On a rank-unimodality conjecture of Morier-Genoud and Ovsienko

Abstract: Let $\alpha=(a,b,\ldots)$ be a composition, that is, a finite sequence of positive integers. Consider the associated partially ordered set $F(\alpha)$, called a fence, whose covering relations are $$ x_1\lhd x_2 \lhd \ldots\lhd x_{a+1}\rhd x_{a+2}\rhd \ldots\rhd x_{a+b+1}\lhd x_{a+b+2}\lhd \ldots\ . $$ We study the associated distributive lattice $L(\alpha)$ consisting of all lower order ideals of $F(\alpha)$. These lattices are important in the theory of cluster algebras and their rank generating functions can be used to define $q$-analogues of rational numbers. We make progress on a recent conjecture of Morier-Genoud and Ovsienko that $L(\alpha)$ is rank unimodal. All terms from the theory of partially ordered sets will be carefully defined. This is joint work with Thomas McConville and Clifford Smyth.

mathematical physicscommutative algebraalgebraic geometrycombinatoricsquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

Online Cluster Algebra Seminar (OCAS)