Giving a manifestly positive formula for the structure constants
arising from the multiplication of two Schubert classes in the
cohomology of a flag manifold is a long-standing open problem in
algebraic combinatorics. This has only seen limited progress. Most
results have restrictions on both classes that are multiplied, and the
only formula with a restriction on only one is the Pieri-type formula
that is 30 years old.
In this talk, I will describe formulas, in both ordinary and
equivariant cohomology, for the coefficients that arise when
multiplying a Schubert class by one pulled back from a Grassmannian of
$2$-planes. This uses geometric constructions including a novel
description of certain positroid varieties. The cohomological formula
identifies each coefficient as particular Littlewood-Richardson
coefficient on a Grassmannian of $k$-planes, for some $k$, not necessarily
equal to $2$.
This is joint work with Changzheng Li and Mingzhi Yang.
In 1943, Hadwiger conjectured that every graph that does not contain the complete graph on $t+1$ vertices as a minor is properly $t$-colorable. This conjecture is considered one of the major open problems in graph theory, and the $t=4$ case implies the Four Color Theorem. Delcourt and Postle made a breakthrough a few years ago, showing that $O(t \log \log t)$ colors are sufficient. In this talk we will show that $O(t \log \log \log t)$ colors are sufficient by building on their result. Joint work with Jason Luo.
A fundamental identity for the number of vacillating tableaux was originally obtained through the representation theory of partition algebras. We extend this identity to arbitrary differential posets and show that it, together with several analogous identities, follows directly from the structural properties of differential posets. Specializing to Young's lattice and its Cartesian powers, we further obtain new bijective proofs via a simple deletion--insertion process. This is a joint work with Derek Wu.
Over any field, we can "add" and "multiply" vector spaces using direct sums and tensor products. Similarly, if we have two representations of a group $G$, we can "add" and "multiply" these by giving their direct sums and tensor products a natural action from $G$. Both of these are examples of tensor categories. On the underlying space, the tensor product of representations is the same as over vector spaces. Over a field of positive characteristic, Benson, Etingof, and Ostrik constructed an infinite family of tensor categories, called the higher Verlinde categories, whose tensor product is fundamentally different from vector spaces. In this talk, we explore the structure of one of these categories, $\text{Ver}_8^+$, by viewing it as modules of some algebra. We give a combinatorial classification of the indecomposable modules by strings and bands, and describe the structure of the unique tensor product of $\text{Ver}_8^+$ by determining how it acts on its Auslander--Reiten quiver. This is joint work with Andrew Aguilar, Wolfgang Allred, and Ian Martin.
