Research pathwaysCategory TheoryPlanned

Categorical representation theory

Optional research direction: representations on categories rather than vector spaces, including tensor and fusion categories, module categories, categorical actions and categorification, such as Chuang–Rouquier sl₂-categorification and Khovanov homology.

math.cat.categorical-representation-theory

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Builds on

Higher categories Abelian categories Representation theory of finite groups

Required prerequisites, all the way down: 42

You will learn to

Explain categorification through examples: from natural numbers to finite sets, from Euler characteristic to homology, and from the Jones polynomial to Khovanov homology.
Define tensor and fusion categories and module categories over them, with Rep(G) as an example.
Read a survey on categorical actions and separate established results from open problems.

Definition

Categorification replaces a set-based structure by a category-based one from which the original can be recovered, often by passing to a Grothendieck group.
A fusion category is a rigid semisimple k-linear monoidal category with finitely many simple objects and a simple unit object.
The Grothendieck group K₀(A) of an abelian category is generated by classes [X] of objects, with [B] = [A] + [C] for every short exact sequence 0 → A → B → C → 0.

Theorem

For a finite group G, Rep(G) over ℂ is a fusion category whose Grothendieck ring is the representation ring of G.
Khovanov homology is a link invariant whose graded Euler characteristic is the (unnormalised) Jones polynomial.
Chuang and Rouquier used sl₂-categorification to prove Broué's abelian defect group conjecture for symmetric groups.

Common misconceptions

Thinking categorification is a single construction; it is a family of methods, and a structure may have many categorifications or none known.
Assuming the Grothendieck group remembers everything about a category; much information is lost.

Optional discoveries

← Quiver representations