Towards a toric analogue of total positivity

Postnikov’s boundary measurement map gives a parametrization of the totally nonnegative Grassmannian using weighted bipartite graphs on a disk. There is a parallel construction for weighted bipartite graphs on a torus: the spectral transform of Kenyon and Okounkov, which produces a Harnack curve together with some extra data. I will explain why this should be viewed as a toric analogue of the boundary measurement map. This raises the question of whether there is a toric analogue of the totally nonnegative Grassmannian, together with a stratification analogous to the positroid stratification. I will describe some partial progress in this direction, including an understanding of the positive tropical points. Based on joint work with Pavel Galashin.

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