Graduation Year

2018

Document Type

Honors Thesis

Degree Name

Bachelor of Arts

Department

Mathematics and Computer Science

Program or Major

Mathematics

Faculty Advisor

Joseph Alfano

Abstract

Schur polynomials are a fundamental object in the field of algebraic combinatorics. The product of two Schur polynomials can be written as a sum of Schur polynomials using non-negative integer coefficients. A simple combinatorial algorithm for generating these coefficients is called the Littlewood-Richardson Rule. Schubert polynomials are generalizations of the Schur polynomials. Schubert polynomials also appear in many contexts, such as in algebraic combinatorics and algebraic geometry. It is known from algebraic geometry that the product of two Schubert polynomials can be written as a sum of Schubert polynomials using non-negative integer coefficients. However, a simple combinatorial algorithm for generating these coefficients is not known in general. Monk’s Rule is a known algorithm that can be used in specific cases. This research seeks to identify more algorithms for the multiplication of Schubert polynomials. In this thesis, I will provide a brief overview of Schur polynomials and Schubert polynomials. Also, I will present diagrams called ’pipe-dreams’ to illustrate Schubert polynomials and establish a connection to Schur polynomials. Our main result is in Schubert polynomial multiplication. I will present two algorithms for Schubert polynomial multiplication, which generalize Monk’s rule in specific cases.

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