Schur-positivity of differences of augmented staircase diagrams

• Main Author: Morin, Ma
• Language: English
• Published: University of British Columbia 2010
• Online Access: http://hdl.handle.net/2429/25747

The Schur functions {s_lambda} and ubiquitous Littlewood-Richardson coefficients are instrumental in describing representation theory, symmetric functions,and even certain areas of algebraic geometry. Determining when two skew diagrams D₁, D₂ have the same skew Schur function or determining when the difference of two such skew Schur functions SD₁ - SD₂ is Schur-positive reveals information about the structures corresponding to these functions. By defining a set of staircase diagrams that we can augment with other (skew) diagrams, we discover collections of skew diagrams for which the question of Schur-positivity among each difference can be resolved. Furthermore, for certain Schur-positive differences we give explicit formulas for computing the coefficients of the Schur functions in the difference. We extend from simple staircases to fat staircases, and carry on to diagrams called sums of fat staircases. These sums of fat staircases can also be augmented with other (skew) diagrams to obtain many instances of Schur positivity. We note that several of our Schur-positive differences become equalities of skew Schur functions when the number of variables is reduced. Finally, we give a factoring identity which allows one to obtain many of the non-trivial finite-variable equalities of skew Schur functions.