Research Statement: Specializations of Macdonald polynomials

Symmetric functions are vital to the study of combinatorics because they provide valuable information about partitions and permutations, topics which constitute the core of the subject. The significance of symmetric function theory is manifest by its connections to other branches of mathematics, including group theory, representation theory, Lie algebras, and algebraic geometry. One important basis for the vector space of symmetric functions is the Schur function basis, which has an elegant combinatorial construction. Many other symmetric function bases are special cases of the Schur functions. Schur functions arise in representation theory as the characters of the irreducible representations of the general linear group. They provide insight into the multiplicative structure of the cohomology ring of the Grassmannian. Combinatorial properties of the Schur functions, such as the Littlewood-Richardson Rule and RSK algorithm, provide powerful tools for attacking problems in combinatorics such as permutation enumeration and plane partitions. Macdonald polynomials are a generalization of symmetric functions which have recently generated a significant amount of excitement. The combinatorial formulas for Macdonald polynomials and nonsymmetric Macdonald polynomials [4], [6] permit a combinatorial description of nonsymmetric functions, called Demazure atoms, which decompose the Schur functions. We explore several properties of these functions and their applications to representation theory [14], [15]. Quasisymmetric functions are objects that bridge the gap between nonsymmetric functions and symmetric functions. In addition to providing combinatorial information about symmetric functions, the rich structure of quasisymmetric functions relates to other algebraic structures as well. For instance, the Hopf algebra of quasisymmetric functions is dual to the Hopf algebra of noncommutative symmetric functions, which appears as the Solomon descent algebra. We introduce a new basis for quasisymmetric functions which decompose the Schur functions and we describe several properties they share with Schur functions [7], [8]. We call these functions quasisymmetric Schur functions because of their close relationship to the Schur functions.