Total positivity, Toeplitz matrices, and loop groups

A real matrix is totally nonnegative if every minor of it is nonnegative. The classical Edrei-Thoma theorem classifies totally nonnegative infinite Toeplitz matrices, and is related to problems in representation theory, combinatorics and probability. I will discuss progress towards two variations on this theorem to block-Toeplitz matrices, and to finite Toeplitz matrices. Both of these variations connect the classical theory to loop groups.

Part of this talk will be based on joint work with Pavlo Pylyavskyy.