Titles, Abstracts, Slides:
Samantha Dahlberg: The Antipode of NCSym
Consider the algebra of formal power series with rational coefficients in non-commuting variables \(\{x_1, x_2,\ldots \}\). The algebra of symmetric functions in non-commuting variables, NCSym, contains functions such that for all permutations \(\pi\), \(f(x_1,x_2,\ldots) = f(x_{\pi(1)},x_{\pi(2)},\ldots)\). A cancellation-free antipode formula in the power sum basis was found by Baker-Jarvis, Bergeron, and Thiem. We also have a formula which appears in a different form and is derived using Takeuchi's formula and a sign-reversing involution. This is a technique which has recently been introduced by Benedetti and Sagan.
Robert Davis: Ehrhart Series of Polytopes Related to Symmetric Doubly-Stochastic Matrices
In Ehrhart theory, the \(h^*\)-vector of a rational polytope often provides insights into properties of the polytope that may be otherwise obscured. As an example, the Birkhoff polytope, also known as the polytope of real doubly-stochastic matrices, has a unimodal \(h^*\)-vector, but when even small modifications are made to the polytope, the same property can be very difficult to prove. In this talk, we examine the \(h^*\)-vectors of a class of polytopes containing real doubly-stochastic symmetric matrices.
Laura Escobar Vega: Toric matrix Schubert varieties
Start with a permutation matrix \(\pi\) and consider all matrices that can be obtained from \(\pi\) by taking downward row operations and rightward column operations; the closure of this set gives the matrix Schubert variety \(X_\pi\). Such a variety can be written as \(X_\pi=Y_\pi\times \mathbb{C}^q\) (where \(q\) is maximal). We characterize when \(Y_\pi\) is toric (with respect to a \(2n-1\)-dimensional torus) and study the associated polytope of its projectivization. We construct regular triangulations of these polytopes which we show are geometric realizations of a family of subword complexes. Based on joint work with Karola Mészáros.
Mathieu Guay-Paquet: A Hopf-algebraic proof of the Shareshian-Wachs conjecture slides
In 1995, Stanley defined the Chromatic Symmetric Function (CSF) of a graph, and for the class of indifference graphs of unit interval orders, the Stanley--Stembridge conjecture (still open) states that the CSF has a positive expansion in the basis of elementary symmetric functions. In 2012, Shareshian and Wachs refined the CSF to include a parameter \(q\) counting ascents, proved (among other things) that the result is palindromic in \(q\), and conjectured that this refined CSF is closely related to Tymoczko's "dot" action of the symmetric group on the cohomology of the regular semisimple Hessenberg flag varieties. This past November, Brosnan and Chow gave gave a geometric proof of the Shareshian-Wachs conjecture. We present an alternative, more combinatorial proof, which relies on expressing the CSF as a map on a new Hopf algebra of Dyck paths.
June Huh: Hard Lefschetz theorem and Hodge-Riemann relations for combinatorial geometries
A conjecture of Read predicts that the coefficients of the chromatic polynomial of a graph form a log-concave sequence for any graph. A related conjecture of Welsh predicts that the number of linearly independent subsets of varying sizes form a log-concave sequence for any configuration of vectors in a vector space. In this talk, I will argue that two main results of Hodge theory, the Hard Lefschetz theorem and the Hodge-Riemann relations, continue to hold in a realm that goes beyond that of Kahler geometry. This implies the above mentioned conjectures and their generalization to arbitrary matroids. Joint work with Karim Adiprasito and Eric Katz.
Tony Iarrobino: Equations for loci of commuting nilpotent matrices (work joint with Mats Boij, Leila Khatami, Bart van Steirtegham, and Rui Zhao) slides
The Jordan type of a nilpotent matrix is the partition giving the sizes of the Jordan blocks of the Jordan matrix in its conjugacy class. Given a stable Jordan type of two parts, \(Q=(u,u-r)\), with \(r\) at least 2, there is a known table \(T(Q)\) of Jordan types \(P\) for \(n\times n\) matrices whose maximum commuting nilpotent Jordan type is \(Q\) (arXiv 1409.2192). Let \(B\) be the Jordan matrix of partition \(Q\), and consider the affine space \(N(B)\) parametrizing nilpotent matrices commuting with \(B\). For a partition \(P\) in \(T(Q)\), the locus \(Z(P)\) of \(P\) is all matrices \(A\) in \(N(B)\) of Jordan type \(P\). In this talk we outline results concerning the equations defining \(Z(P)\). If time permits, we state conjectures (and perhaps results) about loci equations for table partitions of stable \(Q\) having three parts.
Ryan Kaliszewski: Airports: a link between Schubert calculus and Macdonald polynomials
Carly Klivans: The Partitionability Conjecture
In 1979, Stanley made the following conjecture: Every Cohen-Macaulay simplicial complex is partitionable. Motivated by questions in the theory of face numbers of complexes, the conjecture sought to bridge a combinatorial condition and an algebraic condition. Recent work resolves the conjecture in the negative. Due to a result of Herzog, Jahan and Yassemi, our construction also disproves the conjecture that the Stanley depth of a monomial ideal is always at least its depth. I will discuss the history and context of the partitionability conjecture, the counter-examples, the consequences, and the new questions we are now asking.
