Positivity in Real Grassmannians:Combinatorial Formulas.

Total Positivity;Grassmannians;Planar Networks;Mathematics;Science;Mathematics

Talaska, Kelli F.Stembridge, John R. ;

University of Michigan

关键词: Total Positivity;    Grassmannians;    Planar Networks;    Mathematics;    Science;    Mathematics;   

Others  :  https://deepblue.lib.umich.edu/bitstream/handle/2027.42/77689/kellicar_1.pdf?sequence=1&isAllowed=y

瑞士|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

The main results of this dissertation are explicit formulas for a combinatorial approach to the study of totally nonnegative Grassmannians.A totally nonnegative Grassmannian consists of the points in a real Grassmannian where all Plücker coordinates can be taken to be simultaneously nonnegative.The combinatorial approach to the study of totally nonnegative Grassmannians was initiated by Postnikov, who introduced the concept of a boundary measurement matrix associated with a planar network. Using indirect recursive arguments, he showed that these matrices represent points in a totally nonnegative Grassmannian, and that every such point can be obtained in this way. This dissertation strengthens Postnikov;;s results by providing explicit combinatorial formulas whichimmediately imply these two key properties.First, we obtain subtraction-free formulas for the Plücker coordinates of a general boundary measurement matrix, thereby giving a constructive proof that any such matrix defines a totally nonnegative point. Going in reverse, for a point x in a totally nonnegative Grassmannian, we construct a specific network whose boundary measurement matrix represents x.