Emmy Noether Lectures

2014 Lecturer: Georgia Benkart

Walking on Graphs the Representation Theory Way

How many walks of n steps are there from point A to point B on a graph? Often finding the answer involves clever combinatorics or tedious treading. But if the graph is the representation graph of a group, representation theory can facilitate the counting and provide much insight. The simply-laced affine Dynkin diagrams are representation graphs of the finite subgroups of the special unitary group SU(2) by the celebrated McKay correspondence. These subgroups are essentially the symmetry groups of the platonic solids, and the correspondence has been shown to have important connections with diverse subjects including mirror symmetry and the resolution of singularities. Inherent in McKay’s correspondence is a rich combinatorics coming from the Dynkin diagrams. Some of the ideas involved in seeing this go back to Schur, who used them to establish a remarkable duality between the representation theories of the general linear and symmetric groups. There is a similar duality between the SU(2) subgroups and certain algebras that enable us to count walks and solve other combinatorial problems. In this case, the duality leads to connections with the Temperley-Lieb algebras of statistical mechanics, with partitions, with Catalan numbers, and much more.
Brief Biography:
Georgia Benkart is an international leader in the structure and representation theory of Lie algebras and related algebraic structures. A longtime faculty member at the University of Wisconsin, she received her Ph.D. from Yale in 1974 with Nathan Jacobson. She has given hundreds of invited talks worldwide and published over 100 journal articles, mainly within four broad categories: (1) Modular Lie algebras, (2) Combinatorics of Lie algebra representations, (3) Graded algebras and Superalgebras, and (4) Quantum groups and related structures. Many of her most important papers represent breakthroughs. Her work on the classification of the rank one modular Lie algebras and on the “Recognition Theorem” provided the building blocks for the subsequent classification of the finite-dimensional simple modular Lie algebras. The combinatorial tools developed in other papers provided an effective way to study the stability of root and weight multiplicities of finite dimensional as well as infinite dimensional Kac-Moody Lie algebras. Motivated by the creation and annihilation operators in physics, Benkart and Roby introduced a new family of algebras, “down-up algebras” that still inspire current research. Benkart and co-authors introduced crystal bases for representations of general linear quantum superalgebras, and in a series of papers, she, jointly with others, determined the Lie algebras graded by finite root systems. Georgia has given excellent service to the mathematical community, particularly as a former President of AWM and as current AMS Secretary. She has been a superb mentor for her 21 Ph.D. students and postdocs. She won the University of Wisconsin Distinguished Teaching Award in 1987 and the Mid-Career Faculty Research Award in 1996. A fantastic speaker, Georgia was the Mathematical Association of America Polya Lecturer for 2000-2002.