Li Li: Combinatorics of theta bases of Cluster Algebras slides
Cluster Algebra is a new branch in mathematics which grows rapidly and has far-reaching implications in many fields including representation theory, geometry, combinatorics, mirror symmetry of string theory, statistical physics, etc. Lots of research of cluster algebras focuses on construction of their natural bases. Various combinatorial models are discovered in the study of bases, including snake diagrams and perfect matching, Dyck paths and compatible pairs, and in particular, a recent surprising construction of theta bases by Gross, Hacking, Keel and Kontsevich using techniques (scattering diagrams, broken lines) developed in the study of mirror symmetry of string theory. In this talk, I will discuss explicit combinatorial enumeration of broken lines for some special classes of cluster algebras.
Oliver Pechenik: Puzzles and equivariant \(K\)-theory of Grassmannians
The cohomology of the Grassmannian has a basis given by Schubert varieties. The structure coefficients of this ring are the celebrated Littlewood-Richardson coefficients, and are calculated by any of the Littlewood-Richardson rules. This story has been extended to \(K\)-theory by A. Buch (2002) and to torus-equivariant cohomology by A. Knutson-T. Tao (2003). It is natural to unify these theories via a combinatorial rule for structure coefficients in equivariant \(K\)-theory. In 2005, A. Knutson-R. Vakil used puzzles to conjecture such a rule. Recently we proved the first combinatorial rule for these coefficients. Using our new rule, we construct a counterexample to the Knutson-Vakil conjecture and prove a mild correction to it. (Joint work with Alexander Yong)
James Propp: Cyclic actions on Catalan objects slides
A variety of natural invertible operations can be applied to Catalan objects to yield Catalan objects of the same order. By iterating such operations one obtains dynamical systems exhibiting interesting phenomena such as periodicity with surprisingly small period, cyclic sieving, and homomesy. Some of these operations appear to be shadows of piecewise-linear operations on polytopes, and these in turn often appear to be shadows of birational maps.
Vivien Ripoll: Hurwitz-transitivity and shellability in generated groups (joint work with Henri Mühle) slides
Let \(G\) be a group equipped with a fixed set of generators \(A\). Any element \(g\) of \(G\) can be written as a product of minimal length of elements in \(A\) (this is called a reduced decomposition of \(g\)). We can define a natural prefix order on \(G\), so that the reduced \(A\)-decompositions of \(g\) are in bijection with the maximal chains in the interval \([1,g]\) in this prefix order. Assuming \(A\) is closed by conjugacy, there is a natural action of the braid group on \(n\) strands on the set of reduced \(A\)-decompositions of any group element of length n, called the Hurwitz action. We investigate the relations between the shellability property of the poset [1,g] and the transitivity of the Hurwitz action on the reduced decompositions of \(g\). When \(G\) is a finite Coxeter group, \(A\) its whole set of reflections, and \(g\) a Coxeter element, these properties are well-studied and significant in the context of the W-generalized noncrossing partition lattice of Coxeter-Catalan combinatorics. In this case, an important tool is the existence of a nice total order on \(A\), called "compatible order". We study the implications of the existence of such a well-behaved order in a more general context.
Bruce Sagan: Of antipodes and involutions slides
Let \(H\) be a graded, connected Hopf algebra. Then Takeuchi's formula gives an expression for the antipode of \(H\). But this alternating sum usually has lots of cancellation. We will describe a method using sign-reversing involutions to ob tain cancellation-free formulas for various \(H\). This technique displays remark able similarities across the Hopf algebras to which it has been applied. No bac kground about Hopf algebras will be assumed. This is joint work with Carolina Benedetti.
Markus Schmidmeier: Refinements of Littlewood-Richardson Tableaux slides
Often in algebra, when difficult counting problems occur, the Littlewood-Richardson coefficient turns out to be the decisive tool. It counts the number of LR-tableaux of a given shape. Beyond their number, also the entries in the tableaux have precise meaning, in particular in the theory of invariant subspaces of nilpotent linear operators. We discuss how refinements of LR-tableaux (Klein tableaux, partial maps) determine the geometry of invariant subspace varieties. This talk is about a joint project with Justyna Kosakowska from Torun University, Poland.
William Slofstra: Staircase diagrams and smooth Schubert varieties
I will discuss a new combinatorial data structure called staircase diagrams, which (for finite-type Dynkin diagrams) are in bijection with rationally smooth Schubert varieties. Many properties of the Schubert variety can be seen directly from its diagram. In addition, staircase diagrams seem to be a very natural combinatorial object.
Greg Smith: Splendid complexes slides
Syzygies capture subtle geometric properties of a subvariety in projective space. However, when the ambient space is a product of projective spaces or a general smooth toric variety, minimal free resolutions over the Cox ring are too long and contain many geometrically superfluous summands. After illustrating this problem with some simple examples, we will construct some shorter free complexes that better encode the intrinsic geometry. This talk is based on joint work with Daniel Erman and Christine Berkesch Zamaere.