Abstract

A real matrix is totally nonnegative if all its minors are nonnegative. This class of matrices has been studied in the past hundred years, and has connection with combinatorics, probability, etc. This notion of total positivity was generalised by Lusztig in the 1990s to arbitrary flag varieties. In particular, this led to the notion of totally nonnegative grassmannian. In 2006, Postnikov obtained groundbreaking results about a cell decomposition of the totally nonnegative grassmannians. In particular, he described various combinatorial objects parametrising this cell decomposition. Interestingly, this cell decomposition and the associated combinatorial objects have recently found key applications in Integrable systems (work of Kodama-Williams on the KP equation), and in Theoretical Physics (work of Arkani-Hamed and coauthors on scattering amplitudes).

In joint work with Goodearl and Lenagan, we showed that the cell decomposition of the space of totally nonnegative matrices is related to a stratification of the prime spectrum of the algebra of quantum matrices. In this talk, I will review the above results and discuss the case of the (quantum) grassmannians. This is joint work with Lenagan and Nolan.