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Abstract Totally positive matrices are matrices in which each minor is positive.
Lusztig extended the notion to reductive Lie groups. He also proved that
specialization of elements of the dual canonical basis in representation
theory of quantum groups at q=1 are totally non-negative polynomials.
Thus, it is important to investigate classes of functions on matrices that
are positive on totally positive matrices. I will discuss two sourses of
such functions. One has to do with multiplicative determinantal inequalities
(joint work with M. Gekhtman). Another deals with majorizing monotonicity
of symmetrized Fischer's products which are known for hermitian positive
semidefinite case which brings additional motivation to verify if they hold
for totally positive matrices as well (joint work with M. Skandera). The main
tools we employed are network parametrization, Temperley-Lieb and
monomial trace immanants.

The information has also been posted in our seminar web page:
https://sites.math.rutgers.edu/~yzhuang/rci/math/lie-quantum.html