Quantum cluster algebra structures on quantum Grassmannians and their quantum Schubert cells: the finite-type cases

TL;DR

This paper constructs quantum cluster algebra structures on specific quantum Grassmannians and their Schubert cells, focusing on cases where these structures are of finite type, thus extending classical cluster algebra results into the quantum realm.

Contribution

It introduces quantum cluster algebra structures on certain quantum Grassmannians and Schubert cells, specifically in finite type cases, linking quantum and classical cluster algebra frameworks.

Findings

Quantum cluster algebra structures are established on $K_q[Gr(2,n)]$ and select $K_q[Gr(3,6)]$, $K_q[Gr(3,7)]$, $K_q[Gr(3,8)]$.

These quantum structures quantize classical cluster algebra structures found by Scott.

The identified cases are exactly those with finite type quantum cluster algebras.

Abstract

We exhibit quantum cluster algebra structures on quantum Grassmannians $K_q[Gr(2,n)]$ and their quantum Schubert cells, as well as on $K_q[Gr(3,6)]$, $K_q[Gr(3,7)]$ and $K_q[Gr(3,8)]$. These cases are precisely those where the quantum cluster algebra is of finite type and the structures we describe quantize those found by Scott for the classical situation.