Brandeis Combinatorics Seminar -  Fall 2023

The Brandeis Combinatorics Seminar is an introductory seminar for combinatorics.
The talks should be (at least partially) understandable to first year graduate students.

The zoom link is  https://brandeis.zoom.us/j/94622483750

October 3: (in person)

Speaker: Shujian Chen (Brandeis)

Title: Relative projectiveness and reverse Garside element action

Abstract:  Relative projectiveness is a property of reflections in a minimal Coxeter factorization which can be used to define a "signed" factorization that is an ordered W-Catalan object. I will introduce the reverse Garside element action which is derived from the classical braid group action on factorizations and show its interaction with relative protectiveness.

This is joint work with Kiyoshi Igusa.

October 10: (in person)

Speaker: Jiyang Gao (Harvard)

Title: The quantum Bruhat graph and tilted Richardson varieties

Abstract:  The quantum Bruhat graph is a weighted directed graph on a finite Weyl group first defined by Brenti-Fomin-Postnikov. It encodes quantum Monk's rule and can be utilized to study the 3-point Gromov-Witten invariants of the flag variety. In this talk, we provide an explicit formula for the minimal weights between any pair of permutations on the quantum Bruhat graph, and consequently obtain an Ehresmann-like characterization for the tilted Bruhat order. Moreover, for any ordered pair of permutations u and v, we define the tilted Richardson variety T_{u,v}, with a stratification that gives a geometric interpretation to intervals in the tilted Bruhat order. We provide a few equivalent definitions to this new family of varieties that includes Richardson varieties, and establish some fundamental geometric properties including their dimensions and closure relations. This is a joint work with Sihliang Gao and Yibo Gao. 

October 17: (in person)

Speaker: Shizhe Liang (Brandeis)

Title: A Schnyder-type drawing algorithm for 5-connected triangulations

Abstract:  Schnyder woods are defined for plane triangulations as special partitions of inner edges into three trees that cross each other in an orderly fashion. They are the cornerstone of Schnyder's famous barycentric embedding algorithm.

In this talk, I will present some Schnyder-type combinatorial structures, called 5c-woods, on a class of plane triangulations of the pentagon satisfying some connectedness constraints. These structures have three incarnations, as 5-tuples of trees, corner labelings and orientations of some associated graphs. I will provide the existence result and discuss their connections to other previously known structures.

The wood incarnation of 5c-woods associates to each inner vertex a partition of the inner faces into five face-connected regions, which in turn induces Schnyder-type barycentric coordinates for the vertex. I will show that if one fixes the positions of the five outer vertices properly, then this vertex-placement rule produces a planar straight-line drawing.

This is a joint work with Olivier Bernardi and Eric Fusy.

October 24: (in person)

Speaker: Amanda Burcroff (Harvard)

Title:  Faces and Hilbert Bases of Kostka Cones

Abstract:  We'll look at some nice enumerative and structural properties of an object with both combinatorial and representation-theoretic significance.  The r-Kostka cone is generated by pairs of partitions with at most r parts for which the corresponding Kostka coefficient, which counts semistandard Young tableaux with fixed shape and content, is nonzero.  We will see how the d-face structure of the r-Kostka cone can be determined from that of the (3d+1)-Kostka cone, allowing us to characterize its 2-faces and enumerate its d-faces for d at most 4.  These face-counting functions have a positive integer expansion in the binomial coefficient basis, and the coefficients in this expansion count certain cells in the braid arrangement.  We'll provide tight asymptotics for the number of d-faces for arbitrary d and determine the maximum number of extremal rays contained in a d-face for d less than r.  We'll then discuss progress towards a generalization of the Gao-Kiers-Orelowitz-Yong Width Bound on initial entries of Hilbert basis elements of the r-Kostka cone.  Lastly, we'll introduce an open problem involving a curious h-vector phenomenon.

October 31: (in person)

Speaker: Theo Douvropoulos (Brandeis)

Title: Decompositions of parking spaces and reflection Laplacians.

