Series: Mathematics Colloquium

Date: Thursday, October 28, 2004

Time: 4:00 - 5:00 PM

Place: 201 Thomas Building

Host: Yuri Zarhin

Refreshments: 3:15 - 4:00 PM, 321 Whitmore

Speaker: Vladimir Retakh, Rutgers University

Title: What is Noncommutative Total Positivity?

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 factorized 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 enough, one can give explicit formulas for such
  factorizations even for matrices over noncommutative rings.  This is
  a joint paper with A. Berenstein.