Let be a finite group (or a compact Lie group). Prove that if is a faithful finite dimensional complex representation of then any irreducible representation embeds in some tensor product of .

# Foiled by multiple roots

Let be the set of monic complex polynomials of degree 3 with multiple roots, and regard it as a subspace of by looking at the non-leading coefficients. Prove that is homotopy equivalent to , where is the trefoil knot.

# Not an average calculus problem

Prove that for any complex numbers one can find a nonempty subset such that

# An exercise in Linear Algebra

Let be a field of characteristic and be the -vector space of homogeneous degree polynomials in variables. Show that the linear span of the set of -th powers of linear polynomials is the whole space of homogeneous degree polynomials.

In coordinate-free terms: if is a finite-dimensional -vector space then spans .

# Spooky categories

Suppose is an abelian category and is a subcategory which is also abelian. Is a sub-(abelian category) of ?

# Independentist magic

Show that where is the -th Catalan number.

# Categorical noetherian-ness

A ring is noetherian iff the category of finitely generated -modules is a full abelian subcategory of the category of all -modules.