KP solitons and real Grassmannians

Let Gr(k,n) be the real Grassmann manifold defined by the set of all k-dimensional subspaces of R^n. Each point on Gr(k,n) can be represented by a kxn matrix A of rank k. If all the kxk minors of A are nonnegative, the set of all points associated with those matrices forms the totally nonnegative part of the Grassmannian, denoted by Gr(k,n)^+.

In this talk, I show how one can construct a cellular decomposition of Gr(k,n)^+ using the "asymptotic" spatial patterns of certain "regular" solutions of the KP (Kadomtsev-Petviashvili) equation. (In a pre-talk, I give a brief introduction of the KP equations and some special solutions called KP solitons.) This provides a classification theorem of all solitons solutions of the KP equation, showing that each soliton solution is uniquely parametrized by a derrangement of the symmetric group S_n. Each derangement defines a combinatorial object called the Le-diagram (a Young diagram with zeros in particular boxes). Then I show that the Le-diagram provides a classification of the ''entire'' spatial patterns of the KP solitons coming from the Gr(k,n)^+ for asymptotic values of the time. In particular, the spatial patterns of KP solitons for Gr(2,n)^+ provide a KP interpretation of A-type cluster algebras. The talk is based on a collaborative work with Lauren Williams.

If time permits, I will also show how one can compute the integral cohomology of the real Grassmannian using certain "singular" KP solitons.