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This is not an answer, but it's a bit too long for a comment. If I'm not mistaken, any group $G$ which embeds densely into a compact Hausdorff group $H$ has the property that it is finite if and only …

It might be worth explaining why you shouldn't expect $R(G)$ to tell you everything about a group. $R(G)$ is naturally isomorphic to the ring of class functions $G \to \mathbb{C}$ (the functions cons …

Categorification can be thought of as the process of replacing equalities with isomorphisms (in some category). A basic example is replacing a numerical combinatorial identity such as $$2^n = \sum_{k= …

These factorizations are not unique, and there is already a counterexample for the smallest nontrivial group $G = C_2$. The trivial representation has character $(1, 1)$ (where the first entry is the …

Let's first recall the case of finite groups $G$. A projective representation of $G$ is a homomorphism $\rho : G \to PGL_n(\mathbb{C})$. In trying to lift this to a genuine representation $G \to GL_n( …

I don't know if this is what you're looking for, but here's a heuristic argument for this sort of thing being true in great generality. This should be a comment but it got long. It's not hard to con …

Here are the details on Faisal's suggestion in the comments. Lemma (Dixmier): Let $k$ be an algebraically closed field, let $A$ be a $k$-algebra with $\dim A < |k|$, and let $V$ be a simple left $A$- …

I think it is the characteristic. $A_5$ naturally sits in $\text{SO}(3)$ as the group of rotational symmetries of the icosahedron. Pulling $A_5$ back along the double cover $\text{SU}(2) \to \text{SO} …

The point is that there are two ways to describe the restriction functor. The first way is as $\text{Hom}_{\mathbb{C}[G]}(\mathbb{C}[G], -)$, thinking of $\mathbb{C}[G]$ as a $(\mathbb{C}[G], \mathbb{ …

A small generalization. Any doubly transitive action of a group $G$ on a set $X$ has the property that $\frac{1}{|G|} \sum_{g \in G} \text{Fix}(g)^2 = 2$. This is because, by double transitivity, th …

For a partition $\lambda$ let $S^{\lambda}$ be the corresponding Schur functor. Is it true that for every $\lambda$ there exists an irreducible representation $V$ of a finite nonabelian group $G$ suc …

Edit: My original idea doesn't work, but unknown's does. Here are the details. Let $k$ be a field, which is WLOG algebraically closed. Let $V$ be a finite-dimensional representation over $k$ of di …

My understanding is that the "right" way to define the symmetric algebra comes from a braiding that tells you how the symmetric group acts on tensor products. And an easy and general way to get such …

The action of $\mathbb{Z}_n$ on $\mathbb{C}^n$ can be diagonalized: it's conjugate to the action sending a vector $(z_0, z_1, \dots z_{n-1}) \in \mathbb{C}^n$ to $$(z_0, \zeta_n z_1, \zeta_n^2 z_2, \d …

Let L be the lattice of Young diagrams ordered by inclusion and let Ln denote the nth rank, i.e. the Young diagrams of size n. Say that lambda > mu if lambda covers mu, i.e. mu can be obtained from l …