Fusion Rules

Abstract: Fusion rules encode information about tensoring different types of modules (simple, projective) with a finite-dimensional module V, and this information can be recorded in a matrix that depends on V. When the objects are complex simple modules for a finite group, the resulting matrix (often called the McKay matrix due to inspiration from the McKay correspondence), has as its right eigenvectors the columns of the character table of the group, and the eigenvalues are the character of V evaluated on conjugacy class representatives. So in that particular case, the eigenvectors are independent of V. We consider extensions of such results to other settings where tensor products are defined such as finite-dimensional Hopf algebras (e.g. quantum groups at roots of unity, restricted enveloping algebras of Lie algebras in prime characteristic, and Drinfeld doubles). The eigenvectors and eigenvalues have connections with the characters of the Hopf algebra, and in some examples, many connections with Chebyshev polynomials of various kinds. Fusion rule matrices have applications to chip-firing dynamics and to Markov chains.

In this talk I will explain a joint work with Javier Aramayona, Julio Aroca, Rachel Skipper and Xiaolei Wu. We define a new family of groups that are subgroups of the mapping class group $Map(\Sigma_g)$ of a surface $\Sigma_g$ of genus $g$ with a Cantor set removed and we will call these groups Block Mapping Class Groups $B(H)$, where $H$ is a subgroup of $\Sigma_g$ . More visually, this family will be constructed by making a tree-like surface gluing pair of pants and taking homeomorphisms that depend on $H$ with certain preservation properties (it will preserve what we will call a block decomposition of this surface, hence the name of our groups). We will see that this family is closely related to Thompson’s groups and that it has the property of being of type $F_n$ if and only if $H$ is. As a consequence, for every $g\in \mathbb N \cup \{0, \infty\}$ and every $n\ge 1$, we construct a subgroup $G <\Map(\Sigma_g)$ that is of type $F_n$ but not of type $F_{n+1}$, and which contains the mapping class group of every compact surface of genus less or equal to $\g$ and with non-empty boundary. As expected in this workshop, the techniques involve manipulating cube complexes, as the Stein-Farley cube complex.

Mathematics

Audience: researchers in the discipline

CRM-Regional Conference in Lie Theory

Series comments: Registration is free but mandatory:https://www.crm.umontreal.ca/act/form/inscr_lieautomne20_e.shtml