Algebra, Combinatorics and Number Theory seminar

In Spring 2023, the Algebra seminar meets Thursdays 9:30 - 10:30 am online on Zoom. For more information please contact Evan Houston or me.

On Tuesday April 20, 2021, Joe Johnson (NC State University) gave a talk on "Piecewise-Linear Rowmotion in Maximal Rectangles of Moon Polyominoes". The abstract is below.

Rowmotion is a combinatorial bijection from the set of order ideals of a poset to itself. The map has a piecewise-linear lifting to the order polytope which restricts to the original bijection on lattice points. Using properties of rowmotion, we show that a collection of non-lattice polytopes associated to moon polyominoes all have the same Ehrhart series (and volume). We describe a piecewise-linear volume-preserving continuous map between the polytopes in terms of the Robinson-Schensted-Knuth correspondence and promotion on Young tableaux.

We define a notion of deletion and contraction for lattices. The result of a sequence of deletions and contractions is called a minor of the original lattice. The name minors is justified by the fact that the lattice of flats of a graph correspond to the simple minors of the graph when the vertices are labelled (and the edges unlabeled). For each finite lattice we define a poset of minors and show it is Eulerian and a PL sphere. We also obtain inequalities for the cd-indices of these posets of minors.

The Lindström-Gessel-Viennot lemma gives a determinantal formula for the number of collections of nonintersecting paths in a directed graph. One important application of this lemma is the (dual) Jacobi-Trudi identity, which expresses a Schur polynomial as a determinant involving elementary symmetric polynomials. In this talk, we will describe a generalization of this result that gives a determinantal formula for a large class of Schubert polynomials (for permutations avoiding a certain set of patterns of lengths 5 and 6) in terms of elementary symmetric polynomials. This formula also gives the expansion for such a Schubert polynomial in the SEM basis of standard elementary monomials. This is joint work with Hassan Hatam, Joseph Johnson, and Maria Macaulay.

We introduce and prove the n-dimensional Pizza Theorem. Let ℋ be a real n-dimensional hyperplane arrangement. If K is a convex set of finite volume, the pizza quantity of K is the alternating sum of the volumes of the regions obtained by intersecting K with the arrangement ℋ. We prove that if ℋ is a Coxeter arrangement different from A₁ⁿ such that the group of isometries W generated by the reflections in the hyperplanes of ℋ contains the negative of the identity map, and if K is a translate of a convex set that is stable under W and contains the origin, then the pizza quantity of K is equal to zero. Our main tool is an induction formula for the pizza quantity involving a subarrangement of the restricted arrangement on hyperplanes of ℋ that we call the even restricted arrangement. We get stronger results in the case of balls. We prove that the pizza quantity of a ball containing the origin vanishes for a Coxeter arrangement ℋ with |ℋ|-n an even positive integer. This is joint work with Sophie Morel and Margaret Readdy.

On Tuesday February 16, 2021, Richard Ehrenborg (University of Kentucky) gave a talk. The title of his talk is "Three Combinatorial Applications". The abstract is below.

This is not one seminar talk. This is three small seminars. First, we prove a geometric result of Dirichlet using combinatorics. Second, as an application of posets we obtain Sylvester's two coin result. Finally, we present a counting proof when 2 is a quadratic residue in a finite field.

The third topic is joint work with Frits Beukers and Karthik Chandrasekhar.

On Thursday November 19, 2020, Pengyu Liu (Simon Fraser University) gave a talk. The title of his talk is "Polynomial tree-based analysis". The abstract is below.

Tree structures emerge in life sciences, computer science, linguistics and many other fields. Comparing and analyzing tree structures are challenging tasks as both the size and the number of trees are increasing and there are few tools to describe tree structures in a quantitive, accurate, comprehensive and easy to interpret way. Polynomials are important tools in mathematics to study discrete structures, for example, the renowned Tutte polynomial for graphs and Jones polynomial for knots and links. In this talk, we introduce a polynomial for unlabeled trees and show that the polynomial is a complete isomorphism invariant for unlabeled trees, then generalize the polynomial for partially labeled and fully labeled trees. Finally, we present some applications of the polynomial in biology and linguistics.