of Michigan

Organizers: T. Lam, D. Speyer, L. Williams, A. Yong

This conference is a celebration of Sergey Fomin's mathematics (and his 60th birthday). It will be held at the University of Michigan. .

#### Schedule:

 Thursday 11/8 Friday 11/9 Saturday 11/10 Sunday 11/11 freemorning 9:00-9:30: coffee 9:00-9:30: coffee 9:00-9:30: coffee 9:30-10:30: Nathan Reading 9:30-10:30: Dylan Thurston 9:30-10:30: Michael Shapiro 10:30-11:00: break 10:30-11:00: break 10:30-11:00: break 12:30-1:30: Boris Shapiro 11:00-12:00: Karola Meszaros 11:00-12:00: Philippe DiFrancesco 11:00-12:00 Richard Stanley 1:30-2:00: break 12:00-1:30: lunch 12:00-1:30: lunch 12:00-12:15: Concluding remarks by Sergey Fomin 2:00-3:00: Chris Fraser 1:30-2:30: Kyungyong Lee 1:30-2:30: Max Glick goodbye 3:00-3:30: break 2:30-3:00: break 2:30-3:00: break 3:30-4:30: Anna Weigandt 3:00-4:00: Jonah Blasiak 3:00-4:00: William Fulton 4:30-5:00: break 4:00-4:30: break 4:00-4:30: break 5:00-6:00: Patricia Hersh 4:30-5:30: Sara Billey 4:30-5:30: Alex Postnikov conferencedinner

#### General information:

Scientific program: the workshop will start on Thursday at 12:30pm (November 8) and end at 12:30pm on Sunday (November 11).

Location on Thursday and Friday: On Thursday and Friday, the conference will be held in Palmer Commons Forum Hall (4th floor).

Getting to Palmer Commons: Both Google Maps and the walking directions here make references to North Hall, which was demolished in 2015. Instead take a look at THIS MAP.

When walking from the math department to Palmer Commons via East University pedestrian mall, cross North University Ave and continue on, with the School of Dentistry on your left and the new Biological Sciences Building (which replaced North Hall) on your right. Then continue on slightly to the right, with the Undergraduate Science Building on your left and the Life Sciences Institute on your right. As you approach the pedestrian bridge across Washtenaw Ave, Palmer Commons will be on your left. You will enter the building at the 3rd floor. Forum Hall is on the 4th floor.

