This proposal brings together a set of problems aimed at integrating three seemingly different areas in combinatorial representation theory: centralizer algebras, Kronecker products and subalgebras of the symmetric group algebra. A fundamental open problem in combinatorial representation theory is to describe the multiplicities in the decomposition of the tensor product of two irreducible representations of the symmetric group. These multiplicities are called the Kronecker coefficients. The problem of finding a combinatorial interpretation for these coefficients has been open for nearly a hundred years and is one of the main problems in combinatorial representation theory. In a new development the PI and collaborators have established a new connection of this problem to the partition algebra, a centralizer algebra of the symmetric group. The PI will use this new connection to study several outstanding open problems related to the Kronecker coefficients. The PI will also continue her contributions to combinatorial algebras, which are algebras that can be described via combinatorial methods, e.g., the descent and peak algebras. The significance of the descent algebras arise from their applications to noncommutative and quasi-symmetric functions, and from their connections to the ring of representations of the symmetric group. In this project, the PI proposes to investigate new combinatorial algebras and to use centralizer algebras to better understand the existing combinatorial algebras.
This research is in the general area of algebraic combinatorics. Algebraic objects are often complex and difficult to understand. Combinatorics help make abstract algebraic objects more concrete so that they become more accessible and can be more easily used in other fields. This is achieved through efficient algorithms and elementary models which give insight into the properties of algebraic objects. The proposed projects tackle deep and important problems that will advance the understanding of many fields, including quantum information theory, complexity theory, and algorithms. The results will have applications that would span many fields from physics and chemistry to computer science. In addition, the proposed research is of broad interest in mathematics to those studying symmetric functions, Lie theory, quantum groups, algebraic geometry, low dimensional topology and mathematical physics.
