Noncommutative Total Positivity
by Vladimir RETAKH (Rutgers University)
Abstract:
A square matrix with real entries is called totally positive
(totally nonnegative) if all its minors are positive (nonnegative).
Such matrices are important for different applications including
a surprising connection discovered by Lusztig between total
positivity and canonical bases in representation theory.
A celebrated Loewner-Whitney theorem states that a real invertible matrix
is totally nonnegative if and only if it can be factored as a product of
elementary nonnegative Jacobi matrices. Thus, the study of total
positivity can be reduced to a combinatorial problem of studying
"canonical factorizations" of matrices into such products.
Surprisingly, one can give explicit formulas for such
factorizations even for matrices over non-commutative rings.
This talk is based on a joint paper with A. Berenstein.