product iamge

Applications of Jeu de Taquin to Representation Theory and Schubert Calculus

This document was uploaded by one of our users. The uploader already confirmed that they had the permission to publish it. If you are author/publisher or own the copyright of this documents, please report to us by using this DMCA report form.

Download
Search on Amazon

Simply click on the Download Book button.

Yes, Book downloads on Ebookily are 100% Free.

Sometimes the book is free on Amazon As well, so go ahead and hit "Search on Amazon"

Author(s): Daniel S. Hono II
Series: Master thesis at University at Albany, State University of New York
Year: 2019

Language: English

1 Title i
2 Abstract ii
3 Symmetric Functions 1
3.1 The Ring of Symmetric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
3.2 Schur Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
4 Applications to Representation Theory and Schubert Calculus 7
4.1 The Representation Theory of S n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.2 Representation Theory of GL n (C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
4.3 Schubert Calculus on the Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5 Jeu de Taquin and the Littlewood-Richardson Rule 13
5.1 Jeu de Taquin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
5.2 The Littlewood-Richardson Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.3 Growth Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6 Beyond Type A Grassmannians 18
6.1 (Co)Minuscule Flag Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
6.2 A Jeu de Taquin Procedure for (Co)Minuscule Flag Varieties . . . . . . . . . . . . . . . . . . 20
6.3 Example: Root Systems of Type A n−1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
6.4 Example: Root Systems of Type B n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6.5 Example: Root Systems of Type C n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.6 Example: Root Systems of Type D n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32