Visitor parking (at $1.70/hour) is available at the nearby Palmer Parking Structure. (See here for directions.) But (as of 12pm Thursday!) it is currently full. Instead you can park at the Forest Parking at Forest Street and South University. At 12pm on Thursday, Sergey will lead people from the math department (East Hall) to Palmer Commons. If you'd like to join the group, meet at 12pm in the elevator lobby on the 1st floor of the math department (East Hall). Location on Saturday and Sunday: On Saturday and Sunday, the conference will be held in East Hall 1360. Registration: if you're planning to come to the conference please fill out this google poll. Any requests for funding (for early career participants) must be made by October 1st. Accommodation: Conference dinner: the conference dinner will take place on Saturday night. More details will be given later. Travel: The Detroit Metropolitan Airport is about 25 miles from campus. Invited participants: When booking flights, please keep in mind NSF and University of Michigan rules. We will be able to partially cover travel for invited participants (the precise meaning of "partially" will depend on the final number of participants, and their needs). TITLES and ABSTRACTS (with some slides): Sara Billey: Patterns in Standard Young Tableaux and beyond Abstract: Standard Young tableaux are fundamental in combinatorics, representation theory, and geometry. The major index statistic originally defined for permutations has been extended to tableaux and can be interpreted in each of these settings. We consider the probability distribution of the major index on standard tableaux of fixed partition shape chosen uniformly along with the corresponding generating function. We give an explicit hook length style formula for all of the cumulants of these distributions using recent work of Chen--Wang--Wang and Hwang--Zacharovas. The cumulant formula allows us to classify all possible limit laws for any sequence of shapes in terms of a simple auxiliary statistic, aft, generalizing earlier results of Canfield--Janson--Zeilberger, Chen--Wang--Wang, and others. We show that any such sequence of distributions with aft approaching infinity is asymptotically normal. This approach was inspired by earlier work of Josh Swanson on the distribution of modular major index and a conjecture of Sundaram. This leads to a series of questions concerning locations of zero coefficients, unimodality, and asymptotic estimates for the major index generating functions over all standard tableaux of a fixed shape. We settle the first of these by identifying mutation rules in terms of tableaux patterns leading to both a strong and weak poset structure on tableaux ranked by the major index. The classification of zeros can be interpreted as determining which irreducible representations of the symmetric group exist in each homogeneous components of the corresponding coinvariant algebra. By work of Lusztig and Stembridge, the arguments extend to a classification of all nonzero fake degrees of coinvariant algebras for finite complex reflection groups in the infinite family of Shephard--Todd groups. We give conjectured answers concerning unimodality and asymptotic estimates. This talk is based on joint work with Matjaz Konvalinka and Joshua Swanson. Jonah Blasiak: Catalan functions and k-Schur positivity Abstract: Catalan functions are a family of symmetric functions indexed by pairs consisting of a partition contained in the staircase (n-1, ..., 1, 0) and a weight in Z^n. They include the Hall-Littlewood polynomials and their parabolic generalizations. Li-Chung Chen and Mark Haiman conjectured that the k-Schur functions are a subclass of Catalan functions. We settle their conjecture, expose a new miraculous shift invariance property of k-Schur functions, and use this to establish Schur positivity of k-Schur functions. This is joint work with Jennifer Morse, Anna Pun, and Dan Summers. Philippe DiFrancesco: Macdonald operators and cluster algebras Abstract: We introduce generalized (q,t) Macdonald difference operators that interpolate between quantum cluster algebras associated to Q-systems and the Elliptic Hall, or Ding-Iohara-Miki algebra which play a central role in the so-called AGT conjecture. We will show how some of these operators for type A arise naturally from Double-Affine Hecke Algebra representations, and obey triangle relations in the Elliptic Hall algebra as a consequence. The Q-system cluster algebras are recovered in the limit as t goes to infinity. Generalizations to B,C,D types will also be presented. Chris Fraser: Braiding in Grassmannian cluster algebras Abstract: We will give an introduction to the cluster algebra structure on the Grassmannian, and introduce an action of (a version of) the braid group on Gr(k,n). This braid group action preserves cluster combinatorics. Using the action, we give evidence for a conjectural description of cluster variables in Grassmannians due to Fomin and Pylyavskyy. Williams Fulton: Degeneracy Locus Formulas Via Non-Intersecting Paths Absract: Cohomology formulas for many degeneracy loci in types B, C, and D can be given by pfaffians. Stembridge showed that many pfaffians can be given by non-intersecting paths. Carrying this out gives some new combinatorial formulas for these loci, and gives more loci in type D that have pfaffian formulas. This is joint work with Dave Anderson. Max Glick: The limit point of the pentagram map Abstract: The pentagram map is a discrete dynamical system defined on the space of polygons in the plane. In the first paper on the subject, R. Schwartz proved that the pentagram map produces from each convex polygon a sequence of successively smaller polygons that converges exponentially to a point. We investigate the limit point itself, giving an explicit description of its Cartesian coordinates as roots of certain degree three polynomials. Abstract: Anders Björner and Joseph Bernstein raised the question of finding regular CW complexes naturally arising from representation theory having the intervals of Bruhat order as their posets of closure relations. Sergey Fomin and Michael Shapiro conjectured a solution, namely the link of the identity in the totally nonnegative real part of the unipotent radical of a Borel in a semisimple, simply connected algebraic group defined and split over the reals together with a family of related spaces indexed by the different Coxeter group elements. The Fomin-Shapiro conjecture indeed proved to be true, with the proof utilizing an interpretation of these stratified spaces as images of an intriguing family of maps — maps also arising in work of Lusztig related to canonical bases. I will start by reviewing some highlights of this story, then turn to recent joint work with Jim Davis and Ezra Miller regarding the structure of the fibers of these same maps. Kyungyong Lee: Subtraction-free combinatorics on lattice paths Abstract: Let f(x) be a polynomial function. When f(a)-f(b)>0, we want to find a subtraction-free combinatorial expression for the difference. Such an expression can be given in terms of subpaths of lattice paths. Using this combinatorics, we explain positivity of cluster algebras. Karola Meszaros: Abstract: Grothendieck polynomials via root