Schedule for: 17w5012 - Algebraic Combinatorixx 2

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

My main research focus right now is symmetric and quasisymmetric functions. In particular, I am interested in specializations of Macdonald polynomials and generalizations of Schur functions. A symmetric function is a polynomial which remains unchanged when the variables are permuted. The Schur function basis for symmetric functions is related to many different areas of mathematics and can be generated in many ways, but I am most interested in its combinatorial aspects and the properties of semi-standard Young tableaux (boxes containing numbers placed according to certain rules) which can be used to construct Schur functions. Most of my recent work in this area has been on a related collection of polynomials, called "quasisymmetric Schur functions" which can also be constructed using tableaux and which decompose Schur functions in a natural way. These objects come directly from Haglund's recent combinatorial formula for Macdonald polynomials and are a treasure-trove of interesting problems which provide a new perspective for some of the classical problems in this area as well. For example, if a function is symmetric and quasisymmetric Schur-positive, then it is automatically Schur positive, meaning quasisymmetric Schur functions provide a new avenue for investigating Schur positivity. One interesting open problem is to understand the multiplication of quasisymmetric schur functions; there is no known rule for the product of two arbitrary quasisymmetric Schur functions, although a number of special cases are known. Another open problem involves exploring Macdonald polynomial connections. Specifically, since quasisymmetric Schur functions are situated in between Demazure atoms (specializations of nonsymmetric Macdonald polynomials) and Schur functions (specializations of Macdonald polynomials), it is natural to study the quasisymmetric object that sits between nonsymmetric Macdonald polynomials and Macdonald polynomials. This object should generalize quasisymmetric Schur functions in a straightforward manner.

A key area of investigation in chromatic symmetric functions concerns the e-positivity, and/or Schur positivity, of a particular class of chromatic symmetric functions whose underlying graphs are clawfree. The interest in these graphs originates in a paper of Stanley and Stembridge (On immanants of Jacobi-Trudi matrices and permutations with restricted position, JCTA 62 (1993), 261-279) that considers positivity of coefficients of immanants. A natural consideration in their study is a conjecture that states that if a poset is (3+1)-free, then its incomparability graph, which is necessarily clawfree, is e-positive. As a related result, Gasharov (Incomparability graphs of (3+1)-free posets are s-positive, Disc. Math. 157 (1996), 193-197) proved that they are Schur positive. Stanley (Graph colorings and related symmetric functions: ideas and applications, Disc. Math. 193 (1998), 267-286.) has further conjectured that *any* clawfree graph is Schur positive. In parallel to this, in graph theory, much effort has been spent in characterizing the chromatic characteristics of graphs that are H-free, where H is some set of induced subgraphs. A key question in this domain is, can the chromatic number of a H-free graph be determined in polynomial time? This answer is known for large classes of graphs, but, interestingly, the classification for all combinations of subgraphs with 4 vertices is almost, but not quite, complete, Lozin and Malyshev (Vertex coloring of graphs with few obstructions, Disc. Appl. Math. 216 (2017), 273-280). The claw is a graph on 4 vertices. Do the other graphs on 4 vertices have similar positivity results? In graph theory when it is too hard to say something about a single subgraph, the question of multiple subgraphs is often considered (e.g. if we cannot derive results about clawfree graphs, perhaps we can derive results about (claw, 4-cycle)-free graphs), hence the positivity question can similarly be made easier by asking about a set of subgraphs rather than a single one. Just as the question about clawfree graphs was interesting because of the connection to immanant conjectures, so too would questions about other 4 vertex graphs be interesting because of the parallel with studies in graph theory.

Crystal bases were first introduced as a combinatorial tool to understand representations of quantum groups. They can also be understood from a purely combinatorial point of view (see for example the new book by Bump and Schilling on "Crystal bases: Representations and Combinatorics"). In particular, the character of a connected crystal in type A is a Schur function. Hence, knowing the crystal structure on a set that underlies a given symmetric function, will yield the Schur expansion of this symmetric function provided that the expansion is Schur-positive. As an example, Brendon Rhoades in his article “Ordered set partition statistics and the Delta conjecture” investigated the symmetric functions Val_{n,k}(x;0,q). These symmetric functions are Schur-positive (see page 19 of Rhoades' article). It would be interesting to find a crystal structure on the ordered multiset partitions that define these symmetric functions. Crystal bases are very well developed in the open source computer algebra system Sage. So it should be easy to experiment with these structures in Sage. References: * Brendon Rhoades, Ordered set partition statistics and the Delta conjecture arXiv:1605.04007 * Sage * Daniel Bump, Ben Salisbury, Anne Schilling Thematic Tutorial: Lie Methods and Related Combinatorics in Sage

