The twist for positroids. (English. French summary) Zbl 1440.05244
Proceedings of the 28th international conference on formal power series and algebraic combinatorics, FPSAC 2016, Vancouver, Canada, July 4–8, 2016. Nancy: The Association. Discrete Mathematics & Theoretical Computer Science (DMTCS). Discrete Math. Theor. Comput. Sci., Proc., 911-922 (2020).
Summary: There are two reasonable ways to put a cluster structure on a positroid variety. In one, the initial seed is a set of Plücker coordinates. In the other, the initial seed consists of certain monomials in the edge weights of a plabic graph. We will describe an automorphism of the positroid variety, the twist, which takes one to the other. For the big positroid cell, this was already done by R. J. Marsh and J. S. Scott [Commun. Math. Phys. 341, No. 3, 821–884 (2016; Zbl 1341.13009)]; we generalize their results to all positroid varieties. This also provides an inversion of the boundary measurement map which is more general than Talaska’s [K. Talaska, Int. Math. Res. Not. 2008, Article ID rnn081, 19 p. (2008; Zbl 1170.05031)], in that it works for all reduced plabic graphs rather than just Le-diagrams. This is the analogue for positroid varieties of the twist map of Berenstein, Fomin and Zelevinsky for double Bruhat cells. Our construction involved the combinatorics of dimer configurations on bipartite planar graphs.
For the entire collection see [Zbl 1434.05002].
For the entire collection see [Zbl 1434.05002].
MSC:
| 05E16 | Combinatorial aspects of groups and algebras |
| 05C10 | Planar graphs; geometric and topological aspects of graph theory |
| 13F10 | Principal ideal rings |
References:
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