Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6425865 | Advances in Mathematics | 2013 | 40 Pages |
Abstract
We study the relationship between two sets of coordinates on a double Bruhat cell, the cluster variables introduced by Berenstein, Fomin, and Zelevinsky and the X-coordinates defined by the coweight parametrization of Fock and Goncharov. In these coordinates, we show that the generalized Chamber Ansatz of Fomin and Zelevinsky coincides with the cluster ensemble map, a canonical monomial transformation between the cluster variables and X-coordinates defined by a common exchange matrix. We prove this in the setting of an arbitrary symmetrizable Kac-Moody group, generalizing along the way many previous results on the double Bruhat cells of a semisimple algebraic group. In particular, we construct an upper cluster algebra structure on the coordinate ring of any double Bruhat cell in a symmetrizable Kac-Moody group, proving a conjecture of Berenstein, Fomin, and Zelevinsky.
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Physical Sciences and Engineering
Mathematics
Mathematics (General)
Authors
Harold Williams,