G. Barkley

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math.CO2025

Extended Weak Order for the Rank 3 Universal Coxeter Group

Grant Barkley, Colin Defant, Patricia Hersh +3

The weak order is a classical poset structure on a Coxeter group; it is a lattice when the group is finite but merely a meet-semilattice when the group is infinite. Motivated by pr…

math.CO2025

The Affine Tamari Lattice

Grant Barkley, Colin Defant

Given a fixed integer $n \geq 2$ , we construct two new finite lattices that we call the cyclic Tamari lattice and the affine Tamari lattice. The cyclic Tamari lattice is a sublattic…

math.CO2024

Bender--Knuth Billiards in Coxeter Groups

Grant Barkley, Colin Defant, Eliot Hodges +2

Let $(W, S)$ be a Coxeter system, and write $S = {s_{i} : i \in I}$ , where $I$ is a finite index set. Fix a nonempty convex subset $L$ of $W$ . If $W$ is of type $A$ , then $\mat…

math.CO2024

On two notions of total positivity for generalized partial flag varieties of classical Lie types

Grant Barkley, Jonathan Boretsky, Christopher Eur +1

For Grassmannians, Lusztig's notion of total positivity coincides with positivity of the Plucker coordinates. This coincidence underpins the rich interaction between matroid theory…

math.CO2024

Oriented matroid structures on rank 3 root systems

Grant Barkley, Katherine Tung

We show that, given a rank 3 affine root system $I^¦$ with Weyl group $W$ , there is a unique oriented matroid structure on $I^¦$ which is $W$ -equivariant and restricts to the usual m…