Voting theory has traditionally been explored using geometric methods, but a more recent approach has been to look at voting through the lens of representation theory. This algebraic perspective views voting data as elements of a QS_n module, and encodes traditional voting methods such as plurality and Borda count as QS_n module homomorphisms. By applying Schur’s Lemma, one can deduce which voting information determines the election outcome and what information has no influence on the results. Some possible questions to explore include examining voting systems when voters are allowed to respond with any partial order of the candidates and voting to elect committee with representatives from different groups.

The Kronecker coefficients describe the decomposition of the tensor product of two irreducible representations of a symmetric group into irreducible representations. They also the structural constants for the inner multiplication in the algebra of symmetric functions. Despite much interest, very little is known about them. Recently, some results have been obtained by focusing on a related family of coefficients, the reduced Kronecker coefficients. These coefficients, discovered by Murnaghan in the 1930s, seem to be easier to understand. Still, they contain enough information to recover from them the value of the Kronecker coefficients. In recent work, together with E. Briand, L. Colmenarejo, and A. Rattan, the rate of row of some families of reduced Kronecker coefficients has been studied. In particular, it has been showed that the reduced Kronecker coefficients stabilize when we increase the size of their first columns, and grow linearly when we increase the size of their first rows. Also some new combinatorial formulas for some families of Kronecker coefficients have been obtained (in terms of plane partitions) that allows us to compute explicity their rates of growth. Note that these results can be immediately translated to the setting of the Kronecker coefficients I propose to try to find new closed/combinatorial formulas for some new families of dilated Kronecker, and try to use this information to study new instances of the rate of growth of the Kronecker coefficients. And in particular, to explore some conjectures of Ron King related to them.

My research is in enumerative, algebraic, and topological combinatorics. My current specific interests include Coxeter groups, the Bruhat order, reduced words, permutation patterns, pattern avoidance and pattern characterizations, posets, zonotopal and domino tilings, sorting algorithms, optimization, and discrete Morse theory. One theme of my recent work has been a structural analysis of Coxeter systems. This is part of continuing efforts to relate three aspects of these objects: reduced words, patterns, and the Bruhat order. Each area has been extensively studied independently, and there is compelling evidence of interrelationships between them. I have developed a method for translating between permutation patterns and reduced words, and have been able to exploit this relationship this so that one object may be used to answer questions about the other. I also work with various structural questions about permutation patterns, including the poset of avoidance, the coincidence of different pattern definitions, and bijective enumerations of pattern classes, and there are many problems in this area still to be addressed.

In 1904 Steinitz characterized the face vector of 3-dimensional polytopes. Amazingly the question of characterizing the face vector of polytopes is open for dimensions 4 and above. For the subfamily of simplicial polytopes, Stanley used methods of algebraic geometry to prove the necessity of McMullen's conditions for f-vectors of simplicial polytopes and Billera--Lee used geometric constructions for the sufficiency. Bayer and Klapper's introduction of the cd-index, a noncommutative polynomial which encodes the face incidence data of polytopes and Eulerian posets without any linear redundancies, has allowed new advances in the field. Ehrenborg and I have showed that the cd-index has an inherent coalgebraic structure. This structure has enabled my coauthors and I to express the flag vector change due to geometric operations in terms of linear operators on the cd-index, prove new non-trivial inequalities for flag vectors of polytopes, and to simplify our understanding of flag vectors of oriented matroids. Most recently, with Ehrenborg and Goresky, I have been extending the theory to more general settings, including Bruhat graphs, balanced digraphs, Whitney stratified spaces and quasi-graded posets.

Alcoved Polytopes are lattice polytopes whose facet normals are taken from a type A root system. Tropical polytopes are special polyhedral complexes made up of alcoved polytopes. The goal of the project is to study the Ehrhart polynomials and related algebraic structures of alcoved and tropical polytopes.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

TBD

This talk will describe a labeling of the points of the moduli space of genus 0 curves in the complete complex flag variety using the combinatorial machinery of alcove walks. Following Peterson, this geometric labeling explains the "quantum equals affine" phenomenon which relates the quantum cohomology of this flag variety to the homology of the affine Grassmannian. Time permitting, we will discuss how this labeling yields a rational parameterization of certain Gromov-Witten varieties.

