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SYNOPSIS: DUAL EQUIVALENCE GRAPHS, RIBBON TABLEAUX AND MACDONALD POLYNOMIALS
 

Summary: SYNOPSIS: DUAL EQUIVALENCE GRAPHS, RIBBON TABLEAUX AND
MACDONALD POLYNOMIALS
SAMI H. ASSAF
1. Introduction
The primary focus of this dissertation is symmetric function theory. The main objectives are to present
a new combinatorial construction which may be used to establish the symmetry and Schur positivity of a
function expressed in terms of monomials, and to use this method to find a combinatorial description of the
Schur expansion for two important classes of symmetric functions, namely LLT and Macdonald polynomials.
Symmetric function theory plays an important role in many areas of mathematics including algebraic
combinatorics, representation theory, Lie groups and Lie algebras, algebraic geometry and the theory of
special functions. Multiplicities of irreducible components, dimensions of algebraic varieties, and various
other algebraic constructions that require the computation of certain integers may be translated to the
computation of the coefficients in the expansion of certain generalizations of the Schur basis. Often the
coefficients can be identified as generating functions of tableau-like structures, providing a useful and often
insightful combinatorial formula.
Since their introduction in 1988, Macdonald polynomials have been intensely studied and have been found
to have applications in such areas as representation theory, algebraic geometry, group theory, statistics, and
quantum mechanics. Unfortunately, given the indirect definition of these polynomials as the unique functions
satisfying certain conditions, most results require difficult technical machinery. Recent work by Haglund,
Haiman and Loehr has connected the study of Macdonald polynomials to that of LLT polynomials. Though

  

Source: Assaf, Sami H. - Department of Mathematics, Massachusetts Institute of Technology (MIT

 

Collections: Mathematics