Quantizing Grassmannians, Schubert cells and cluster algebras

The quantum Grassmannians and their quantum Schubert cells are
well-known and important examples in the study of quantum groups and
quantum geometry.  It has been known for some time that their
classical counterparts admit cluster algebra structures, which are
closely related to positivity properties.  Recently we have shown
that in the finite-type cases quantum Grassmannians admit quantum
cluster algebra structures, as introduced by Berenstein and
Zelevinsky.  We will describe these structures explicitly and also
show that they naturally induce quantum cluster algebra structures on
the quantum Schubert cells.

This is joint work with S. Launois.