Search Results

This is a different take on Steven Landsburg's answer. The short version is that conjugacy classes and irreducible representations should be thought of as being dual to each other. Fix an algebraica …

A simple bound on the largest dimension of a complex irreducible representation (which is either equal to or half of the largest dimension of a real irreducible representation) is the following: we kn …

Let $G$ be a finite subgroup of $F^{\ast}$ of order $n$. Then all the elements of $G$ satisfy $x^n = 1$ in $F$. Since polynomials of degree $n$ over a field have at most $n$ roots, it follows that the …

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 answer to questions 0 and 1 is yes. Here is a generalization. Claim: Let $\pi$ be a finitely generated group and $G$ be a finite group. Then $$\frac{|\text{Hom}(\pi \times \mathbb{Z}, G)|}{|G|}$$ …

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_{ …

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 …

Not a complete answer. Your definition of an extendible map says that $\beta$ is an endomorphism of the forgetful functor $U$ from the under category $G \downarrow \mathrm{FinGrp}$ to $\mathrm{Set}$ s …

This is maybe stretching it a little bit, but Tim Gowers' quasi-random groups describes and references some extremal combinatorial properties of graphs constructed from the groups $PSL_2(\mathbb{F}_q) …

As mentioned in the comments, for general $n$ this is pretty hopeless. For $n = 4$ we can take advantage of the fact that $SO(4)$ is double covered by $Spin(4) \cong SU(2) \times SU(2)$, which more or …

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 …

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 …

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, …

Artin's Algebra has a good chapter on representations of finite groups. The exercises are nice.

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 $SU(2)$ …