Refined symmetric functions and affine analogs in combinatorics

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This proposal is to explore a connection between the type-A affine Weyl group and symmetric functions called k-Schur functions. The k-Schur functions arose in a study of the Macdonald polynomial basis, a basis for the symmetric function space that plays an important role in geometry, representation theory, and physics. As the k-Schur functions were investigated more deeply, it came to light that these functions may not only offer an approach to difficult problems in the theory of Macdonald polynomials, but also provide a fundamental basis for a subspace of the symmetric space. A wealth of beautiful combinatorial conjectures that extend and refine classical ideas in symmetric function theory came out of the study of k-Schur functions.This proposal is to work on these conjectures and to further investigatethe k-Schur functions. For example, recent work with the k-Schurfunctions suggests a connection between the Macdonald polynomials andthe affine symmetric group. Combinatorics is a vast area of mathematics loosely described as the study of counting collections of objects that satisfy specified criteria. As such, combinatorics plays an integral role in the development of many fields, and combinatorial methods are employed by scientists ranging from biologists to theoretical physicists.The PI seeks to understand a natural refinement for the beautiful combinatorics that arises in the theory of symmetric functions - a classical part of mathematics with a wide variety of applications in fields including physics, engineering, and computer science.