Cluster algebra associated to open Richardson varieties : an algorithm to compute initial seed

Abstract: In his paper of 2016, Leclerc wanted to study the total nonnegativity criteria on flag variety in the same way as Fomin and Zelevinsky studied in '99 the total nonnegativity on $GL_n(mathbb{R})$ by stratification via double Bruhat cells. In this setting he wanted to study the cluster algebra structure on the open Richardson varieties stratifying the flag variety.

But in order to study this cluster algebra he used an additive categorification of the open Richardson variety $mathcal{R}_{v,w}$ by the category $mathcal{C}_{v,w}$. He proved that there is a cluster structure (in the sense of Buan, Iyama, Reiten, Scott) but hadn't given a way to explicitly build a seed for this cluster structure.

My PhD work was to design a prove an algorithm to explictly build such a seed starting from a seed for the cluster structure on the category $mathcal{C}_wsupset mathcal{C}_{v,w}$. I will explain the principle, the concrete usage of this algorithm and draw a sketch of the proof.

If time allows it, I will also introduce the Sage implementation I have written during my PhD.

mathematical physicscommutative algebraalgebraic geometrycombinatoricsquantum algebrarings and algebrasrepresentation theory

Audience: researchers in the topic

Online Cluster Algebra Seminar (OCAS)