We prove that three spaces of importance in topological combinatorics are homeomorphic to closed balls: the totally nonnegative Grassmannian, the compactification of the space of electrical networks, and the cyclically symmetric amplituhedron.
Lusztig [Section 4] LusIntro used a flow similar to @math to show that @math is contractible. Our flow can be thought of as an affine (or loop group) analogue of his flow, and is closely related to the whirl matrices of @cite_29 . We also remark that Ayala, Kliemann, and San Martin @cite_6 used the language of control theory to give an alternative development in type @math of Lusztig's theory of total positivity. In that context, @math ( @math ) lies in the interior of the compression semigroup of @math , and @math is its attractor .
@cite_29 @cite_6
2963963365 1988370440
Abstract 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 G L n ( R [ t , t − 1 ] ) , and for the formal loop group G L n ( 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 Painleve dynamical systems and Berenstein–Kazhdan’s study of geometric crystals. The objective of this article is to bring together two different mathematical subjects, namely totally positive matrices and control sets. It describes the control sets of the totally positive matrices and the sign-regular matrices in flag manifolds. In particular, the classical result by Gantmacher and Krein, follows from Theorem 9.7. One expects that with this description some theorems proved by combinatorial techniques have a geometric or dynamic interpretation.