and flow polytopes The normalized volumes of certain root and flow polytopes equal the number of reduced pipedreams of certain permutations. Moreover, the Ehrhart series of the aforementioned polytopes can be expressed through specializations of Grothendieck polynomials. We explain these results by establishing a connection between triangulations of root and flow polytopes and the combinatorial expression of Grothendieck polynomials in terms of pipedreams. We then show that the Newton polytope of the Schubert polynomial for any permutation is a generalized permutahedron. Moreover, we prove the analogous statement for all homogeneous parts of Grothendieck polynomials of certain permutations. We achieve this by exploiting the connections between triangulations of root and flow polytopes and integer points of generalized permutahedra. This talk is based on joint works with Laura Escobar, Alex Fink and Avery St. Dizier. Alex Postnikov: Positroids and polytopes Abstract: The nonnegative Grassmannian is a cell complex with rich geometric, algebraic, and combinatorial structures. Its cells correspond to combinatorial objects, called positroids. Initially, this study was motivated by Sergey Fomin and Andrei Zelevinsky's theory of double Bruhat cells, and their theory of cluster algebras. Remarkably, these structures also appeared in many other areas of mathematics and physics, e.g., the study of scattering amplitudes, solitons, etc. We'll discuss old and new results on positroids. We'll survey earlier results, including work of the speaker, joint works with David Speyer and Lauren Williams and with Suho Oh and David Speyer. We'll mention a recent result of Pavel Galashin, Steven Karp, and Thomas Lam on regularity of the nonnegative Grassmannian. We'll talk about a project with Thomas Lam on polypositroids, where positroids are explained in terms of polyhedral geometry. They are closely related to generalized permutohedra and alcoved polytopes. We'll discuss discrete Plateau's problem on minimal surfaces, and extend positroids in general Lie theoretic language using combinatorics of root systems and Coxeter elements. We'll discuss the relations with Billera-Sturmfels' fiber polytopes and the Baues poset of polyhedral subdivisions. Nathan Reading: A fan for every cluster algebra Abstract: In this talk, I will tell a story that starts with the classification of cluster algebras of finite type by Fomin and Zelevinsky. The combinatorial essence of cluster algebras of finite type is a fan (in fact, the normal fan of a polytope--a generalized associahedron). The fan records the d-vectors or g-vectors of clusters, and in infinite type, there is an analogous infinite fan of d-vectors or g-vectors. These infinite fans are not complete, however, and it became clear from work in rank 2 and in the surfaces case that the space outside the g-vector/d-vector fan is vital to understanding the cluster algebra. Recently, Gross, Hacking, Keel, and Kontsevich constructed cluster scattering diagrams and used them to prove many of Fomin and Zelevinsky's structural conjectures on cluster algebras. They showed that the cluster scattering diagram (a collection of codimension-1 cones) "cuts out" the g-vector fan. More recently, I showed that the cluster scattering diagram cuts out a complete fan containing the g-vector fan as a subfan. This fan is the most general "generalized associahedron fan", the combinatorial essence of a general cluster algebra. The cluster scattering diagram arises from a non-constructive (in practice) existence theorem. Thus combinatorial models for cluster scattering diagrams and fans are crucially needed. After reviewing some of the backstory, I will describe work in progress to construct cluster scattering diagrams and fans in affine type (with Stella) and in the surfaces case (with Muller and Viel). Boris Shapiro: Several algebras associated to a (multi)graph Abstract: I recall the definitions of two natural algebras associated to a given (multi)graph which were originally introduced about 20 years ago by M.Shapiro and myself and later extended jointly with A.Postnikov. I survey some recent developments in the study of these and several related algebras obtained mainly by A.Kirillov, G.Nenashev. (joint with G. Nenashev, A. Postnikov, and M. Shapiro) Michael Shapiro: Non-commutative pentagram map Abstract: A pentagram map is an integrable map in the configuration space of n points in a projective plane. Its generalization to collection of points in Grassmannian was described by Felipe and Mari Beffa. This construction can be viewed as a usual pentagram map with coefficients in k x k matrices. We discuss a pentagram map with fully non-commutative coefficients, and show a non-commutative Liouville integrability. After a work by N.Ovenhouse. Richard Stanley: I. Stern's diatomic array and beyond II. A conjecture on the weak order of the symmetric group Abstract: This talk will have two unrelated parts. I. Stern's diatomic array is a certain array of numbers somewhat analogous to Pascal's triangle. We consider a slight variation whose nth row is the sequence of coefficients of the polynomial$\prod_{i=0}^{n-1}\left(1+x^{2^i}+x^{2^{i+1}}\right)$. After reviewing the basic properties of this array and the closely related diatomic sequence of Stern, we consider the problem of computing the sum of the$r$th powers of the elements of the array. For instance, the sum of the cubes of the elements of row n >= 1 is$3\cdot 7^{n-1}$. We then consider some generalizations, some of which have especially elegant properties. II. A finite graded poset P is said to have the Sperner property if the largest level of P is an antichain of P of maximum size. (It is always an antichain.) It is unknown whether the weak (Bruhat) order on the symmetric group$S_n$has the Sperner property. We define some matrices whose entries are specializations of Schubert polynomials and give an explicit conjectured value for their determinant. If this conjecture is true, then the weak order on$S_n\$ does indeed have the Sperner property.

Dylan Thurston: Divides, plabic graphs, and quasipositive links

Abstract: From every plabic graph, one can construct, on the one hand, a cluster algebra and, on the other hand, a link of a special type (quasipositive). The links one can construct in this way include all links of singularities, and conjecturally the cluster algebra determines the link of the singularity. This is joint work with S. Fomin, P. Pylyavskyy, and E. Shustin.

Anna Weigandt: Bumpless Pipe Dreams and Alternating Sign Matrices

Abstract: Lam, Lee, and Shimozono introduced bumpless pipe dreams to study back stable Schubert calculus. In particular, Schubert polynomials can be expressed as a weighted sum over bumpless pipe dreams in a square grid. Working from a different perspective, Lascoux gave a formula for Grothendieck polynomials as a sum over alternating sign matrices. We show that alternating sign matrices are in natural bijection with bumpless pipe dreams. Restricting to the lowest degree terms of Lascoux's formula recovers the LLS formula for Schubert polynomials. Along the way, we discuss how to use the pipe dream perspective to compute keys of ASMs.

Workshop supported by NSF DMS-1818766, as well as