# Faithful representations and tensor products

Let $G$ be a finite group (or a compact Lie group). Prove that if $\rho:G\to\mathrm{GL}(V)$ is a faithful finite dimensional complex representation of $G$ then any irreducible representation embeds in some tensor product of $V$.

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# Foiled by multiple roots

Let $\Sigma_3$ be the set of monic complex polynomials of degree 3 with multiple roots, and regard it as a subspace of $\mathbb{C}^3$ by looking at the non-leading coefficients. Prove that $\mathbb{C}^3\setminus \Sigma_3$ is homotopy equivalent to $S^3\setminus K$, where $K$ is the trefoil knot.

# Not an average calculus problem

Prove that for any complex numbers $z_1,\dots, z_n$ one can find a nonempty subset $I\subseteq \{1,2,\dots, n\}$ such that

$\left|\sum_{i\in I}z_i\right|\geq \frac{1}{\pi}\sum_{i=1}^n |z_i|$

# An exercise in Linear Algebra

Let $k$ be a field of characteristic $0$ and $k[x_1,\ldots,x_n]^{(d)}$ be the $k$-vector space of homogeneous degree $d$ polynomials in $n$ variables. Show that the linear $k$ span of the set of $d$-th powers of linear polynomials $(a_1 x_1+\ldots+a_n x_n)^d$ is the whole space of homogeneous degree $d$ polynomials.

In coordinate-free terms: if $V$ is a finite-dimensional $k$-vector space then $\{ v^n : v\in V\}$ spans $\mathrm{Sym}^dV$.

# Spooky categories

Suppose $C$ is an abelian category and $B\subseteq C$ is a subcategory which is also abelian. Is $B$ a sub-(abelian category) of $C$?

# Independentist magic

Show that $\displaystyle\int_{0}^{1} (2\cos(\pi x))^{2n} (2\sin(\pi x))^2 dx = C_n$ where $C_n = \frac{1}{n+1}\binom{2n}{n}$ is the $n$-th Catalan number.

# Categorical noetherian-ness

A ring $A$ is noetherian iff the category $_A\mathrm{mod}$ of finitely generated $A$-modules is a full abelian subcategory of the category $_A\mathrm{Mod}$ of all $A$-modules.