Abstract:  In the early 90's Haiman introduced the parking space Park(n), a module of coinvariants for a natural diagonal action of the symmetric group S_n. It has now become a central object in algebraic combinatorics, representation theory, and algebraic geometry. The ungraded module Park(n) is isomorphic to the space of parking functions, whose S_n-orbits are naturally indexed by Dyck paths (which form a Catalan family); this determines a parabolic decomposition of the parking space.

Already Haiman but later also Gordon, Cherednik, Oblomkov and many more, have generalized the invariant theoretic approach to arbitrary reflection groups W and defined algebraic parking spaces Park(W). The Coxeter-Combinatorics community has constructed an (ungraded) model for Park(W) which is based on a generalization of noncrossing partitions, another Catalan object, but the only proof of this W-isomorphism relied on the classification of reflection groups and it has been a long open problem to give a case-free proof.

The talk will present recent work, in part joint with Matthieu Josuat-Vergès, where we give such a case-free proof; our approach proceeds via comparing natural recursions on the combinatorial objects with their representation-theoretic counterparts. The main technical ingredient is a spectral study of the W-Laplacian operator, which allows us to prove complicated relations involving the Coxeter numbers of W and its parabolic subgroups. The story will be presented mostly in terms of the symmetric group and if there is time, we will conclude with some open questions regarding graded versions of these decompositions.

November 7:  (in person)

Speaker: Melissa Sherman-Bennett (MIT)

Title: Investigating the amplituhedron

Abstract:   Physicists Arkani-Hamed and Trnka introduced the amplituhedron to better understand scattering amplitudes in N=4 super Yang-Mills theory. The amplituhedron is the image of the totally nonnegative Grassmannian under the "amplituhedron map". Examples of amplituhedra include cyclic polytopes, the totally nonnegative Grassmannian itself, and cyclic hyperplane arrangements. I'll discuss recent work on tilings of the m=2 amplituhedron (with Parisi and Williams) and the m=4 amplituhedron (with Even-Zohar, Lakrec, Parisi, Tessler and Williams). The former reveals a surprising connection to the hypersimplex, a polytope, while the latter resolves one of the initial conjectures of Arkani-Hamed and Trnka.

November 14: (in person)

Speaker: Olivier Bernardi (Brandeis)

Title: Bijections for hyperplane arrangements of Coxeter type.

Abstract: A hyperplane arrangements of braid type is a collection of hyperplanes in R^n of the form {x_i-x_j=s}, where i,j are indices in [n] and s is an integer. Classical families include the Catalan, Shi, Semi-order and Linial arrangements. 

We will present bijections between regions of (well behaved) braid type arrangements, and some families of labeled plane trees. We will then extend the discussion to hyperplane arrangements of Coxeter type B,C,D.  This is joint work with Te Cao. 

November 28: (in person)

Speaker: Sergi Elizalde (Dartmouth College)

Title: Descents on noncrossing and nonnesting permutations

Abstract:  Stirling permutations were introduced by Gessel and Stanley to give a combinatorial interpretation of certain polynomials related to Stirling numbers, which count set partitions with a given number of blocks. A natural extension of Stirling permutations are noncrossing (also called quasi-Stirling) permutations, which are in bijection with labeled rooted plane trees. Archer et al. introduced these permutations, and conjectured that there are $(n+1)^{n-1}$ such permutations of size $n$ having $n$ descents.

In this talk we prove this conjecture and, more generally, we find the generating function for noncrossing permutations by the number of descents. We show that some of the properties of descents on usual permutations and on Stirling permutations also hold for noncrossing permutations.

Finally, we consider a nonnesting analogue, and we show that the polynomial giving the distribution of the number of descents on nonnesting permutations is a product of an Eulerian polynomial and a Narayana polynomial. It follows that, rather unexpectedly, this polynomial is palindromic.

Links to previous semesters: Spring2023, Fall 2022, Spring 2022, Fall 2021, Spring 2020, Fall 2019, Fall 2018, Spring 2018, Fall 2017, Spring 2017, Fall 2016, Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013.