Positivity in Algebraic Combinatorics

Speaker: Olga Azenhas (Universidade de Coimbra, Portugal)
Title: The involutive nature of Littlewood Richardson (LR) commutativity bijection
Abstract: I shall introduce two involutions, one on LR tableaux and another on LR hives, making manifest the commutativity of LR coefficients, $c_mu,nu^lambda=c_nu,mu^lambda$. Both maps proceed by providing sequences consisting of successively
smaller LR tableaux and of successively smaller LR hives in which the sequences of inner shapes and left-hand boundary edges, respectively, determine the corresponding LR partners. They do this by specifying completely a same    Gelfand-Tsetlin pattern of type $mu$ associated with the LR partner.
I shall also discuss the relationship between these  LR commutativity bijections and others known up to date. In particular, the involutive nature of the present bijections is exhibited without recourse to the Schützenberger involution.
This is a joint work with R.C. King and I. Terada.

Speaker: Nantel Bergeron (York University, Canada)
Title: Antipode formulas for Linearizable Hopf Monoids
Abstract: [Joint work with Carolina Benedetti]
A Hopf monoid is linearizable if one has a basis that is a set species and the Hopf structure is a linearization of the structure on the set-species. This typically happen when the multiplication and comultiplication is encode by combinatorial objects.
In such context, I will give a multiplicity free and cancellation free formula for the antipode.

Speaker: Soojin Cho (Ajou University, Korea)
Title: A Pieri rule for K-theoretic Schur P-functions
Abstract: K-theoretic analogues of Schur P-functions were defined by T. Ikeda and H. Naruse in 2013, which represent K-theoretic Schubert classes of the maximal orthogonal Grassmannians. We use set-valued decomposition tableaux to describe a Pieri rule for K-theoretic Schur P-functions, different from the Pieri rule by A. Buch and V. Ravikumar that is generalized to Littlewood-Richardson rule by E. Clifford, H. Thomas and A. Yong. We also state a conjecture for Littlewood-Richardson rule of K-theoretic Schur P-functions in terms of set valued decomposition tableaux.
The first positive equivariant Pieri rules for isotropic Grassmannians are proved by C. Li and V. Ravikumar, and we describe another Pieri rule for factorial Schur P-functions in terms of barred decomposition tableaux.
This talk is based upon the work done with Takeshi Ikeda.

Speaker: Sergey Fomin (University of Michigan, USA)
Title: Noncommutative Schur functions, switchboards, and Schur positivity
Abstract: The problem of expanding various families of symmetric functions in the basis of Schur functions arises in many mathematical contexts. Several instances of this problem (e.g. those related to Schubert, Grothendieck, LLT, and Macdonald polynomials) can be reformulated, and sometimes solved, using noncommutative analogues of Schur functions. This is joint work with Jonah Blasiak.

Speaker: Ira Gessel (Brandeis University, USA)
Title: Amdeberhan's Conjecture and Plethystic Inverses
Abstract: Motivated by Billey, Konvalinka, and Matsen's formula for unlabeled tanglegrams, Tewodros Amdeberhan conjectured that a generalization of their formula always gives integers. I will prove Amdeberhan's conjecture using the fact that the plethystic inverse of an integral symmetric function is integral, and I will discuss the conjectured Schur positivity of these symmetric functions, which are related to the primitive necklace symmetric functions.

Speaker: Jim Haglund (University of Pennsylvania, USA)
Title: LLT polynomials and the chromatic symmetric function of unit interval orders
Abstract: Shareshian and Wachs have recently conjectured that a certain symmetric function, which depends on a Dyck path and a parameter t, has positive coefficients when expressed as a polynomial in the elementary symmetric functions. Their conjecture implies an earlier conjecture of Stanley and Stembridge. We show how some elements of the preprint of Carlsson and Mellit "A proof of the shuffle conjecture" imply that the Shareshian-Wachs symmetric function can be expressed, via a plethystic substitution, in terms of LLT polynomials, specifically LLT products of single cells. As corollaries we obtain combinatorial formulas for the expansion of Jack polynomials into the Schur basis, and also the power-sum basis. These formulas are signed, not always positive, but perhaps could be simplified. "This is joint work with Per Alexandersson, Greta Panova, and Andy Wilson".

