Combinatorial Group Actions and Applications to Geometry, Knot Theory, and Representation Theory

Rotating a molecule without changing the chemical bonds it can form; braiding strands of hair, which is how quantum mechanics describes the structure of certain elementary particles; rearranging the nucleotides that make up the building blocks of a strand of DNA. All of these—rotations, braids, rearrangements—are examples of group actions, namely families of symmetries satisfying structural constraints that arise from the context of each application.

This project studies group actions using combinatorial tools, including graphs and Young tableaux, in order to characterize their underlying algebraic structures. Applications include other areas of mathematics (like knot theory and algebraic geometry) and other fields (like mathematical biology). The broader impacts of the project include continued leadership of the Center for Women in Math at Smith College, mentoring of students at all levels through the PI's Math Research Lab as well as professional development for graduate students and continued career advice for alumni of the Center for Women in Math.

More technically, this project addresses a set of interconnected questions about two graph-theoretic actions of permutations: 1) a skein-type action of permutations on a family of graphs called webs (of which noncrossing matchings are the best-known example), and 2) a representation constructed from graph automorphisms for a family of subgraphs of the Cayley graph of the permutation group. Each action is the nexus between several mathematical fields, giving rise to a cycle of related questions.

In the first case, these are: combinatorially analyzing the transition matrices between two important bases of irreducible symmetric-group representations; describing the geometry and topology of components of Springer fibers; and constructing a basis for a vector space arising from questions in knot- and representation-theory. In the second case, they are: attacking the Stanley-Stembridge conjecture in combinatorics using a newly-established geometric correspondence; computing the (equivariant) cohomology rings of Hessenberg varieties; and describing generalized splines for different edge-labeled graphs.

Recent work of Brosnan and Chow, Rhoades, and others mean that the field is poised for new advances. The project exploits a synergistic strategy by attacking these interrelated questions simultaneously, and uses computational aspects of the work to incorporate mentoring and practical training of students.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.