The PI will study quantum groups, Hecke algebras, and the efficient description of particular bases of these which were introduced by Kazhdan, Lusztig, and Kashiwara. These algebra bases, which facilitate representation theoretic computations, currently have somewhat cumbersome recursive descriptions. Improved descriptions will employ combinatorial interpretation, nonnegativity properties, and cluster algebras.
The importance of quantum group representations has been widely recognized since the 1980s, when Sklyanin and others found that these representations provide solutions to physical problems arising in quantum field theory and statistical mechanics. Reformulating questions in different mathematical terms, Drinfeld and Jimbo then inspired a truly enormous volume of research in mathematics and physics. Physicists contributed substantial mathematical results and applied these to study a variety of physical phenomena. Examples include rotations and vibrations in atomic nuclei, and, rather recently, mutations in genetic code. Mathematicians too used their areas of expertise not only to help explain quantum group representations, but to apply new knowledge to other mathematical problems. The PI will continue work, initiated by Kauffman and others, to efficiently describe quantum group representations by replacing certain cumbersome recursive procedures with combinatorial diagrams of wires and knots. He will also continue his past work to construct, from these diagrams, mathematical tools for studying inequalities satisfied by symmetric functions.
