This is the first of a series of papers where we develop a theory of total positivity for loop groups. In this paper, we completely describe the totally nonnegative part of the polynomial loop group GLn(R[t,t-1]), and for the formal loop group GLn(R((t))) we describe the totally nonnegative points which are not totally positive. Furthermore, we make the connection with networks on the cylinder. Our approach involves the introduction of distinguished generators, called whirls and curls, and we describe the commutation relations amongst them. These matrices play the same role as the poles and zeros of the Edrei-Thoma theorem classifying totally positive functions (corresponding to our case n=1). We give a solution to the "factorization problem" using limits of ratios of minors. This is in a similar spirit to the Berenstein-Fomin-Zelevinsky Chamber Ansatz where ratios of minors are used. A birational symmetric group action arising in the commutation relation of curls appeared previously in Noumi-Yamada's study of discrete Painlevé dynamical systems and Berenstein-Kazhdan's study of geometric crystals.
Bibliographical noteFunding Information:
T.L. was partially supported by NSF grants DMS-0600677 , DMS-0652641 and DMS-0901111 , and by a Sloan Fellowship. P.P. was partially supported by NSF grant DMS-0757165 . Part of this work was completed during a stay at MSRI. We thank Alexei Borodin and Bernard Leclerc for discussing this work with us. We are grateful to Michael Shapiro for familiarizing us with some of the ideas in  . We also thank Sergey Fomin for many helpful comments, and for stimulating this project at its early stage.
Copyright 2012 Elsevier B.V., All rights reserved.
- Loop groups
- Total positivity