Abstract
Given a point A in the real Grassmannian, it is well-known that one can construct a soliton solution u A (x,y,t) to the KP equation. The contour plot of such a solution provides a tropical approximation to the solution when the variables x, y, and t are considered on a large scale and the time t is fixed. In this paper we give an overview of our work on the combinatorics of such contour plots. Using the positroid stratification and the Deodhar decomposition of the Grassmannian (and in particular the combinatorics of Go-diagrams), we completely describe the asymptotics of these contour plots when y or t go to ±∞. Other highlights include: a surprising connection with total positivity and cluster algebras; results on the inverse problem; and the characterization of regular soliton solutions—that is, a soliton solution u A (x,y,t) is regular for all times t if and only if A comes from the totally non-negative part of the Grassmannian.
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- 1.
The convention of [25] was to place the boundary vertices in clockwise order.
- 2.
This forbidden pattern is in the shape of a backwards L, and hence is denoted
and pronounced “Le.” - 3.
Actually Postnikov’s convention was to set π :(i)=j above, so the decorated permutation we are associating is the inverse one to his.
- 4.
Our numbering differs from that in [22] in that the rows of our matrices in SL n are numbered from the bottom.
- 5.
Since
-diagrams are a special case of Go-diagrams, one might also refer to them as Lego diagrams. - 6.
Recall from Definition 2.11 that our convention is to label boundary vertices of a plabic graph 1,2,…,n in counterclockwise order. If one chooses the opposite convention, then one must replace the word counterclockwise in Definition 9.7 by clockwise.
and pronounced “Le.”
-diagrams are a special case of Go-diagrams, one might also refer to them as Lego diagrams.References
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Acknowledgements
The first author (YK) was partially supported by NSF grants DMS-0806219 and DMS-1108813. The second author (LW) was partially supported by an NSF CAREER award and an Alfred Sloan Fellowship.
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Kodama, Y., Williams, L. (2013). Combinatorics of KP Solitons from the Real Grassmannian. In: Buan, A., Reiten, I., Solberg, Ø. (eds) Algebras, Quivers and Representations. Abel Symposia, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39485-0_8
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