It is well known that the number of non-isomorphic unit interval orders on $[n]$ equals the $n$-th Catalan number. Using work of Skandera and Reed and work of Postnikov, we show that each unit interval order on $[n]$ naturally induces a rank $n$ positroid on $[2n]$. We call the positroids produced in this fashion \emph{unit interval positroids}. We characterize the unit interval positroids by describing their associated decorated permutations, showing that each one must be a $2n$-cycle encoding a Dyck path of length $2n$. We also give a combinatorial description of the $f$-vectors of unit interval orders. This is joint work with Felix Gotti.

Stirling numbers of the second kind $S(n,k)$ enumerate the number of set partitions of $n$ elements into k disjoint blocks. We give a compact expression for the $q$-Stirling numbers as a $q$-$(1+q)$-analogue. We show there is a poset explanation for this phenomenon as well as a homological interpretation. This is joint work with Margaret Readdy.

Consider the set of all triangulations of a convex $(n+3)$-gon. These triangulations are related to one another by diagonal flips, and the graph defined by these flips is the 1-skeleton of the familiar $n$-dimensional polytope known as the associahedron. The $n$-dimensional cyclohedron is constructed analogously using centrally-symmetric triangulations of a regular $(2n+2)$-gon, with relations given by centrally-symmetric diagonal flips. Modding out by the symmetry, we may equivalently view the cyclohedron as arising from ``triangulations" of an orbifold: the $(n+1)$-gon with a single two-fold branch point at the center.\\ In this talk we will introduce orbifold-resection, a simple combinatorial operation which maps the ``once-orbifolded" $(n+1)$-gon to the $(n+3)$-gon. More generally, orbifold-resection maps a triangulated orbifold to a triangulated surface while preserving the number of diagonals and respecting adjacencies. This induces a relationship on the signed adjacency matrices of the triangulations, called dominance, which gives rise to many interesting phenomena. For example, the normal fan of the cyclohedron refines that of the associahedron; work is in progress to show that such fan refinement holds generally in the case of orbifold-resection. If time allows, we will explore other dominance phenomena in the context of the surfaces-and-orbifolds model.

I will discuss a certain minimal factorization of the elements in a finite lattice L called the canonical join representation. The canonical join representation of an element w is the unique ``lowest'' subset of L whose join is equal to w (where ``lowest'' will be made precise). When each element in L has a canonical join representation, we define the canonical join complex to be the abstract simplicial complex of subsets A such that the join of A is a canonical join representation. I will characterize the class of finite lattices whose canonical join complex is flag, and discuss several examples.

Let $\mathbb{K}$ denote an algebraically closed field and let $V$ denote a vector space over $\mathbb{K}$ with finite positive dimension. In this talk, we consider a tridiagonal pair $A,A^*$ on $V$ which has $q$-Racah type. We will introduce the linear transformations $\psi:V\to V$, $\Delta:V \to V$, and $\mathcal{M}:V\to V$, each of which acts on the split decompositions of $V$ in an attractive way. We will show that $\Delta$ can be factored into a $q^{-1}$-exponential in $\psi$ times a $q$-exponential in $\psi$. We view $\Delta$ as a transition matrix from the first split decomposition of $V$ to the second. Consequently, we view the $q^{-1}$-exponential in $\psi$ as a transition matrix from the first split decomposition to a decomposition of $V$ which we interpret as a kind of half-way point. This half-way point turns out to be the eigenspace decomposition of $\mathcal{M}$. We will discuss the eigenspace decomposition of $\mathcal{M}$ and give the actions of various operators on this decomposition.

Remmel and Whitney provided an algorithmic procedure for determining the Littlewood-Richardson coefficients that appear in the Schur function expansion of a product of Schur functions. Haglund et al. introduced the quasisymmetric Schur functions as a basis for QSym. We adapt Remmel and Whitney's approach in order to determine the coefficients that appear in the quasisymmetric Schur function expansion of the product of a quasisymmetric Schur function and a (symmetric) Schur function. Versions for both column and row strict quasisymmetric Schur functions are presented.

