The U(n) Gelfand-Zeitlin system as a tropical limit of Ginzburg-Weinstein diffeomorphisms

Wednesday, 4 April, 2018

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We show that the Ginzburg-Weinstein diffeomorphism \mathfrak{u}(n)^* \to U(n)^* of Alekseev-Meinrenken admits a scaling tropical limit. The target of the limit map is a product \mathcal{C} \times T, where C is a cone and T is a torus, and it carries an integrable system with natural action-angle coordinates. The pull-back of these coordinates to \mathfrak{u}(n)^* recovers the Gelfand-Zeitlin integrable system of Guillemin-Sternberg.
As a by-product of our proof, we show that the Lagrangian tori of the Flaschka-Ratiu integrable system on the set of upper triangular matrices meet the set of totally positive matrices for sufficiently large action coordinates.


Anton Alekseev
Jeremy Lane
Yanpeng Li