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With regards to Q1 (and part of Q4), the numbers of the form $R(G)$ are dense in $\mathbb{R}_{\ge 0}$ even when $G$ is restricted to be abelian. Some first results. $R(G \times H) = R(G) R(H)$ if $\g …

The fusion ring, as a ring with basis, contains the same information as the character table. So your question, phrased in language more familiar to finite group theorists, is: Is a finite simple g …

It seems we get one from any symmetry of the diagram; are these all of them? No and no. The $A_n$ diagrams have a diagram symmetry of order $2$ for every $n$, but the induced automorphism of $S_{ …

The rings $\mathbb{Z}/n$ are the only examples. I assume that "primitive character" just means that it is faithful, or equivalently that it does not factor through a proper quotient; this is the meani …

The general classification of extensions $1 \to N \to G \to H \to 1$ with $N$ and $H$ fixed (which here are $N = \text{SL}_2(\mathbb{F}_q)$ and $H = \mathbb{Z}_2$) is that they correspond to equivalen …

In this context representations are completely determined by their character, so two $G$-sets $M, N$ have this property if and only if the number of fixed points of any $g \in G$ acting on $M, N$ are …

Edit: I think the answer to Question 2 is yes. Let $R = \mathbb{Z}[x, y, z]/I$ where $$I = (2x, 2y, 2z, z - xz^2, x - x^2 z, z - yz^2, y - y^2 z).$$ Then I think the embedding $$(12) \mapsto \lef …

Here are some things you probably know. For a representation $W$ of $G$, let $\text{Inv}(W)$ denote the subspace of $G$-invariants. For an irreducible representation $V$ with character $\chi$, the F-S …

As mentioned in the comments, conjecturally almost all finite groups are $2$-step nilpotent $2$-groups, so conjecturally the answers to 1) and 3) are that the limits both exist and both equal $1$; tha …

Everything that's been written so far about the classification of finite groupoids reducing to the classification of finite groups is true but, I think, misleading. In order to actually produce a list …

This computation could in principle be done using Clifford theory. Clifford theory tells you how to describe the representation theory of a group $G$ given that it can be described as an extension $$ …

This is probably really easy, but I just need someone to help me get mentally unstuck. As part of a description of the McKay correspondence, I want to show that if $G$ is a finite subgroup of $SU(2)$ …

Kurokawa's Zeta functions of categories contains the following definitions. Definition 1: In a category $C$ with a zero object, a simple object is an object $X$ such that, for every object $Y \in C$, …

Here is a more conceptual approach to Clifford theory. Let me work with a slightly more general setup: namely, suppose we have a short exact sequence $$1 \to N \to G \to H \to 1$$ of finite groups, …

For what it's worth, I had exactly the same question as you and worked out the proof that Noah sketches in some detail at The orthogonality relations for representations of finite groups, although I d …