The clustering properties of Jack polynomials are involved in the theoretical study of the fractional Hall states as Bernevig and Haldane described. For simplicity, these properties imply that the polynomials factorize completely for some specialization of the variables. In the aim to understand this phenomenon, we adopted the following strategy. First, we consider Macdonald polynomials instead of Jack polynomials. Macdonald polynomials are a deformation with two parameters which degenerates to Jack polynomials. The computational and combinatorial aspects of these polynomials are fully exploited when using non symmetric and/or non homogeneous version of this polynomials. This work is the sequel of a previous paper of J-G. Luque and C. F. Dunkl in which they prove a conjecture of Peter Forrester by investigating the vanishing properties of non symmetric/non homogeneous singular Macdonald polynomials via the Yang-Baxter graph. We prove that these polynomials have very nice factorization properties under some specialization of the variables which generalize the conjecture of Forrester. We illustrate our methods by giving some factorization formulas for staircase or quasi-staircase Macdonald polynomials. This is a work in progress with J-G. Luque and C. F. Dunkl.

Discussion on issues related to gender in academia and mathematics (Do you have a subtopic of interest? Do you have something you feel compelled to share on this topic? Are you interested in moderating such a discussion? Feel free to let us know!)

The set of upper-triangular nilpotent matrices with entries in a finite field $\mathbb{F}_q$ has Jordan canonical forms indexed by the partitions $\lambda\vdash n$. We present a combinatorial formula for computing the number $F_\lambda(q)$ of matrices of Jordan type $\lambda$ as a weighted sum over standard Young tableaux. We then discuss connections between these matrices, non-attacking rook placements, and set partitions, which lead to a refinement of the formula for $F_\lambda(q)$.

In this work, we construct elliptic analogues of the rook numbers and file numbers by attaching elliptic weights to the cells in a board. We show that our elliptic rook and file numbers satisfy elliptic extensions of corresponding factorization theorems which in the classical case was established by Goldman, Joichi and White and by Garsia and Remmel in the file number case. This factorization theorem can be used to define elliptic analogues of various kinds of Stirling numbers of the first and second kind, and Abel numbers. We also give analogous results for matchings of graphs, elliptically extending the result of Haglund and Remmel. This is a joint work with Michael Schlosser.

In recent work, Robin Sulzgruber describes a bijection between multisets of rim hooks of a partition shape and reverse plane partitions of the same shape in the form of an insertion algorithm. We show that this bijection is natural in the context of quiver representations. This project is joint with Hugh Thomas and Alexander Garver.

The celebrated hook-length formula of Frame, Robinson and Thrall from 1954 gives a product formula for the number of standard Young tableaux of straight shape. No such product formula exists for skew shapes. In 2014, Naruse announced a formula for skew shapes as a positive sum of products of hook-lengths using "excited diagrams" [ Ikeda-Naruse, Kreiman, Knutson-Miller-Yong]. We will show combinatorial and aglebraic proof of this formula, leading to a bijection between SSYTs or reverse plane partitions of skew shape and certain integer arrays that gives two q-analogues of the formula. We show how excited diagrams give asymptotic results for the number of skew Standard Young Tableaux in various regimes of convergence for both partitions. We will also show a multivariate versions of the hook formula with consequences to exact product formulas for certain skew SYTs and lozenge tilings with multivariate weights. We will also exhibit other curious phenomena emerging from there techniques like product formulas for various classes of skew SYTs and relations with reduced decompositions of permutations. Joint work with Alejandro Morales and Igor Pak.

Given $k (TCPL 201)

In Stanley's seminal 1995 paper on the chromatic symmetric function, he stated that there was no known graph that was not contractible to the claw and whose chromatic symmetric function was not e-positive, namely, not a positive linear combination of elementary symmetric functions. We resolve this by giving infinite families of graphs that are not contractible to the claw and whose chromatic symmetric functions are not e-positive. Moreover, one such family is additionally claw-free, thus establishing that the e-positivity of chromatic symmetric functions in general is not dependent on the claw.

(joint work with Evita Nestoridi) The random to random shuffle on a deck of cards is given by at each step choosing a random card from the deck, removing it, and replacing it in a random location. We show an upper bound for the mixing time of the walk of $3/4n\log(n) +cn$ steps. Together with matching lower bound of Subag (2013), this shows the walk mixes with cutoff at $3/4n\log(n)$ steps, answering a conjecture of Diaconis. We use the diagonalization of the walk by Dieker and Saliola (2015), which relates the eigenvalues to skew partitions and desarrangement tableaux.

Idea exchange between faculty at teaching oriented and research oriented institutions: How can we help each other? (Do you have a subtopic of interest? Do you have something you feel compelled to share on this topic? Are you interested in moderating such a discussion? Feel free to let us know!)

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.