Abstract

A recent trend in mathematics is a renewed focus on the idea of “categorification”, in which some useful mathematical structure is replaced by a category that models the original structure in some way, such that the original structure is recovered by taking isomorphism classes of objects. For example, the category of Sets categorifies the natural numbers N. The notion of monoidal category (also known as tensor category) is a categorification of monoid. Tensor categories have been studied since MacLane and others in the 1960s, but there is renewed interest in them. Indeed, there is a recent book on the subject.

Deligne (2007) constructed a tensor category Rep(S_t), analogous to the category Rep(S_n) of complex representations of the symmetric group S_n, except that t is allowed to be any complex number. You might ask how can a non-existent thing have representations? I will try to answer this question, using combinatorial gadgets called partition diagrams (which are related to the partition algebra independently discovered by V. Jones and P.P. Martin in the 1990s). This talk is largely expository and I will mainly follow a 2011 paper of Comes and Ostrik.