Date of Award

1-1-2019

Language

English

Document Type

Master's Thesis

Degree Name

Master of Arts (MA)

College/School/Department

Department of Mathematics and Statistics

Content Description

1 online resource (iii, 34 pages) : illustrations.

Dissertation/Thesis Chair

Cristian Lenart

Keywords

Symmetric functions, Fifteen puzzle, Schur functions, Symmetry groups, Representations of groups, Geometry, Algebraic

Subject Categories

Physical Sciences and Mathematics

Abstract

We describe some applications of the \emph{jeu de taquin} algorithm on standard Young tableaux of skew shape $\lambda/\mu$ for $\lambda$ and $\mu$ partitions. We first briefly survey the relevant background on symmetric functions with a focus on the \emph{Schur functions}. We then introduce the Littlewood-Richardson coefficients in terms of Schur functions and survey some of the applications of the corresponding Littlewood-Richardson rule to representation theory and Schubert calculus on the Grassmannian. A reformulation of the Littlewood-Richardson rule in terms of the jeu de taquin algorithm and \emph{growth diagrams} is then surveyed. We illustrate this formulation with some examples. Finally, we discuss some work by Hugh Thomas and Alexander Yong in extending the Littlewood-Richardson rule to the more general setting of (co)minuscule flag varieties. In this setting we describe another reformulation of growth diagrams in terms of chains in Bruhat order and describe some examples of the generalized jeu de taquin for root systems of types $A_{n-1}$, $B_n$, $C_{n}$, and $D_{n}$.

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