Calculating fusion rules for $\operatorname{Rep}(G)$ and $G_{k}$ [reference request]

Question

Could somebody please direct me to textbooks / literature that (perhaps lay the foundations for and) detail the method for determining the fusion rules for categories such as

• $\operatorname{Rep}(G)$ i.e. the category of representations for a finite group $G$; I've also seen references to things like $\operatorname{Rep}(G)/H$ for a group $H$ in the literature — I'm not sure how they're defined, but this is also of interest.
• $G_{k}$ i.e. a Lie group $G$ at level $k$ e.g. $(G_{2})_2$.
• $\mathfrak{g}_{k}$ i.e. a Lie algebra $\mathfrak{g}$ at level $k$ e.g. $\mathfrak{su}(n)_{k}$

I'm happy to assume the base field is $\mathbb{C}$ where possible. Full disclosure, I'm not even sure what ‘at level $k$’ means in the last two points. Roadmaps for learning and elementary/pedagogical references are greatly appreciated. If it helps, I have a pretty basic understanding of the representation theory of finite groups and Lie algebras. I have a physics background (so I'm familiar with fusion categories in a sense, but not via the routes/perspectives above — a situation that I think would be useful to rectify!).

Answer 1

In general you should not expect to have explicit formulas for the decomposition of tensor products. The situation is a bit better for Lie algebras than for finite groups. As Ehud Meir pointed out, even calculating the Kronecker coefficients for symmetric groups

https://en.wikipedia.org/wiki/Kronecker_coefficient

is very hard. If you take other finite groups (except for very small ones), you won't find any general rules. For small groups you can certainly do this via character computations. Good introductory texts are the books by Serre and Fulton-Harris on representation theory.

I don't know such a thing as a Lie group or Lie algebra at level k. What you probably mean are level $k$ integrable highest weight modules over some affine Lie algebra. But you need to have a good understanding of the classical theory first.

For semisimple Lie algebras there are several ways to compute the tensor product decomposition. One approach is using the Weyl character formula and $ch(V \otimes W) = ch(V) \cdot ch(W)$. In this way you can get some complicated formulas for the fusion coefficients (see the book of Humphreys on Lie algebras (Chapter 24, "Steinberg's formula") for an example).

For $\mathfrak{sl}(n,\mathbb{C})$ the character of an irreducible representation $L(\lambda)$ is given by the Schur function $s_{\lambda}$. The structure coefficients $c_{\lambda,\mu}^{\nu}$ in $s_{\lambda} \cdot s_{\mu} = \sum_{\nu} c_{\lambda,\mu}^{\nu} s_{\nu}$ are known as Littlewood-Richardson coefficients, and the combinatorial algorithm to compute them is the Littlewood-Richardson rule

https://en.wikipedia.org/wiki/Littlewood%E2%80%93Richardson_rule

Similar combinatorial descriptions exist for $\mathfrak{so}(n,\mathbb{C})$ and $\mathfrak{sp}(2n,\mathbb{C})$.

A very general rule is the Littelmann path model

https://en.wikipedia.org/wiki/Littelmann_path_model

which can be used to get fusion rules for representations of symmetrizable Kac-Moody algebras.

If you move away from semisimple Lie algebras and go to non-semisimple representation categories such as representations of a quantum group at a root of unity, or representations of a supergroup, or representation theory in positive characteristic, very little general formulas are known.

Since you mention fusion categories and level $k$: What you are ultimately interested in might be the theory of fusion categories arising from a quantum group and the Kazhdan-Lusztig equivalence. But this is quite advanced and if you are not familiar with enough with the classical theory, the books of Humphreys and Fulton-Harris might be better in the beginning. An introduction to representations of quantum groups can be found in the book on Quantum Groups by Chari-Pressley (Chapter 10, 11). A good introduction to fusion rings is Sawin's article on Quantum groups at roots of unity and modularity (which includes the Racah formula for tensor products)

https://arxiv.org/abs/math/0308281

As for the Kazhdan-Lusztig equivalence you can have a look at these mathoverflow answers to see what it is about:

Why should affine lie algebras and quantum groups have equivalent representation theories?

What's the state of affairs concerning the identification between quantum group reps at root of unity, and positive energy affine Lie algebra reps?