Speaker: Ryan Kaliszewski (Drexel University, USA)
Title: New Formulas for Macdonald Polynomials
Abstract: Macdonald polynomials, a remarkable set of multivariate symmetric functions in two parameters, have been found to have applications in math, physicis, computer science, and statistics. While it has been proven by Haiman that these functions expand positively in the Schur basis, a combinatorial description of the coecients has so far eluded researchers.
In joint work with Morse, we have found that the Macdonald polynomials are the (q, t)-generating functions for a new combinatorial structure, using familiar statistics to derive the q and t parameters. Our description allows powerful techniques such as crystal bases and tableaux combinatorics to be applied to long-standing open problems regarding the Schur expansion. This has led us to proofs of several properties such as (q, t)-symmetry and a combinatorial description when q = 1. In addition, it has allowed us to reveal connections among the cocharge, -cocharge, and major index statistics, thus relating the q = 0 and q = 1 cases to the general monomial expansion.

Speaker: Thomas Lam (University of Michigan, USA)
Title: Cohomology of cluster varieties
Abstract: I will discuss joint work with David Speyer where we study the singular cohomology of cluster varieties. One of our main results is a curious Lefschetz duality for these cohomologies. I will also discuss the related problem of counting the number of points of a cluster variety over a finite field. A potential application is to open Richardson varieties whose cohomology computes the higher
extension groups of Verma modules in the principal block of Category O.

Speaker: Jae-Ho Lee (Tohoku University, Japan)
Title: Distance-regular graphs and double affine Hecke algebras
Abstract: The Q-polynomial property of distance-regular graphs was introduced in 1973 by Delsarte in his work on coding theory. These graphs are thought of as finite/ combinatorial analogues of compact symmetric spaces of rank one. Double affine Hecke algebras (DAHAs) were introduced in 1992 by Cherednik. The theory of DAHAs has been connected to many other mathematical topics such as algebraic geometry, Lie theory, and orthogonal polynomials. In this talk, we will present a relationship between Q-polynoimal distance-regular graphs and a DAHA of rank one. As an application we give a combinatorial interpretation of a nonsymmetric version of Askey-Wilson polynomials.

Speaker: Li Li (Oakland University, USA)
Title: Positivity Phenomenon of Bases of Cluster Algebras
Abstract: Positivity is a main motivating phenomenon in the study of cluster algebras and their bases. A desirable property of a good basis of a cluster algebra is that all elements are universally positive and that all structure constants are positive. In this talk, I will survey some of the combinatorial, representation-theoretical, and geometric methods that are used to study the positivity of various natural bases, in particular of greedy bases (and theta function bases in general) and triangular bases.

Speaker: Ricky Liu (North Carolina State University, USA)
Title: Positive expressions for skew divided difference operators
Abstract: For any pair of permutations, Macdonald defines a skew divided difference operator and shows how these operators can be used to compute the structure constants for Schubert polynomials. We will show that any skew divided difference operator can be written explicitly as a polynomial in the degree 1 divided difference operators with positive coefficients, which settles a conjecture of Kirillov. The proof relies on various tools from the braided Hopf algebra structure of the Fomin-Kirillov algebra.

Speaker: Brendon Rhoades (University of California, San Diego, USA)
Title: Ordered set partitions and the Delta Conjecture
Abstract: The Shuffle Conjecture of Haglund, Haiman, Loehr, Remmel, and Ulyanov (which was recently proven by Carlsson and Mellit) gives an expansion of the Frobenius character of the diagonal coinvariant module in the fundamental quasisymmetric function basis. The Delta Conjecture of Haglund, Remmel, and Wilson is a recent generalization of the Shuffle Conjecture. I will explain how some equidistribution results on ordered (multi-)set partitions can be used to prove a special case of the Delta Conjecture.

Speaker: Anne Schilling (University of California, Davis, USA)
Title: Positivity of k-Schur functions
Abstract: I will present new and old results and conjectures related to positivity of k-Schur functions.